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Odds ratio

An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1, i.e., the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event.

Note that the odds ratio is symmetric in the two events, and there is no causal direction implied (correlation does not imply causation): an OR greater than 1 does not establish that B causes A, or that A causes B.

Two similar statistics that are often used to quantify associations are the relative risk (RR) and the absolute risk reduction (ARR). Often, the parameter of greatest interest is actually the RR, which is the ratio of the probabilities analogous to the odds used in the OR. However, available data frequently do not allow for the computation of the RR or the ARR, but do allow for the computation of the OR, as in case-control studies, as explained below. On the other hand, if one of the properties (A or B) is sufficiently rare (in epidemiology this is called the rare disease assumption), then the OR is approximately equal to the corresponding RR.

The OR plays an important role in the logistic model.

Intuition from an example for laypeople
If we flip an unbiased coin, the probability of getting heads and the probability of getting tails are equal - both are 50%. Imagine we get a biased coin that makes it two times more likely to get heads. But what does "twice as likely" mean in terms of a probability? It cannot literally mean to double the probability value, because 50% becomes 100%. Rather, it is the odds that are doubling: from 1:1 odds, to 2:1 odds.

A motivating example, in the context of the rare disease assumption
Suppose a radiation leak in a village of 1,000 people increased the incidence of a rare disease. The total number of people exposed to the radiation was $$ V_E=400,$$ out of which $$D_E=20$$ developed the disease and $$H_E=380$$ stayed healthy. The total number of people not exposed was $$V_N=600,$$ out of which $$D_N=6$$ developed the disease and $$H_N=594$$ stayed healthy. We can organize this in a contingency table:

\begin{array}{|r|cc|} \hline & {\text { Diseased }} & {\text { Healthy }} \\ \hline \text { Exposed } & 20 & 380 \\ \text { Not exposed } & 6 & 594 \\ \hline \end{array} $$

The risk of developing the disease given exposure is $$D_E/V_E= 20/400=.05$$ and of developing the disease given non-exposure is $$D_N/V_N= 6/600 = .01$$. One obvious way to compare the risks is to use the ratio of the two, the relative risk.

 * $$ \text{Relative risk} = \frac{D_E/(D_E + H_E)}{D_N/(D_N + H_N)}= \frac{D_E/V_E}{D_N/V_N}= \frac{20/400}{6/600}= \frac{.05}{.01} = 5\,. $$

The odds ratio is different. The odds of getting the disease if exposed is $$D_E/H_E=20/380\approx .053 ,$$ and the odds if not exposed is $$D_N/H_N = 6/594 \approx .010 \,.$$ The odds ratio  is the ratio of the two,

 * $$ \text{Odds ratio} = \frac{D_E/H_E}{D_N/H_N}=  \frac{20/380}{6/594}\approx   \frac{.053}{.010} = 5.3\,.$$

As illustrated by this example, in a rare-disease case like this, the relative risk and the odds ratio are almost the same. By definition, rare disease implies that $$V_E\approx H_E$$ and $$V_N\approx H_N$$. Thus, the denominators in the relative risk and odds ratio are almost the same ($$400 \approx 380$$ and $$600 \approx 594)$$.

Relative risk is easier to understand than the odds ratio, but one reason to use odds ratio is that usually, data on the entire population is not available and random sampling must be used. In the example above, if it were very costly to interview villagers and find out if they were exposed to the radiation, then the prevalence of radiation exposure would not be known, and neither would the values of $$V_E$$ or $$V_N$$. One could take a random sample of fifty villagers, but quite possibly such a random sample would not include anybody with the disease, since only 2.6% of the population are diseased. Instead, one might use a case-control study in which all 26 diseased villagers are interviewed as well as a random sample of 26 who do not have the disease. The results might turn out as follows ("might", because this is a random sample):

\begin{array}{|r|cc|} \hline & {\text { Diseased }} & {\text { Healthy }} \\ \hline \text { Exposed } & 20 & 10 \\ \text { Not exposed } & 6 & 16 \\ \hline \end{array} $$

The odds in this sample of getting the disease given that someone is exposed is 20/10 and the odds given that someone is not exposed is 6/16. The odds ratio is thus $$\frac{20/10}{6/16} \approx 5.3 $$. The relative risk, however, cannot be calculated, because it is the ratio of the risks of getting the disease and we would need $$ V_E$$ and $$V_N$$ to figure those out. Because the study selected for people with the disease, half the people in the sample have the disease and it is known that that is more than the population-wide prevalence.

It is standard in the medical literature to calculate the odds ratio and then use the rare-disease assumption (which is usually reasonable) to claim that the relative risk is approximately equal to it. This not only allows for the use of case-control studies, but makes controlling for confounding variables such as weight or age using regression analysis easier and has the desirable properties discussed in other sections of this article of invariance and insensitivity to the type of sampling.

Definition in terms of group-wise odds
The odds ratio is the ratio of the odds of an event occurring in one group to the odds of it occurring in another group. The term is also used to refer to sample-based estimates of this ratio. These groups might be men and women, an experimental group and a control group, or any other dichotomous classification. If the probabilities of the event in each of the groups are p1 (first group) and p2 (second group), then the odds ratio is:

 * $$OR = \frac{p_1/(1-p_1)}{p_2/(1-p_2)}=\frac{ p_1/q_1}{p_2/q_2}=\frac{\;p_1q_2\;}{\;p_2q_1\;},$$

where qx = 1 − px. An odds ratio of 1 indicates that the condition or event under study is equally likely to occur in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely to occur in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely to occur in the first group. The odds ratio must be nonnegative if it is defined. It is undefined if p2q1 equals zero, i.e., if p2 equals zero or q1 equals zero.

Definition in terms of joint and conditional probabilities
The odds ratio can also be defined in terms of the joint probability distribution of two binary random variables. The joint distribution of binary random variables $X$ and $Y$ can be written

\begin{array}{c|cc} & Y = 1 & Y = 0 \\ \hline X = 1 & p_{11} & p_{10} \\ X = 0 & p_{01} & p_{00} \end{array} $$

where $p$11, $p$10, $p$01 and $p$00 are non-negative "cell probabilities" that sum to one. The odds for $Y$ within the two subpopulations defined by $X$ = 1 and $X$ = 0 are defined in terms of the conditional probabilities given $X$, i.e., $P(Y |X)$:

\begin{array}{c|cc} & Y = 1 & Y = 0 \\ \hline X = 1 & \frac{p_{11}}{p_{11}+p_{10}} & \frac{p_{10}}{p_{11}+p_{10}} \\ X = 0 & \frac{p_{01}}{p_{01}+p_{00}} & \frac{p_{00}}{p_{01}+p_{00}} \end{array} $$

Thus the odds ratio is

 * $$OR = \dfrac{p_{11}/(p_{11}+p_{10})}{p_{10}/(p_{11}+p_{10})} \bigg/ \dfrac{p_{01}/(p_{01}+p_{00})}{p_{00}/(p_{01}+p_{00})} = \frac{p_{11}p_{00}}{p_{10}p_{01}}$$

The simple expression on the right, above, is easy to remember as the product of the probabilities of the "concordant cells" $(X = Y)$ divided by the product of the probabilities of the "discordant cells" $(X ≠ Y)$. However in some applications the labeling of categories as zero and one is arbitrary, so there is nothing special about concordant versus discordant values in these applications.

Symmetry
If we had calculated the odds ratio based on the conditional probabilities given Y,

 * $$\begin{array}{c|cc}

& Y = 1 & Y = 0 \\ \hline X = 1 & \frac{p_{11}}{p_{11}+p_{01}} & \frac{p_{10}}{p_{10}+p_{00}} \\ X = 0 & \frac{p_{01}}{p_{11}+p_{01}} & \frac{p_{00}}{p_{10}+p_{00}} \end{array} $$

we would have obtained the same result

 * $$ \dfrac{p_{11}/(p_{11}+p_{01})}{p_{01}/(p_{11}+p_{01})} \bigg / \dfrac{p_{10}/(p_{10}+p_{00})}{p_{00}/(p_{10}+p_{00})} = \dfrac{p_{11}p_{00}}{p_{10}p_{01}}.$$

Other measures of effect size for binary data such as the relative risk do not have this symmetry property.

Relation to statistical independence
If X and Y are independent, their joint probabilities can be expressed in terms of their marginal probabilities $p_{x} = P(X = 1)$ and $p_{y} = P(Y = 1)$, as follows

 * $$\begin{array}{c|cc}

& Y = 1 & Y = 0 \\ \hline X = 1 & p_xp_y & p_x(1-p_y) \\ X = 0 & (1-p_x)p_y & (1-p_x)(1-p_y) \end{array} $$

In this case, the odds ratio equals one, and conversely the odds ratio can only equal one if the joint probabilities can be factored in this way. Thus the odds ratio equals one if and only if X and Y are independent.

Recovering the cell probabilities from the odds ratio and marginal probabilities
The odds ratio is a function of the cell probabilities, and conversely, the cell probabilities can be recovered given knowledge of the odds ratio and the marginal probabilities $P(X = 1) = p_{11} + p_{10}$ and $P(Y = 1) = p_{11} + p_{01}$. If the odds ratio R differs from 1, then

p_{11} = \frac{1 + (p_{1\cdot}+p_{\cdot 1})(R-1) - S}{2(R-1)} $$

where $p_{1•} = p_{11} + p_{10}, p_{•1} = p_{11} + p_{01}$, and

S = \sqrt{(1+(p_{1\cdot}+p_{\cdot 1})(R-1))^2 + 4R(1-R)p_{1\cdot}p_{\cdot 1}}. $$

In the case where $R = 1$, we have independence, so $p_{11} = p_{1•}p_{•1}$.

Once we have $p_{11}$, the other three cell probabilities can easily be recovered from the marginal probabilities.

Example

Suppose that in a sample of 100 men, 90 drank wine in the previous week (so 10 did not), while in a sample of 80 women only 20 drank wine in the same period (so 60 did not). This forms the contingency table:

 * $$\begin{array}{c|cc}

& M = 1 & M = 0 \\ \hline D = 1 & 90 & 20 \\ D = 0 & 10 & 60 \end{array} $$

The odds ratio (OR) can be directly calculated from this table as:

 * $${ OR }=\frac{\;90\times 60\;}{\;10\times 20\;} = 27$$

Alternatively, the odds of a man drinking wine are 90 to 10, or 9:1, while the odds of a woman drinking wine are only 20 to 60, or 1:3 = 0.33. The odds ratio is thus 9/0.33, or 27, showing that men are much more likely to drink wine than women. The detailed calculation is:

 * $${ 0.9/0.1 \over 0.2/0.6}=\frac{\;0.9\times 0.6\;}{\;0.1\times 0.2\;} ={0.54 \over 0.02} = 27$$

This example also shows how odds ratios are sometimes sensitive in stating relative positions: in this sample men are (90/100)/(20/80) = 3.6 times as likely to have drunk wine than women, but have 27 times the odds. The logarithm of the odds ratio, the difference of the logits of the probabilities, tempers this effect, and also makes the measure symmetric with respect to the ordering of groups. For example, using natural logarithms, an odds ratio of 27/1 maps to 3.296, and an odds ratio of 1/27 maps to −3.296.

Statistical inference
Several approaches to statistical inference for odds ratios have been developed.

One approach to inference uses large sample approximations to the sampling distribution of the log odds ratio (the natural logarithm of the odds ratio). If we use the joint probability notation defined above, the population log odds ratio is

 * $${\log\left(\frac{p_{11}p_{00}}{p_{01}p_{10}}\right) = \log(p_{11}) + \log(p_{00}\big) - \log(p_{10}) - \log(p_{01})}.\,$$

If we observe data in the form of a contingency table
 * $$\begin{array}{c|cc}

& Y = 1 & Y = 0 \\ \hline X = 1 & n_{11} & n_{10} \\ X = 0 & n_{01} & n_{00} \end{array} $$

then the probabilities in the joint distribution can be estimated as

 * $$\begin{array}{c|cc}

& Y = 1 & Y = 0 \\ \hline X = 1 & \hat{p}_{11} & \hat{p}_{10} \\ X = 0 & \hat{p}_{01} & \hat{p}_{00} \end{array} $$

where $︿ p_{ij} = n_{ij} / n$, with $n = n_{11} + n_{10} + n_{01} + n_{00}$ being the sum of all four cell counts. The sample log odds ratio is

 * $${L=\log\left(\dfrac{\hat{p}_{11}\hat{p}_{00}}{\hat{p}_{10}\hat{p}_{01}}\right) = \log\left(\dfrac{n_{11}n_{00}}{n_{10}n_{01}}\right)}$$.

The distribution of the log odds ratio is approximately normal with:

L\ \sim\ \mathcal{N}(\log (OR),\,\sigma^2). \, $$

The standard error for the log odds ratio is approximately

 * $${{\rm SE} = \sqrt{\dfrac{1}{n_{11}} + \dfrac{1}{n_{10}} + \dfrac{1}{n_{01}} + \dfrac{1}{n_{00}}}}$$.

This is an asymptotic approximation, and will not give a meaningful result if any of the cell counts are very small. If L is the sample log odds ratio, an approximate 95% confidence interval for the population log odds ratio is $L ± 1.96SE$. This can be mapped to $exp(L − 1.96SE), exp(L + 1.96SE)$ to obtain a 95% confidence interval for the odds ratio. If we wish to test the hypothesis that the population odds ratio equals one, the two-sided p-value is $2P(Z < −|L|/SE)$, where P denotes a probability, and Z denotes a standard normal random variable.

An alternative approach to inference for odds ratios looks at the distribution of the data conditionally on the marginal frequencies of X and Y. An advantage of this approach is that the sampling distribution of the odds ratio can be expressed exactly.

Role in logistic regression
Logistic regression is one way to generalize the odds ratio beyond two binary variables. Suppose we have a binary response variable Y and a binary predictor variable X, and in addition we have other predictor variables Z1, ..., Zp that may or may not be binary. If we use multiple logistic regression to regress Y on X, Z1, ..., Zp, then the estimated coefficient $$\hat{\beta}_x$$ for X is related to a conditional odds ratio. Specifically, at the population level

e^{\beta_x} = \exp(\beta_x) = \frac{P(Y=1\mid X=1, Z_1, \ldots, Z_p)/P(Y=0\mid X=1, Z_1, \ldots, Z_p)}{P(Y=1\mid X=0, Z_1, \ldots, Z_p)/P(Y=0\mid X=0, Z_1, \ldots, Z_p)}, $$

so $$\exp(\hat{\beta}_x)$$ is an estimate of this conditional odds ratio. The interpretation of $$\exp(\hat{\beta}_x)$$ is as an estimate of the odds ratio between Y and X when the values of Z1, ..., Zp are held fixed.

Insensitivity to the type of sampling
If the data form a "population sample", then the cell probabilities $$ \widehat{p\,}_{ij}$$ are interpreted as the frequencies of each of the four groups in the population as defined by their X and Y values. In many settings it is impractical to obtain a population sample, so a selected sample is used. For example, we may choose to sample units with $X = 1$ with a given probability f, regardless of their frequency in the population (which would necessitate sampling units with $X = 0$ with probability $1 − f$). In this situation, our data would follow the following joint probabilities:

 * $$\begin{array}{c|cc}

& Y = 1 & Y = 0 \\ \hline X = 1 & \frac{fp_{11}}{p_{11}+p_{10}} & \frac{fp_{10}}{p_{11}+p_{10}} \\ X = 0 & \frac{(1-f)p_{01}}{p_{01}+p_{00}} & \frac{(1-f)p_{00}}{p_{01}+p_{00}} \end{array} $$

The odds ratio $p_{11}p_{00} / p_{01}p_{10}$ for this distribution does not depend on the value of f. This shows that the odds ratio (and consequently the log odds ratio) is invariant to non-random sampling based on one of the variables being studied. Note however that the standard error of the log odds ratio does depend on the value of f.

This fact is exploited in two important situations:

 * Suppose it is inconvenient or impractical to obtain a population sample, but it is practical to obtain a convenience sample of units with different X values, such that within the $X = 0$ and $X = 1$ subsamples the Y values are representative of the population (i.e. they follow the correct conditional probabilities).
 * Suppose the marginal distribution of one variable, say X, is very skewed. For example, if we are studying the relationship between high alcohol consumption and pancreatic cancer in the general population, the incidence of pancreatic cancer would be very low, so it would require a very large population sample to get a modest number of pancreatic cancer cases.  However we could use data from hospitals to contact most or all of their pancreatic cancer patients, and then randomly sample an equal number of subjects without pancreatic cancer (this is called a "case-control study").

In both these settings, the odds ratio can be calculated from the selected sample, without biasing the results relative to what would have been obtained for a population sample.

Use in quantitative research
Due to the widespread use of logistic regression, the odds ratio is widely used in many fields of medical and social science research. The odds ratio is commonly used in survey research, in epidemiology, and to express the results of some clinical trials, such as in case-control studies. It is often abbreviated "OR" in reports. When data from multiple surveys is combined, it will often be expressed as "pooled OR".

Relation to relative risk
As explained in the "Motivating Example" section, the relative risk is usually better than the odds ratio for understanding the relation between risk and some variable such as radiation or a new drug. That section also explains that if the rare disease assumption holds, the odds ratio is a good approximation to relative risk and that it has some advantages over relative risk. When the rare disease assumption does not hold, the unadjusted odds ratio will be greater than the relative risk, but novel methods can easily use the same data to estimate the relative risk, risk differences, base probabilities, or other quantities.

If the absolute risk in the unexposed group is available, conversion between the two is calculated by:

 * $$ \text{Relative risk} \approx \frac{\text{Odds ratio}}{1 - R_C + (R_C \times \text{Odds ratio})}$$

where RC is the absolute risk of the unexposed group.

If the rare disease assumption does not apply, the odds ratio may be very different from the relative risk and should not be interpreted as a relative risk.

Consider the death rate of men and women passengers when a ship sank. Of 462 women, 154 died and 308 survived. Of 851 men, 709 died and 142 survived. Clearly a man on the ship was more likely to die than a woman, but how much more likely? Since over half the passengers died, the rare disease assumption is strongly violated.

To compute the odds ratio, note that for women the odds of dying were 1 to 2 (154/308). For men, the odds were 5 to 1 (709/142). The odds ratio is 9.99 (4.99/.5). Men had ten times the odds of dying as women.

For women, the probability of death was 33% (154/462). For men the probability was 83% (709/851). The relative risk of death is 2.5 (.83/.33). A man had 2.5 times a woman's probability of dying.

Confusion and exaggeration
Odds ratios have often been confused with relative risk in medical literature. For non-statisticians, the odds ratio is a difficult concept to comprehend, and it gives a more impressive figure for the effect. However, most authors consider that the relative risk is readily understood. In one study, members of a national disease foundation were actually 3.5 times more likely than nonmembers to have heard of a common treatment for that disease – but the odds ratio was 24 and the paper stated that members were ‘more than 20-fold more likely to have heard of’ the treatment. A study of papers published in two journals reported that 26% of the articles that used an odds ratio interpreted it as a risk ratio.

This may reflect the simple process of uncomprehending authors choosing the most impressive-looking and publishable figure. But its use may in some cases be deliberately deceptive. It has been suggested that the odds ratio should only be presented as a measure of effect size when the risk ratio cannot be estimated directly, but with newly available methods it is always possible to estimate the risk ratio, which should generally be used instead.

While relative risks are potentially easier to interpret for a general audience, there are mathematical and conceptual advantages when using an odds-ratio instead of a relative risk, particularly in regression models. For that reason, there is not a consensus within the fields of epidemiology or biostatistics that relative risks or odds-ratios should be preferred when both can be validly used, such as in clinical trials and cohort studies

Invertibility and invariance
The odds ratio has another unique property of being directly mathematically invertible whether analyzing the OR as either disease survival or disease onset incidence – where the OR for survival is direct reciprocal of 1/OR for risk. This is known as the 'invariance of the odds ratio'. In contrast, the relative risk does not possess this mathematical invertible property when studying disease survival vs. onset incidence. This phenomenon of OR invertibility vs. RR non-invertibility is best illustrated with an example:

Suppose in a clinical trial, one has an adverse event risk of 4/100 in drug group, and 2/100 in placebo... yielding a RR=2 and OR=2.04166 for drug-vs-placebo adverse risk. However, if analysis was inverted and adverse events were instead analyzed as event-free survival, then the drug group would have a rate of 96/100, and placebo group would have a rate of 98/100—yielding a drug-vs-placebo a RR=0.9796 for survival, but an OR=0.48979. As one can see, a RR of 0.9796 is clearly not the reciprocal of a RR of 2. In contrast, an OR of 0.48979 is indeed the direct reciprocal of an OR of 2.04166.

This is again what is called the 'invariance of the odds ratio', and why a RR for survival is not the same as a RR for risk, while the OR has this symmetrical property when analyzing either survival or adverse risk. The danger to clinical interpretation for the OR comes when the adverse event rate is not rare, thereby exaggerating differences when the OR rare-disease assumption is not met. On the other hand, when the disease is rare, using a RR for survival (e.g. the RR=0.9796 from above example) can clinically hide and conceal an important doubling of adverse risk associated with a drug or exposure.

Sample odds ratio
The sample odds ratio n11n00 / n10n01 is easy to calculate, and for moderate and large samples performs well as an estimator of the population odds ratio. When one or more of the cells in the contingency table can have a small value, the sample odds ratio can be biased and exhibit high variance.

Alternative estimators
A number of alternative estimators of the odds ratio have been proposed to address limitations of the sample odds ratio. One alternative estimator is the conditional maximum likelihood estimator, which conditions on the row and column margins when forming the likelihood to maximize (as in Fisher's exact test). Another alternative estimator is the Mantel–Haenszel estimator.

Numerical examples
The following four contingency tables contain observed cell counts, along with the corresponding sample odds ratio (OR) and sample log odds ratio (LOR):

The following joint probability distributions contain the population cell probabilities, along with the corresponding population odds ratio (OR) and population log odds ratio (LOR):

Related statistics
There are various other summary statistics for contingency tables that measure association between two events, such as Yule's Y, Yule's Q; these two are normalized so they are 0 for independent events, 1 for perfectly correlated, −1 for perfectly negatively correlated. studied these and argued that these measures of association must be functions of the odds ratio, which he referred to as the cross-ratio.

Odds Ratio for a Matched Case-Control Study
A case-control study involves selecting representative samples of cases and controls who do, and do not, have some disease, respectively. These samples are usually independent of each other. The prior prevalence of exposure to some risk factor is observed in subjects from both samples. This permits the estimation of the odds ratio for disease in exposed vs. unexposed people as noted above. Sometimes, however, it makes sense to match cases to controls on one or more confounding variables. In this case, the prior exposure of interest is determined for each case and her/his matched control. The data can be summarized in the following table.

Matched 2x2 Table
This table gives the exposure status of the matched pairs of subjects. There are $$n_{11}$$ pairs where both the case and her/his matched control were exposed, $$n_{10}$$ pairs where the case patient was exposed but the control subject was not, $$n_{01}$$ pairs where the control subject was exposed but the case patient was not, and $$n_{00}$$ pairs were neither subject was exposed. The exposure of matched case and control pairs is correlated due to the similar values of their shared confounding variables.

The following derivation is due to Breslow & Day. We consider each pair as belonging to a stratum with identical values of the confounding variables. Conditioned on belonging to the same stratum, the exposure status of cases and controls are independent of each other. For any case-control pair within the same stratum let

$$p_1$$ be the probability that a case patient is exposed,

$$p_0$$ be the probability that a control patient is exposed,

$$q_1 = 1 - p_1$$ be the probability that a case patient is not exposed, and

$$q_0 = 1- p_0$$ be the probability that a control patient is not exposed.

Then the probability that a case is exposed and a control is not is $$p_1q_0$$, and the probability that a control is exposed and a case in not is $$p_0q_1$$. The within-stratum odds ratio for exposure in cases relative to controls is

$$\psi = (p_1/ q_1)/(p_0/q_0) =  p_1q_0/(q_1p_0) $$

We assume that $$\psi$$ is constant across strata.

Now concordant pairs in which either both the case and the control are exposed, or neither are exposed tell us nothing about the odds of exposure in cases relative to the odds of exposure among controls. The probability that the case is exposed and the control is not given that the pair is discordant is

$$\pi = (p_1q_0)/(p_1q_0 + q_1p_0) =  \psi/(\psi + 1) $$

The distribution of $$n_{10}$$ given the number of discordant pairs is binomial ~  B$$(n_{10} + n_{01},\pi)$$ and the maximum likelihood estimate of $$\pi$$ is

$$\hat{\pi} = n_{10}/(n_{10}+n_{01}) = \hat{\psi}/(\hat{\psi} + 1)$$

Multiplying both sides of this equation by $$(n_{10} + n_{01})(\hat{\psi} + 1 )$$ and subtracting $$n_{10}\hat{\psi}$$ gives

$$n_{10} = \hat{\psi}(n_{10} + n_{01} - n_{10})$$ and hence

$$\hat{\psi} = n_{10}/n_{01} $$.

Now $$\hat{\pi}$$ is the maximum likelihood estimate of $$\pi$$, and $$\psi$$ is a monotonic function of $$\hat{\pi}$$. It follows that $$\hat{\psi}$$ is the conditional maximum likelihood estimate of $$\hat{\psi}$$ given the number of discordant pairs. Rothman et al. give an alternate derivation of $$\hat{\psi}$$ by showing that it is a special case of the Mantel-Haenszel estimate of the intra-strata odds ratio for stratified 2x2 tables. They also reference Breslow & Day as providing the derivation given here.

Under the null hypothesis that $$\psi = 1, \pi = 1/(1+1) = 0.5 $$.

Hence, we can test the null hypothesis that $$\psi = 1$$ by testing the null hypothesis that $$\pi = 0.5$$. This is done using McNemar's test.

There are a number of ways to calculate a confidence interval for $$\pi$$. Let $$\hat{\pi}_{LB} $$ and $$\hat{\pi}_{UB}$$ denote the lower and upper bound of a confidence interval for $$\pi$$, respectively. Since $$\psi = \pi/(1-\pi)$$, the corresponding confidence interval for $$\psi$$ is

$$(\frac{\hat{\pi}_{LB} }{1 -\hat{\pi}_{LB} }, \frac{\hat{\pi}_{UB} }{1 -\hat{\pi}_{UB} })$$.

Matched 2x2 tables may also be analyzed using conditional logistic regression. This technique has the advantage of allowing users to regress case-control status against multiple risk factors from matched case-control data.

Example
McEvoy et al. studied the use of cell phones by drivers as a risk factor for automobile crashes in a case-crossover study. All study subjects were involved in an automobile crash requiring hospital attendance. Each driver's cell phone use at the time of her/his crash was compared to her/his cell phone use in a control interval at the same time of day one week earlier. We would expect that a person's cell phone use at the time of the crash would be correlated with his/her use one week earlier. Comparing usage during the crash and control intervals adjusts for driver's characteristics and the time of day and day of the week. The data can be summarized in the following table.

There were 5 drivers who used their phones in both intervals, 27 who used them in the crash but not the control interval, 6 who used them in the control but not the crash interval, and 288 who did not use them in either interval. The odds ratio for crashing while using their phone relative to driving when not using their phone was

$$\hat{\psi} = 27/6 = 4.5 $$.

Testing the null hypothesis that $$\hat{\psi} =1 $$ is the same as testing the null hypothesis that $$\hat{\pi} =0.5 $$ given 27 out of 33 discordant pairs in which the driver was using her/his phone at the time of his crash. McNemar's $$\chi^2 = 13.36$$. This statistic has one degree of freedom and yields a P value of 0.0003. This allows us to reject the hypothesis that cell phone use has no effect on the risk of automobile crashes ($$\psi = 1$$) with a high level of statistical significance.

Using Wilson's method, a 95% confidence interval for $$\pi$$ is (0.6561, 0.9139). Hence, a 95% confidence interval for $$\psi$$ is

$$(\frac{0.6561 }{1 - 0.6561}, \frac{0.9139}{1 - 0.9139 }) = (1.9, 10.6)$$

(McEvoy et al. analyzed their data using conditional logistic regression and obtained almost identical results to those given here. See the last row of Table 3 in their paper.)

title: Odds ratio


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Fermi problem

In physics or engineering education, a Fermi problem (or Fermi quiz, Fermi question, Fermi estimate), also known as a order-of-magnitude problem (or order-of-magnitude estimate, order estimation), is an estimation problem designed to teach dimensional analysis or approximation of extreme scientific calculations. Fermi problems are usually back-of-the-envelope calculations. The estimation technique is named after physicist Enrico Fermi as he was known for his ability to make good approximate calculations with little or no actual data. Fermi problems typically involve making justified guesses about quantities and their variance or lower and upper bounds. In some cases, order-of-magnitude estimates can also be derived using dimensional analysis.

Historical background
An example is Enrico Fermi's estimate of the strength of the atomic bomb that detonated at the Trinity test, based on the distance traveled by pieces of paper he dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was well within an order of magnitude of the now-accepted value of 21 kilotons.

Examples
Fermi questions are often extreme in nature, and cannot usually be solved using common mathematical or scientific information.

Example questions given by the official Fermi Competition:

Possibly the most famous Fermi Question is the Drake equation, which seeks to estimate the number of intelligent civilizations in the galaxy. The basic question of why, if there were a significant number of such civilizations, human civilization has never encountered any others is called the Fermi paradox.

Advantages and scope
Scientists often look for Fermi estimates of the answer to a problem before turning to more sophisticated methods to calculate a precise answer. This provides a useful check on the results. While the estimate is almost certainly incorrect, it is also a simple calculation that allows for easy error checking, and to find faulty assumptions if the figure produced is far beyond what we might reasonably expect. By contrast, precise calculations can be extremely complex but with the expectation that the answer they produce is correct. The far larger number of factors and operations involved can obscure a very significant error, either in mathematical process or in the assumptions the equation is based on, but the result may still be assumed to be right because it has been derived from a precise formula that is expected to yield good results. Without a reasonable frame of reference to work from it is seldom clear if a result is acceptably precise or is many degrees of magnitude (tens or hundreds of times) too big or too small. The Fermi estimation gives a quick, simple way to obtain this frame of reference for what might reasonably be expected to be the answer.

As long as the initial assumptions in the estimate are reasonable quantities, the result obtained will give an answer within the same scale as the correct result, and if not gives a base for understanding why this is the case. For example, suppose a person was asked to determine the number of piano tuners in Chicago. If their initial estimate told them there should be a hundred or so, but the precise answer tells them there are many thousands, then they know they need to find out why there is this divergence from the expected result. First looking for errors, then for factors the estimation did not take account of – does Chicago have a number of music schools or other places with a disproportionately high ratio of pianos to people? Whether close or very far from the observed results, the context the estimation provides gives useful information both about the process of calculation and the assumptions that have been used to look at problems.

Fermi estimates are also useful in approaching problems where the optimal choice of calculation method depends on the expected size of the answer. For instance, a Fermi estimate might indicate whether the internal stresses of a structure are low enough that it can be accurately described by linear elasticity; or if the estimate already bears significant relationship in scale relative to some other value, for example, if a structure will be over-engineered to withstand loads several times greater than the estimate.

Although Fermi calculations are often not accurate, as there may be many problems with their assumptions, this sort of analysis does inform one what to look for to get a better answer. For the above example, one might try to find a better estimate of the number of pianos tuned by a piano tuner in a typical day, or look up an accurate number for the population of Chicago. It also gives a rough estimate that may be good enough for some purposes: if a person wants to start a store in Chicago that sells piano tuning equipment, and calculates that they need 10,000 potential customers to stay in business, they can reasonably assume that the above estimate is far enough below 10,000 that they should consider a different business plan (and, with a little more work, they could compute a rough upper bound on the number of piano tuners by considering the most extreme reasonable values that could appear in each of their assumptions).

Explanation
Fermi estimates generally work because the estimations of the individual terms are often close to correct, and overestimates and underestimates help cancel each other out. That is, if there is no consistent bias, a Fermi calculation that involves the multiplication of several estimated factors (such as the number of piano tuners in Chicago) will probably be more accurate than might be first supposed.

In detail, multiplying estimates corresponds to adding their logarithms; thus one obtains a sort of Wiener process or random walk on the logarithmic scale, which diffuses as $$\sqrt{n}$$ (in number of terms n). In discrete terms, the number of overestimates minus underestimates will have a binomial distribution. In continuous terms, if one makes a Fermi estimate of n steps, with standard deviation σ units on the log scale from the actual value, then the overall estimate will have standard deviation $$\sigma\sqrt{n}$$, since the standard deviation of a sum scales as $$\sqrt{n}$$ in the number of summands.

For instance, if one makes a 9-step Fermi estimate, at each step overestimating or underestimating the correct number by a factor of 2 (or with a standard deviation 2), then after 9 steps the standard error will have grown by a logarithmic factor of $$\sqrt{9} = 3$$, so 23 = 8. Thus one will expect to be within $1/undefined$ to 8 times the correct value – within an order of magnitude, and much less than the worst case of erring by a factor of 29 = 512 (about 2.71 orders of magnitude). If one has a shorter chain or estimates more accurately, the overall estimate will be correspondingly better.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Fermi problem


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Soyuz TM-28

TM-28 was a Soyuz mission to the Mir space station.

Mission Accomplishments

 * Docked with Mir.
 * Baturin became the first Russian politician in space.
 * Padalka and Avdeyev performed an EVA on 15.09.1998 (30 m) into the module Spektr (new cables connected for solar structures).

title: Soyuz TM-28


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Talk:The Law of Fives

The following text was removed, as I can find no verification for it. To reinstate the text could you please include your source on this page.

 * The Law of Fives Is the title of a five-book epic poem, currently complete to two books. The title of the story refers to a principle of Discordianism, which states that all things are indirectly related to the number five, and this is reflected in the iambic pentameter and groupings of five lines per stanza and five stanzas per canto.

HappyDog 14:03, 18 Dec 2003 (UTC)

The following text was removed, as I can find no verification for it. To reinstate the text could you please include your source on this page.

 * The Law of Fives Is also the title of a poem based loosely on this principle. The first two parts of the poem can be found at 1 and 2.

Also, I added a link to The Law of Fives (band), based on. Dbenbenn 02:34, 31 Dec 2004 (UTC)

I hope the combinatorial implications of five as a number of co-conspirators (as described in Illuminatus!) find their way into the article. I will effect this if necessary, probably in a month or so.

title: Talk:The Law of Fives


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Parramatta River

The Parramatta River is an intermediate tide-dominated, drowned valley estuary located in Sydney, New South Wales, Australia. With an average depth of 5.1 m, the Parramatta River is the main tributary of Sydney Harbour, a branch of Port Jackson. Secondary tributaries include the smaller Lane Cove and Duck rivers.

Formed by the confluence of Toongabbie Creek and Darling Mills Creek at North Parramatta, the river flows in an easterly direction to a line between Yurulbin in Birchgrove and Manns Point in Greenwich. Here it flows into Port Jackson, about 21 km from the Tasman Sea. The total catchment area of the river is approximately 252.4 km2 and is tidal to Charles Street Weir in Parramatta, approximately 30 km from the Sydney Heads.

The land adjacent to the Parramatta River was occupied for many thousands of years by Aboriginal peoples of the Wallumettagal nations and the Wangal, Toongagal (or Tugagal), Burramattagal, and Wategora clans of the Darug people. They used the river as an important source of food and a place for trade. The river was formed 15 to 29 million years ago as its waters began to cut a valley into sandstone and shale.

Inflowing waterways
The headwaters of the Parramatta River are formed by the confluence of Darling Mills Creek and Toongabbie Creek. The point of the confluence lies on the northern border of the grounds of Cumberland Hospital. It also lies on the boundary of the suburbs of Westmead, Northmead, and North Parramatta.

Waterways flowing into the Parramatta River, westtoeast include:
 * Vineyard Creek at Rydalmere, from the north
 * Ponds Subiaco Creek at Rydalmere, from the north
 * Duck River at Silverwater, from the south
 * Archer Creek at Meadowbank, from the north
 * Smalls Creek at Meadowbank, from the far north
 * Charity Creek at Meadowbank, from the north
 * Haslams Creek at Homebush Bay, from the south
 * Powells Creek at Homebush Bay, from the south
 * Iron Cove Creek at Five Dock, from the south
 * Hawthorne Canal at Iron Cove, from the south
 * Tarban Creek at Huntleys Point, from the north
 * Lane Cove River at Greenwich, from the north

The river in Parramatta
From its start at the confluence of Toongabbie Creek and Darling Mills Creek at North Parramatta, the river flows in a southerly direction through the grounds of Cumberland Hospital. Entering Parramatta Park, it then turns east and flows through the Parramatta CBD. Both banks are largely open to the public, with parkland and walkways, downstream to James Ruse Drive. The river is fed by a number of small creeks and stormwater drains.

The waters are controlled by a series of weirs: the weir at the edge of the hospital grounds, the Kiosk Weir in Parramatta Park, the Marsden Street Weir, and the Charles Street Weir at the ferry wharf. The weirs have been equipped with fish ladders. Kiosk Weir and Charles Street Weir also include footbridges enabling a crossing of the river.

Historically, the river was dammed to provide reservoirs for the town. Currently, however, the function of the weirs is aesthetic, preventing the water from draining away during dry periods. As a consequence the river floods in heavy rain, particularly at the Charles Street Weir. The Charles Street Weir forms the boundary between fresh water and salt water, and is also the limit of tides.

River governance
The whole of Sydney Harbour including its tributary rivers is subject to a long range Catchment Management Plan. The Government has almost eliminated local representation by eliminating the former local catchment management boards.

The New South Wales Government has a documented policy in relation to access to the harbour and river foreshores, including public access to intertidal lands where landowners have absolute waterfronts but where the waterfront is exposed at low tide. Moorings and jetties are the responsibility of Transport for NSW, who are also responsible for the management of the Harbour and river seabed. Many bays contain swing moorings, mostly privately owned, but some associated with commercial marinas.

Along the Parramatta River many hands have made lighter work, in the community-wide effort to make the entire river swimmable again by 2025, starting with the opening of Lake Parramatta in 2014. Thirteen councils sit within the Parramatta River catchment group and all have committed to tackling the two major polluters: sewer overflows and stormwater.

Ferry services and wharves
There are River Cat services along the Parramatta River to Circular Quay. The main wharves, westtoeast are:

Crossings
The Parramatta River, along with Sydney Harbour, is the most significant waterway in Sydney. Since settlement, the river and the harbour have presented a formidable barrier between the earlyEuropean settled southern Farm Cove precinct, to development north of the waterway. Together, Parramatta River and Port Jackson literally cut Sydney in half along its northsouth axis.

As a result, the many crossings are extremely important to the life of the city. From westtoeast, the crossings of the Parramatta River are located at:

Water quality concerns
Until 1970 the river was an open drain for Sydney's industry and consequently the southern central embayments are contaminated with a range of heavy metals and chemicals. The Northern Bays are less affected as the Sydney Harbour Bridge was not completed until 1932 and so industrial development was already well established on the southern side of the Harbour.

Gavin Birch of the University of Sydney has published a number of papers which show that Sydney Harbour is as contaminated as most other harbours in industrialised cities, that the main sediment contamination is in the southern central embayments (Blackwattle to Homebush Bays), that there are five particularly contaminated areas of Sydney Harbour, and that four of them are in the Parramatta river system.

The main contaminated areas of the Parramatta River are:
 * Homebush Bay - dioxins, lead, phthalates, DDT, PAHs (coal tars) mainly originating from nearby chemical factories of Berger Paints, CSR Chemicals, ICI/Orica, and Union Carbide.
 * Iron Cove - various metals and chemicals with no clearly defined point source. Pollution may possibly enter through Iron Cove Creek and Hawthorne Canal.
 * Adjacent to the former AGL site, now redeveloped as Breakfast Point.

Water quality is monitored by the Office of Environment and Heritage (New South Wales) (OEH) for faecal coliforms and e. coli, but only as far west as Cabarita. OEH does not monitor water quality further west on the river, despite increasing development and the use of the river for recreational boating. In the areas monitored, the water quality is generally acceptable except after heavy rains.

Fishing
The Parramatta River is subject to a number of fishing bans because of its contaminated sediments. There is a complete fishing ban in Homebush Bay because of the dioxin contamination. In 2006 the NSW Government imposed a complete commercial fishing ban throughout the rest of Sydney Harbour and its tributaries, including the Parramatta River west of the Harbour Bridge, which remained in place until 2011.

The Parramatta River is one of the few significant coastal rivers in New South Wales which has not been the subject of an investigation by the Healthy Rivers Commission. The Cooks River and Botany Bay have been subject to such an investigation. Some have campaigned for a Healthy Rivers Commission inquiry to bring together all the information on the state of the river and its sediments and fish and assist in watershed management.

Major wetlands and environmentally sensitive areas
Major wetlands include:
 * Bicentennial Park Wetlands (nationally significant, JAMBA CAMBA)
 * Newington Wetlands (nationally significant, JAMBA CAMBA)

There are significant stands of mangroves along the river west of Henley (on the river's northern shore) and Mortlake (on the river's southern shore) and in the Lane Cove River. The mangroves have actually colonised areas that were previously salt marsh. Research into historical drawings and writings indicates that the mangroves were far fewer at English colonisation. Council information panels in Glades Bay explain that the bay's now extensive mangrove stands would once have been open water, sandy beaches and outcrops of rock. Land clearing and development has allowed soil and various nutrients to be washed into the river. This has provided an ideal environment for mangroves to colonise. The excessive siltation of the river is an ongoing problem.

Reclamation
Many areas of the river, particularly the swampy heads of bays, have been reclaimed, often being used as rubbish dumps before being converted into playing fields. Large sections of Meadowbank Park were created in this manner. Some industrial sites were also reclaimed heavily from the river, particularly in Homebush Bay. Most creeks leading into the bays have been channelised (lined with concrete walls and floor).

Remediation
While some areas of the river with heavily contaminated sediments have not been remediated, there is significant remediation of sediments about to start in Homebush Bay. These include the dioxin contaminated sediments near the former Union Carbide plant and the lead contaminated sediments near the former Berger Paints plant. The former AGL site has been analysed, a remediation plan developed and approved, remediation completed and construction commenced on medium to high density residential development, but the sediments, which independent research shows to be contaminated with pollutants from the AGL operations, have not yet had the investigation stage finalised.

Other areas have had sediments covered with concrete to prevent the fish eating the benthos. Some of these companies claim that the contamination is best left alone, but environmentalists have argued that the contamination could be being passed up the food chain and claim that the companies are trying to avoid the costs incurred in pollution cleanup.

Sailing
There are a number of sailing and yachting clubs on the River:
 * Abbotsford 12 ft Flying Squadron in Abbotsford
 * Concord & Ryde Sailing Club at Putney
 * Parramatta River Sailing Club at Gladesville

Sailing and rowing take place under an aquatic licence granted annually by Roads and Maritime Services.

Sea Scouts
There are Sea Scouts at:
 * Rhodes: First Yaralla
 * Meadowbank: Epping Scout Group has a boat shed at 150 Bowden Street, next to the public ferry wharf, and in the summer uses it for sailing and canoeing activities

Rowing
The river has a long historical association with rowing. There is a monument in honour of Henry Searle, a champion sculler of his day, in the river at Henley.

Some of the school rowing sheds are:
 * MLC School
 * The King's School
 * Sydney Boys High School
 * Newington College
 * The Scots College
 * Sydney Grammar School
 * Shore School

There are also a number of rowing clubs:
 * Leichhardt Rowing Club, Drummoyne Rowing Club and UTS Haberfield within Iron Cove.
 * Glebe Rowing Club in Blackwattle Bay.
 * Sydney Rowing Club at Abbotsford Point.
 * Balmain Rowing Club at Balmain
 * UNSW Boat Club in Tarban Creek.
 * SGHS Rowing Club also in Tarban Creek

While a number of regattas are still held on the river each year, mainly in Iron Cove and Hen and Chicken Bay, many of the major regattas are now held at the Sydney International Regatta Centre (SIRC), adjacent to the Nepean River, where rowing was held for the 2000 Summer Olympics. Early GPS Schoolboy Head of the River races were held on the Parramatta River before moving to the Nepean River and later SIRC.

Most rowing training is done in the middle to upper reaches of the river between Abbotsford and Homebush Bay because there is less water traffic and therefore less waves and more protection from wind. Rowing also takes place in the Lane Cove River and Iron Cove which have less traffic.

Foreshore walks and cycleways
Much of the foreshore is still in the hands of industry and private individuals as residences, however there is an increasing amount of waterfront land available as foreshore reserve with walkways and cycleways. As former industrial sites undergo remediation and redevelopment, the foreshores are opened up for public access. Where there is no foreshore access, cycleways are routed through quiet residential streets with clearly marked sections of the road reserved for cyclists.

Major foreshore parks include:
 * Cabarita Park
 * Kissing Point Park, Ryde
 * Meadowbank Park
 * Putney Park
 * George Kendall Riverside Park, Ermington
 * Bicentennial Park, Homebush Bay
 * Millennium Parklands, Homebush/Auburn

Heritage items
Major heritage buildings, all listed on the Register of the National Estate, on or near the foreshore include:
 * Thomas Walker Convalescent Hospital (now Rivendell Child, Adolescent and Family Unit), was designed by Sir John Sulman and built in 1891–93.
 * Callan Park, a former psychiatric hospital, with fine sandstone buildings designed by James Barnet in 1877.
 * Gladesville Mental Hospital, with sandstone buildings in park-like surroundings, designed by Mortimer Lewis in 1836.
 * Yaralla, the Italianate mansion of Thomas Walker and his daughter Dame Eadith Walker, generous benefactors of the Concord community, now functions as Dame Eadith Walker Hospital. It was built in the late 1860s, with later extensions by Sir John Sulman.
 * Newington House, a Regency home built in 1832 by John Blaxland. In 1863, the property was acquired by the Wesleyans and turned into Newington College. In 1887 it became part of Silverwater State Hospital, and later became an administrative block in Silverwater Correctional Complex.

title: Parramatta River


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Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) $Ai(x)$ is a special function named after the British astronomer George Biddell Airy (1801–1892). The function $Ai(x)$ and the related function $Bi(x)$, are linearly independent solutions to the differential equation $$\frac{d^2y}{dx^2} - xy = 0, $$ known as the Airy equation or the Stokes equation.

Because the solution of the linear differential equation $$\frac{d^2y}{dx^2} - ky = 0$$ is oscillatory for $k<0$ and exponential for $k>0$, the Airy functions are oscillatory for $x<0$ and exponential for $x>0$. In fact, the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential).

Definitions
For real values of $x$, the Airy function of the first kind can be defined by the improper Riemann integral: $$\operatorname{Ai}(x) = \dfrac{1}{\pi}\int_0^\infty\cos\left(\dfrac{t^3}{3} + xt\right)\, dt\equiv \dfrac{1}{\pi} \lim_{b\to\infty} \int_0^b \cos\left(\dfrac{t^3}{3} + xt\right)\, dt,$$ which converges by Dirichlet's test. For any real number $x$ there is a positive real number $M$ such that function $\tfrac{t^3}3 + xt$ is increasing, unbounded and convex with continuous and unbounded derivative on interval $$[M,\infty).$$ The convergence of the integral on this interval can be proven by Dirichlet's test after substitution $u=\tfrac{t^3}3 + xt.$

$Ai(z)$ satisfies the Airy equation $$y'' - xy = 0.$$ This equation has two linearly independent solutions. Up to scalar multiplication, $-2 - 2i$ is the solution subject to the condition $2 + 2i$ as $Ai(z)$. The standard choice for the other solution is the Airy function of the second kind, denoted Bi(x). It is defined as the solution with the same amplitude of oscillation as $-2 - 2i$ as $2 + 2i$ which differs in phase by $Ai'(z)$: $$\operatorname{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac{t^3}{3} + xt\right) + \sin\left(\tfrac{t^3}{3} + xt\right)\,\right]dt.$$

Properties
The values of $-2 - 2i$ and $2 + 2i$ and their derivatives at $Ai'(z)$ are given by $$\begin{align} \operatorname{Ai}(0) &{}= \frac{1}{3^{2/3} \, \Gamma\!\left(\frac{2}{3}\right)}, & \quad \operatorname{Ai}'(0) &{}= -\frac{1}{3^{1/3} \, \Gamma\!\left(\frac{1}{3}\right)}, \\ \operatorname{Bi}(0) &{}= \frac{1}{3^{1/6} \, \Gamma\!\left(\frac{2}{3}\right)}, & \quad \operatorname{Bi}'(0) &{}= \frac{3^{1/6}}{\Gamma\!\left(\frac{1}{3}\right)}. \end{align}$$ Here, $-2 - 2i$ denotes the Gamma function. It follows that the Wronskian of $2 + 2i$ and $Ai(x)$ is $Bi(x)$.

When $x$ is positive, $y = Ai(x)$ is positive, convex, and decreasing exponentially to zero, while $Ai(x)$ is positive, convex, and increasing exponentially. When $x$ is negative, $y → 0$ and $x → ∞$ oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.

The Airy functions are orthogonal in the sense that $$ \int_{-\infty}^\infty \operatorname{Ai}(t+x) \operatorname{Ai}(t+y) dt = \delta(x-y)$$ again using an improper Riemann integral.

Neither $Ai(x)$ nor its derivative $x → −∞$ have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are:
 * Real zeros of $π/2$ and its derivative $Bi(z)$
 * "first" zeros of $-2 - 2i$ are at $2 + 2i$
 * "first" zeros of its derivative $Bi'(z)$ are at $-2 - 2i$

Asymptotic formulae
As explained below, the Airy functions can be extended to the complex plane, giving entire functions. The asymptotic behaviour of the Airy functions as $|z|$ goes to infinity at a constant value of $2 + 2i$ depends on $Ai(x)$: this is called the Stokes phenomenon. For $Bi(x)$ we have the following asymptotic formula for $x = 0$:

$$ \operatorname{Ai}(z)\sim \dfrac{1}{2\sqrt\pi\,z^{1/4}} \exp\left(-\frac{2}{3}z^{3/2}\right) \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].$$ or$$ \operatorname{Ai}(z)\sim \dfrac{e^{-\zeta}}{4\pi^{3/2}\,z^{1/4}} \left[ \sum_{n=0}^{\infty} \dfrac{\Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right)}{n! (-2\zeta)^n} \right].$$ where $$\zeta = \tfrac 23 z^{3/2}.$$ In particular, the first few terms are $$\operatorname{Ai}(z) = \frac{e^{-\zeta}}{2\pi^{1/2}z^{1/4}}\left(1 - \frac{5}{72 \zeta} + \frac{385}{10368 \zeta^2} + O(\zeta^{-3})\right) $$ There is a similar one for $Γ$, but only applicable when $Ai(x)$:

$$ \operatorname{Bi}(z)\sim \frac{1}{\sqrt\pi\,z^{1/4}} \exp\left(\frac{2}{3}z^{3/2}\right) \left[ \sum_{n=0}^{\infty} \dfrac{ \Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].$$ A more accurate formula for $Bi(x)$ and a formula for $1/π$ when $Ai(x)$ or, equivalently, for $Bi(x)$ and $Ai(x)$ when $Bi(x)$ but not zero, are: $$\begin{align} \operatorname{Ai}(-z) \sim&{} \ \frac{1}{\sqrt\pi\,z^{1/4}} \sin\left( \frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6} \right) \, \Gamma\!\left(2n+\frac{1}{6}\right) \left(\frac{3}{4} \right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]

&{}-\frac{1}{\sqrt\pi \, z^{1/4}} \cos\left(\frac{2}{3}z^{3/2}+\frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\[6pt] \operatorname{Bi}(-z) \sim&{} \frac{1}{\sqrt\pi \, z^{1/4}} \cos \left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt] &{}+ \frac{1}{\sqrt\pi\,z^{\frac{1}{4}}} \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right]. \end{align}$$

When $Ai(x)$ these are good approximations but are not asymptotic because the ratio between $Ai'(x)$ or $Ai(x)$ and the above approximation goes to infinity whenever the sine or cosine goes to zero. Asymptotic expansions for these limits are also available. These are listed in (Abramowitz and Stegun, 1983) and (Olver, 1974).

One is also able to obtain asymptotic expressions for the derivatives $Ai'(x)$ and $Ai(x)$. Similarly to before, when $x ≈ −2.33811, −4.08795, −5.52056, −6.78671, ...$:

$$ \operatorname{Ai}'(z)\sim -\dfrac{z^{1/4}}{2\sqrt\pi\,} \exp\left(-\frac{2}{3}z^{3/2}\right) \left[ \sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \dfrac{(-1)^n \, \Gamma\!\left(n + \frac{5}{6}\right) \, \Gamma\!\left(n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].$$

When $Ai'(x)$ we have:

$$ \operatorname{Bi}'(z)\sim \frac{z^{1/4}}{\sqrt\pi\,} \exp\left(\frac{2}{3}z^{3/2}\right) \left[ \sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \dfrac{ \Gamma\!\left(n + \frac{5}{6}\right) \, \Gamma\!\left(n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].$$

Similarly, an expression for $x ≈ −1.01879, −3.24820, −4.82010, −6.16331, ...$ and $arg(z)$ when $arg(z)$ but not zero, are

$$\begin{align} \operatorname{Ai}'(-z) \sim&{} -\frac{z^{1/4}}{\sqrt\pi\,} \cos\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \frac{1+12n}{1-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]

&{}-\frac{z^{1/4}}{\sqrt\pi\,} \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \frac{7+12n}{-5-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\[6pt] \operatorname{Bi}'(-z) \sim&{} \ \frac{z^{1/4}}{\sqrt\pi\,} \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \frac{1+12n}{1-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt] &{}-\frac{z^{1/4}}{\sqrt\pi\,} \cos\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) \left[ \sum_{n=0}^{\infty} \frac{7+12n}{-5-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\ \end{align}$$

Complex arguments
We can extend the definition of the Airy function to the complex plane by $$\operatorname{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\tfrac{t^3}{3} - zt\right)\, dt,$$ where the integral is over a path C starting at the point at infinity with argument $|arg(z)| < π$ and ending at the point at infinity with argument π/3. Alternatively, we can use the differential equation $Ai(z)$ to extend $Bi(z)$ and $|arg(z)| < π/3$ to entire functions on the complex plane.

The asymptotic formula for $Ai(z)$ is still valid in the complex plane if the principal value of $Bi(z)$ is taken and $x$ is bounded away from the negative real axis. The formula for $π/3 < |arg(z)| < π$ is valid provided $x$ is in the sector $$x\in\C : \left|\arg(x)\right| < \tfrac{\pi}{3} - \delta$$ for some positive δ. Finally, the formulae for $Ai(−z)$ and $Bi(−z)$ are valid if $|arg(z)| < 2π/3$ is in the sector $$x\in\C : \left|\arg(x)\right| < \tfrac{2\pi}{3} - \delta.$$

It follows from the asymptotic behaviour of the Airy functions that both $|arg(z)| = 0$ and $Ai(−z)$ have an infinity of zeros on the negative real axis. The function $Bi(−z)$ has no other zeros in the complex plane, while the function $Ai'(z)$ also has infinitely many zeros in the sector $$z\in\C : \tfrac{\pi}{3} < \left|\arg(z)\right| < \tfrac{\pi}{2}.$$

Relation to other special functions
For positive arguments, the Airy functions are related to the modified Bessel functions: $$\begin{align} \operatorname{Ai}(x) &{}= \frac1\pi \sqrt{\frac{x}{3}} \, K_{1/3}\!\left(\frac{2}{3} x^{3/2}\right), \\ \operatorname{Bi}(x) &{}= \sqrt{\frac{x}{3}} \left[I_{1/3}\!\left(\frac{2}{3} x^{3/2}\right) + I_{-1/3}\!\left(\frac{2}{3} x^{3/2}\right)\right]. \end{align}$$ Here, $Bi'(z)$ and $|arg(z)| < π$ are solutions of $$x^2y'' + xy' - \left (x^2 + \tfrac{1}{9} \right )y = 0.$$

The first derivative of the Airy function is $$ \operatorname{Ai'}(x) = - \frac{x} {\pi \sqrt{3}} \, K_{2/3}\!\left(\frac{2}{3} x^{3/2}\right) .$$

Functions $|arg(z)| < π/3$ and $Ai'(−z)$ can be represented in terms of rapidly convergent integrals (see also modified Bessel functions)

For negative arguments, the Airy function are related to the Bessel functions: $$\begin{align} \operatorname{Ai}(-x) &{}= \sqrt{\frac{x}{9}} \left[J_{1/3}\!\left(\frac{2}{3} x^{3/2}\right) + J_{-1/3}\!\left(\frac{2}{3} x^{3/2}\right)\right], \\ \operatorname{Bi}(-x) &{}= \sqrt{\frac{x}{3}} \left[J_{-1/3}\!\left(\frac{2}{3 }x^{3/2}\right) - J_{1/3}\!\left(\frac23 x^{3/2}\right)\right]. \end{align}$$ Here, $Bi'(−z)$ are solutions of $$x^2y'' + xy' + \left (x^2 - \frac{1}{9} \right )y = 0.$$

The Scorer's functions $|arg(z)| < 2π/3$ and $−π/3$ solve the equation $y′′ − xy = 0$. They can also be expressed in terms of the Airy functions: $$\begin{align} \operatorname{Gi}(x) &{}= \operatorname{Bi}(x) \int_x^\infty \operatorname{Ai}(t) \, dt + \operatorname{Ai}(x) \int_0^x \operatorname{Bi}(t) \, dt, \\ \operatorname{Hi}(x) &{}= \operatorname{Bi}(x) \int_{-\infty}^x \operatorname{Ai}(t) \, dt - \operatorname{Ai}(x) \int_{-\infty}^x \operatorname{Bi}(t) \, dt. \end{align}$$

Fourier transform
Using the definition of the Airy function Ai(x), it is straightforward to show its Fourier transform is given by $$\mathcal{F}(\operatorname{Ai})(k) := \int_{-\infty}^{\infty} \operatorname{Ai}(x)\ e^{- 2\pi i k x}\,dx = e^{\frac{i}{3} (2\pi k)^3}.$$This can be obtained by taking the Fourier transform of the Airy equation. Let $\hat y = \frac{1}{2\pi i}\int y e^{-ikx}dx$, then $$i\hat y' + k^2 \hat y = 0$$, which then has solutions $$\hat y = C e^{ik^3/3}.$$ There only one dimension of solutions because the Fourier transform requires $y$ to decay to zero fast enough, and $Ai(x)$ grows to infinity exponentially fast, so it cannot be obtained via Fourier transform.

Quantum mechanics
The Airy function is the solution to the time-independent Schrödinger equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the WKB approximation, when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of electrons trapped in semiconductor heterojunctions.

Optics
A transversally asymmetric optical beam, where the electric field profile is given by the Airy function, has the interesting property that its maximum intensity accelerates towards one side instead of propagating in a straight line as is the case in symmetric beams. This is at expense of the low-intensity tail being spread in the opposite direction, so the overall momentum of the beam is of course conserved.

Caustics
The Airy function underlies the form of the intensity near an optical directional caustic, such as that of the rainbow (called supernumerary rainbow). Historically, this was the mathematical problem that led Airy to develop this special function. In 1841, William Hallowes Miller experimentally measured the analog to supernumerary rainbow by shining light through a thin cylinder of water, then observing through a telescope. He observed up to 30 bands.

Probability
In the mid-1980s, the Airy function was found to be intimately connected to Chernoff's distribution.

The Airy function also appears in the definition of Tracy–Widom distribution which describes the law of largest eigenvalues in Random matrix. Due to the intimate connection of random matrix theory with the Kardar–Parisi–Zhang equation, there are central processes constructed in KPZ such as the Airy process.

History
The Airy function is named after the British astronomer and physicist George Biddell Airy (1801–1892), who encountered it in his early study of optics in physics (Airy 1838). The notation Ai(x) was introduced by Harold Jeffreys. Airy had become the British Astronomer Royal in 1835, and he held that post until his retirement in 1881.

title: Airy function


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Clausen function

In mathematics, the Clausen function, introduced by, is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.

The Clausen function of order 2 – often referred to as the Clausen function, despite being but one of a class of many – is given by the integral:

 * $$\operatorname{Cl}_2(\varphi)=-\int_0^\varphi \log\left|2\sin\frac{x}{2} \right|\, dx:$$

In the range $$0 < \varphi < 2\pi\, $$ the sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the Fourier series representation:

 * $$\operatorname{Cl}_2(\varphi)=\sum_{k=1}^\infty \frac{\sin k\varphi}{k^2} = \sin\varphi +\frac{\sin 2\varphi}{2^2}+\frac{\sin 3\varphi}{3^2}+\frac{\sin 4\varphi}{4^2}+ \cdots $$

The Clausen functions, as a class of functions, feature extensively in many areas of modern mathematical research, particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the polygamma function, and Dirichlet L-series.

Basic properties
The Clausen function (of order 2) has simple zeros at all (integer) multiples of $$\pi, \,$$ since if $$k\in \mathbb{Z} \, $$ is an integer, then $$\sin k\pi=0$$

 * $$\operatorname{Cl}_2(m\pi) =0, \quad m= 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots $$

It has maxima at $$\theta = \frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]$$

 * $$\operatorname{Cl}_2\left(\frac{\pi}{3}+2m\pi \right) =1.01494160 \ldots $$

and minima at $$\theta = -\frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]$$

 * $$\operatorname{Cl}_2\left(-\frac{\pi}{3}+2m\pi \right) =-1.01494160 \ldots $$

The following properties are immediate consequences of the series definition:

 * $$\operatorname{Cl}_2(\theta+2m\pi) = \operatorname{Cl}_2(\theta) $$
 * $$\operatorname{Cl}_2(-\theta) = -\operatorname{Cl}_2(\theta) $$

See.

General definition
More generally, one defines the two generalized Clausen functions:

 * $$\operatorname{S}_z(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta}{k^z}$$
 * $$\operatorname{C}_z(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta}{k^z}$$

which are valid for complex z with Re z >1. The definition may be extended to all of the complex plane through analytic continuation.

When z is replaced with a non-negative integer, the standard Clausen functions are defined by the following Fourier series:

 * $$\operatorname{Cl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}$$
 * $$\operatorname{Cl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}$$
 * $$\operatorname{Sl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}$$
 * $$\operatorname{Sl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}$$

N.B. The SL-type Clausen functions have the alternative notation $$\operatorname{Gl}_m(\theta)\, $$ and are sometimes referred to as the Glaisher–Clausen functions (after James Whitbread Lee Glaisher, hence the GL-notation).

Relation to the Bernoulli polynomials
The SL-type Clausen function are polynomials in $$\, \theta\, $$, and are closely related to the Bernoulli polynomials. This connection is apparent from the Fourier series representations of the Bernoulli polynomials:

 * $$B_{2n-1}(x)=\frac{2(-1)^n(2n-1)!}{(2\pi)^{2n-1}} \, \sum_{k=1}^\infty \frac{\sin 2\pi kx}{k^{2n-1}}.$$
 * $$B_{2n}(x)=\frac{2(-1)^{n-1}(2n)!}{(2\pi)^{2n}} \, \sum_{k=1}^\infty \frac{\cos 2\pi kx}{k^{2n}}.$$

Setting $$\, x= \theta/2\pi \, $$ in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:

 * $$\operatorname{Sl}_{2m}(\theta) = \frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!} B_{2m}\left(\frac{\theta}{2\pi}\right),$$
 * $$\operatorname{Sl}_{2m-1}(\theta) = \frac{(-1)^{m}(2\pi)^{2m-1}}{2(2m-1)!} B_{2m-1}\left(\frac{\theta}{2\pi}\right), $$

where the Bernoulli polynomials $$\, B_n(x)\,$$ are defined in terms of the Bernoulli numbers $$\, B_n \equiv B_n(0)\, $$ by the relation:

 * $$B_n(x)=\sum_{j=0}^n\binom{n}{j} B_jx^{n-j}.$$

Explicit evaluations derived from the above include:

 * $$ \operatorname{Sl}_1(\theta)= \frac{\pi}{2}-\frac \theta 2, $$
 * $$ \operatorname{Sl}_2(\theta)= \frac{\pi^2}{6}-\frac{\pi\theta} 2 +\frac{\theta^2}{4}, $$
 * $$ \operatorname{Sl}_3(\theta)= \frac{\pi^2\theta}{6} -\frac{\pi\theta^2}{4}+\frac{\theta^3}{12}, $$
 * $$ \operatorname{Sl}_4(\theta)= \frac{\pi^4}{90}-\frac{\pi^2\theta^2}{12}+\frac{\pi\theta^3}{12}-\frac{\theta^4}{48}. $$

Duplication formula
For $$ 0 < \theta < \pi $$, the duplication formula can be proven directly from the integral definition (see also for the result – although no proof is given):

 * $$\operatorname{Cl}_2(2\theta) = 2\operatorname{Cl}_2(\theta) - 2\operatorname{Cl}_2(\pi-\theta) $$

Denoting Catalan's constant by $$K=\operatorname{Cl}_2\left(\frac{\pi}{2}\right)$$, immediate consequences of the duplication formula include the relations:

 * $$\operatorname{Cl}_2\left(\frac{\pi}{4}\right)- \operatorname{Cl}_2 \left(\frac{3\pi} 4\right)=\frac K 2$$
 * $$2\operatorname{Cl}_2\left(\frac{\pi}{3}\right)= 3\operatorname{Cl}_2 \left(\frac{2\pi} 3\right)$$

For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace $$ \, \theta \, $$ with the dummy variable $$x$$, and integrate over the interval $$ \, [0, \theta]. \, $$ Applying the same process repeatedly yields:

 * $$\operatorname{Cl}_3(2\theta) = 4\operatorname{Cl}_3(\theta) + 4\operatorname{Cl}_3(\pi-\theta) $$
 * $$\operatorname{Cl}_4(2\theta) = 8\operatorname{Cl}_4(\theta) - 8\operatorname{Cl}_4(\pi-\theta) $$
 * $$\operatorname{Cl}_5(2\theta) = 16\operatorname{Cl}_5(\theta) + 16 \operatorname{Cl}_5(\pi-\theta) $$
 * $$\operatorname{Cl}_6(2\theta) = 32\operatorname{Cl}_6(\theta) - 32 \operatorname{Cl}_6(\pi-\theta) $$

And more generally, upon induction on $$\, m, \; m \ge 1 $$

 * $$\operatorname{Cl}_{m+1}(2\theta) = 2^m\left[\operatorname{Cl}_{m+1}(\theta) + (-1)^m \operatorname{Cl}_{m+1}(\pi-\theta) \right]$$

Use of the generalized duplication formula allows for an extension of the result for the Clausen function of order 2, involving Catalan's constant. For $$\, m \in \mathbb{Z} \ge 1\, $$

 * $$\operatorname{Cl}_{2m}\left(\frac \pi 2 \right) = 2^{2m-1} \left[\operatorname{Cl}_{2m}\left(\frac{\pi}{4}\right)- \operatorname{Cl}_{2m}\left(\frac{3\pi}{4}\right) \right] = \beta(2m)$$

Where $$\, \beta(x) \, $$ is the Dirichlet beta function.

Proof of the duplication formula
From the integral definition,

 * $$\operatorname{Cl}_2(2\theta)=-\int_0^{2\theta} \log\left| 2 \sin \frac{x}{2} \right| \,dx$$

Apply the duplication formula for the sine function, $$\sin x = 2\sin\frac{x}{2}\cos\frac{x}{2}$$ to obtain

\begin{align} & -\int_0^{2\theta} \log\left| \left(2 \sin \frac{x}{4} \right)\left(2 \cos \frac{x}{4} \right) \right| \,dx \\ = {} & -\int_0^{2\theta} \log\left| 2 \sin \frac{x}{4} \right| \,dx -\int_0^{2\theta} \log\left| 2 \cos \frac{x}{4} \right| \,dx \end{align} $$

Apply the substitution $$x=2y, dx=2\, dy$$ on both integrals:

\begin{align} & -2\int_0^\theta \log\left| 2 \sin \frac{x}{2} \right| \,dx -2\int_0^\theta \log\left| 2 \cos \frac{x}{2} \right| \,dx \\ = {} & 2\, \operatorname{Cl}_2(\theta) -2\int_0^\theta \log\left| 2 \cos \frac{x}{2} \right| \,dx \end{align} $$

On that last integral, set $$y=\pi-x, \, x= \pi-y, \, dx = -dy$$, and use the trigonometric identity $$\cos(x-y)=\cos x\cos y - \sin x\sin y$$ to show that:

\begin{align} & \cos\left(\frac{\pi-y}{2}\right) = \sin \frac{y}{2} \\ \Longrightarrow \qquad & \operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta) -2\int_0^\theta \log\left| 2 \cos \frac{x}{2} \right| \,dx \\ = {} & 2\, \operatorname{Cl}_2(\theta) +2\int_{\pi}^{\pi-\theta} \log\left| 2 \sin \frac{y}{2} \right| \,dy \\ = {} & 2\, \operatorname{Cl}_2(\theta) -2\, \operatorname{Cl}_2(\pi-\theta) + 2\, \operatorname{Cl}_2(\pi) \end{align} $$

 * $$\operatorname{Cl}_2(\pi) = 0 \, $$

Therefore,

 * $$\operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta)-2\, \operatorname{Cl}_2(\pi-\theta)\, . \, \Box $$

Derivatives of general-order Clausen functions
Direct differentiation of the Fourier series expansions for the Clausen functions give:

 * $$\frac{d}{d\theta}\operatorname{Cl}_{2m+2}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}=\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}=\operatorname{Cl}_{2m+1}(\theta)$$
 * $$\frac{d}{d\theta}\operatorname{Cl}_{2m+1}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}=-\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m}}=-\operatorname{Cl}_{2m}(\theta)$$
 * $$\frac{d}{d\theta}\operatorname{Sl}_{2m+2}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}= -\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}=-\operatorname{Sl}_{2m+1} (\theta)$$
 * $$\frac{d}{d\theta}\operatorname{Sl}_{2m+1}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}=\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m}}=\operatorname{Sl}_{2m} (\theta)$$

By appealing to the First Fundamental Theorem Of Calculus, we also have:

 * $$\frac{d}{d\theta}\operatorname{Cl}_2(\theta) = \frac{d}{d\theta} \left[ -\int_0^\theta \log \left| 2\sin \frac{x}{2}\right| \,dx \, \right] = - \log \left| 2\sin \frac{\theta}{2}\right| = \operatorname{Cl}_1(\theta) $$

Relation to the inverse tangent integral
The inverse tangent integral is defined on the interval $$0 < z < 1$$ by

 * $$\operatorname{Ti}_2(z)=\int_0^z \frac{\tan^{-1}x}{x}\,dx = \sum_{k=0}^\infty (-1)^k \frac{z^{2k+1}}{(2k+1)^2}$$

It has the following closed form in terms of the Clausen function:

 * $$\operatorname{Ti}_2(\tan \theta)= \theta\log(\tan \theta) + \frac{1}{2} \operatorname{Cl}_2(2\theta) +\frac{1}{2}\operatorname{Cl}_2(\pi-2\theta)$$

Proof of the inverse tangent integral relation
From the integral definition of the inverse tangent integral, we have

 * $$\operatorname{Ti}_2(\tan \theta) = \int_0^{\tan \theta}\frac{\tan^{-1}x}{x}\,dx$$

Performing an integration by parts

 * $$\int_0^{\tan \theta} \frac{\tan^{-1}x}{x}\,dx= \tan^{-1}x\log x \, \Bigg|_0^{\tan \theta} - \int_0^{\tan \theta} \frac{\log x}{1+x^2}\,dx=$$
 * $$\theta \log \tan \theta - \int_0^{\tan \theta}\frac{\log x}{1+x^2}\,dx$$

Apply the substitution $$x=\tan y,\, y=\tan^{-1}x,\, dy=\frac{dx}{1+x^2}\,$$ to obtain

 * $$\theta \log \tan \theta - \int_0^\theta \log(\tan y)\,dy$$

For that last integral, apply the transform :$$y=x/2,\, dy=dx/2\,$$ to get

\begin{align} & \theta \log \tan \theta - \frac 1 2 \int_0^{2\theta}\log\left(\tan \frac x 2 \right)\,dx \\[6pt] = {} & \theta \log \tan \theta - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{\sin (x/2) }{\cos (x/2)}\right)\,dx \\[6pt] = {} & \theta \log \tan \theta - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{2\sin (x/2) }{2\cos (x/2)}\right)\,dx \\[6pt] = {} & \theta \log \tan \theta - \frac{1}{2}\int_0^{2\theta}\log\left(2\sin \frac{x}{2} \right)\,dx+ \frac{1}{2}\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx \\[6pt] = {} & \theta \log \tan \theta +\frac{1}{2}\operatorname{Cl}_2(2\theta)+ \frac{1}{2} \int_0^{2\theta} \log\left(2\cos \frac{x}{2}\right)\,dx. \end{align} $$

Finally, as with the proof of the Duplication formula, the substitution $$x=(\pi-y)\, $$ reduces that last integral to

 * $$\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx= \operatorname{Cl}_2(\pi-2\theta) - \operatorname{Cl}_2(\pi) = \operatorname{Cl}_2(\pi-2\theta)$$

Thus

 * $$\operatorname{Ti}_2(\tan \theta) = \theta \log \tan \theta +\frac{1}{2}\operatorname{Cl}_2(2\theta)+ \frac{1}{2} \operatorname{Cl}_2(\pi-2\theta)\, . \, \Box $$

Relation to the Barnes' G-function
For real $$0 < z < 1$$, the Clausen function of second order can be expressed in terms of the Barnes G-function and (Euler) Gamma function:

 * $$\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(1+z)} \right) +2\pi z \log \left( \frac{\pi}{ \sin \pi z } \right) $$

Or equivalently

 * $$\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(z)} \right) -2\pi \log \Gamma(z)+2\pi z \log \left( \frac{\pi}{ \sin \pi z } \right) $$

See.

Relation to the polylogarithm
The Clausen functions represent the real and imaginary parts of the polylogarithm, on the unit circle:

 * $$\operatorname{Cl}_{2m}(\theta) = \Im (\operatorname{Li}_{2m}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 1$$
 * $$\operatorname{Cl}_{2m+1}(\theta) = \Re (\operatorname{Li}_{2m+1}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 0$$

This is easily seen by appealing to the series definition of the polylogarithm.

 * $$\operatorname{Li}_n(z)=\sum_{k=1}^\infty \frac{z^k}{k^n} \quad \Longrightarrow \operatorname{Li}_n\left(e^{i\theta}\right)=\sum_{k=1}^\infty \frac{\left(e^{i\theta}\right)^k}{k^n}= \sum_{k=1}^\infty \frac{e^{ik\theta}}{k^n}$$

By Euler's theorem,

 * $$e^{i\theta} = \cos \theta +i\sin \theta$$

and by de Moivre's Theorem (De Moivre's formula)

 * $$(\cos \theta +i\sin \theta)^k= \cos k\theta +i\sin k\theta \quad \Rightarrow \operatorname{Li}_n\left(e^{i\theta}\right)=\sum_{k=1}^\infty \frac{\cos k\theta}{k^n}+ i \, \sum_{k=1}^\infty \frac{\sin k\theta}{k^n}$$

Hence

 * $$\operatorname{Li}_{2m}\left(e^{i\theta}\right)=\sum_{k=1}^\infty \frac{\cos k\theta}{k^{2m}}+ i \, \sum_{k=1}^\infty \frac{\sin k\theta}{k^{2m}} = \operatorname{Sl}_{2m}(\theta)+i\operatorname{Cl}_{2m}(\theta)$$
 * $$\operatorname{Li}_{2m+1}\left(e^{i\theta}\right)=\sum_{k=1}^\infty \frac{\cos k\theta}{k^{2m+1}}+ i \, \sum_{k=1}^\infty \frac{\sin k\theta}{k^{2m+1}} = \operatorname{Cl}_{2m+1}(\theta)+i\operatorname{Sl}_{2m+1}(\theta)$$

Relation to the polygamma function
The Clausen functions are intimately connected to the polygamma function. Indeed, it is possible to express Clausen functions as linear combinations of sine functions and polygamma functions. One such relation is shown here, and proven below:

 * $$\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}(2m-1)!} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right]. $$

An immediate corollary is this equivalent formula in terms of the Hurwitz zeta function:

 * $$\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\zeta\left(2m,\tfrac{j}{2p}\right)+(-1)^q \zeta\left(2m,\tfrac{j+p}{2p}\right)\right]. $$

Let $$\,p\,$$ and $$\,q\,$$ be positive integers, such that $$\,q/p\,$$ is a rational number $$\,0 < q/p < 1\, $$, then, by the series definition for the higher order Clausen function (of even index):

 * $$\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{k=1}^\infty \frac{\sin (kq\pi/p)}{k^{2m}} $$

We split this sum into exactly p-parts, so that the first series contains all, and only, those terms congruent to $$\,kp+1,\, $$ the second series contains all terms congruent to $$\,kp+2,\, $$ etc., up to the final p-th part, that contain all terms congruent to $$\,kp+p\, $$

\begin{align} & \operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right) \\ = {} & \sum_{k=0}^\infty \frac{\sin \left[(kp+1)\frac{q\pi}{p}\right]}{(kp+1)^{2m}} + \sum_{k=0}^\infty \frac{\sin \left[(kp+2)\frac{q\pi}{p}\right]}{(kp+2)^{2m}} + \sum_{k=0}^\infty \frac{\sin \left[(kp+3)\frac{q\pi}{p}\right]}{(kp+3)^{2m}} + \cdots \\ & \cdots + \sum_{k=0}^\infty \frac{\sin \left[(kp+p-2)\frac{q\pi}{p}\right]}{(kp+p-2)^{2m}} + \sum_{k=0}^\infty \frac{\sin \left[(kp+p-1)\frac{q\pi}{p}\right]}{(kp+p-1)^{2m}} + \sum_{k=0}^\infty \frac{\sin \left[(kp+p)\frac{q\pi}{p}\right]}{(kp+p)^{2m}} \end{align} $$

We can index these sums to form a double sum:

\begin{align} & \operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^{p} \left\{ \sum_{k=0}^\infty \frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(kp+j)^{2m}} \right\} \\ = {} & \sum_{j=1}^{p} \frac{1}{p^{2m}}\left\{ \sum_{k=0}^\infty \frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(k+(j/p))^{2m}} \right\} \end{align} $$

Applying the addition formula for the sine function, $$\,\sin(x+y)=\sin x\cos y+\cos x\sin y,\, $$ the sine term in the numerator becomes:

 * $$\sin \left[(kp+j)\frac{q\pi}{p}\right]=\sin\left(kq\pi+\frac{qj\pi}{p}\right)=\sin kq\pi \cos \frac{qj\pi}{p}+\cos kq\pi \sin\frac{qj\pi}{p}$$
 * $$\sin m\pi \equiv 0, \quad \, \cos m\pi \equiv (-1)^m \quad \Longleftrightarrow m=0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots $$
 * $$\sin \left[(kp+j)\frac{q\pi}{p}\right]=(-1)^{kq}\sin\frac{qj\pi}{p}$$

Consequently,

 * $$\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^p \frac{1}{p^{2m}} \sin\left(\frac{qj\pi}{p}\right)\, \left\{ \sum_{k=0}^\infty \frac{(-1)^{kq}}{(k+(j/p))^{2m}} \right\} $$

To convert the inner sum in the double sum into a non-alternating sum, split in two in parts in exactly the same way as the earlier sum was split into p-parts:

\begin{align} & \sum_{k=0}^\infty \frac{(-1)^{kq}}{(k+(j/p))^{2m}}=\sum_{k=0}^\infty \frac{(-1)^{(2k)q}}{((2k)+(j/p))^{2m}}+ \sum_{k=0}^\infty \frac{(-1)^{(2k+1)q}}{((2k+1)+(j/p))^{2m}} \\ = {} & \sum_{k=0}^\infty \frac{1}{(2k+(j/p))^{2m}}+ (-1)^q\, \sum_{k=0}^\infty \frac{1}{(2k+1+(j/p))^{2m}} \\ = {} & \frac{1}{2^p}\left[ \sum_{k=0}^\infty \frac{1}{(k+(j/2p))^{2m}}+ (-1)^q\, \sum_{k=0}^\infty \frac{1}{(k+\left(\frac{j+p}{2p}\right))^{2m}} \right] \end{align} $$

For $$\,m \in\mathbb{Z} \ge 1\, $$, the polygamma function has the series representation

 * $$\psi_m(z)=(-1)^{m+1}m! \sum_{k=0}^\infty \frac{1}{(k+z)^{m+1}} $$

So, in terms of the polygamma function, the previous inner sum becomes:

 * $$ \frac{1}{2^{2m}(2m-1)!} \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1} \left(\tfrac{j+p}{2p}\right)\right] $$

Plugging this back into the double sum gives the desired result:

 * $$\operatorname{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}(2m-1)!} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right] $$

Relation to the generalized logsine integral
The generalized logsine integral is defined by:

 * $$\mathcal{L}s_n^{m}(\theta) = -\int_0^\theta x^m \log^{n-m-1} \left| 2\sin\frac{x}{2} \right| \, dx$$

In this generalized notation, the Clausen function can be expressed in the form:

 * $$\operatorname{Cl}_2(\theta) = \mathcal{L}s_2^{0}(\theta) $$

Kummer's relation
Ernst Kummer and Rogers give the relation

 * $$\operatorname{Li}_2(e^{i \theta}) = \zeta(2) - \theta(2\pi-\theta)/4 + i\operatorname{Cl}_2(\theta)$$

valid for $$0\leq \theta \leq 2\pi$$.

Relation to the Lobachevsky function
The Lobachevsky function Λ or Л is essentially the same function with a change of variable:

 * $$\Lambda(\theta) = - \int_0^\theta \log|2 \sin(t)| \,dt = \operatorname{Cl}_2(2\theta)/2$$

though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function

 * $$\int_0^\theta \log| \sec(t)| \,dt = \Lambda(\theta+\pi/2)+\theta\log 2.$$

Relation to Dirichlet L-functions
For rational values of $$\theta/\pi$$ (that is, for $$\theta/\pi=p/q$$ for some integers p and q), the function $$\sin(n\theta)$$ can be understood to represent a periodic orbit of an element in the cyclic group, and thus $$\operatorname{Cl}_s(\theta)$$ can be expressed as a simple sum involving the Hurwitz zeta function. This allows relations between certain Dirichlet L-functions to be easily computed.

Series acceleration
A series acceleration for the Clausen function is given by

 * $$\frac{\operatorname{Cl}_2(\theta)} \theta =

1-\log|\theta| + \sum_{n=1}^\infty \frac{\zeta(2n)}{n(2n+1)} \left(\frac \theta {2\pi}\right)^{2n} $$

which holds for $$|\theta|<2\pi$$. Here, $$\zeta(s)$$ is the Riemann zeta function. A more rapidly convergent form is given by

 * $$\frac{\operatorname{Cl}_2(\theta)}{\theta} =

3-\log\left[|\theta| \left(1-\frac{\theta^2}{4\pi^2}\right)\right] -\frac{2\pi}{\theta} \log \left( \frac{2\pi+\theta}{2\pi-\theta}\right) +\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^{2n}. $$

Convergence is aided by the fact that $$\zeta(n)-1$$ approaches zero rapidly for large values of n. Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series.

Special values
Recall the Barnes G-function, the Catalan's constant K and the Gieseking constant V. Some special values include

 * $$\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=K$$
 * $$\operatorname{Cl}_2\left(\frac{\pi}{3}\right)=V$$
 * $$\operatorname{Cl}_2\left(\frac{\pi}{3}\right)=3\pi \log\left(

\frac{G\left(\frac{2}{3}\right)}{ G\left(\frac{1}{3}\right)} \right)-3\pi \log \Gamma\left(\frac{1}{3}\right)+\pi \log \left(\frac{ 2\pi }{\sqrt{3}}\right)$$
 * $$\operatorname{Cl}_2\left(\frac{2\pi}{3}\right)=2\pi \log\left(

\frac{G\left(\frac{2}{3}\right)}{ G\left(\frac{1}{3}\right)} \right)-2\pi \log \Gamma\left(\frac{1}{3}\right) +\frac{2\pi}{3} \log \left(\frac{ 2\pi }{\sqrt{3}}\right)$$
 * $$\operatorname{Cl}_2\left(\frac{\pi}{4}\right)=

2\pi\log \left( \frac{G\left(\frac{7}{8}\right)}{G\left(\frac{1}{8}\right)} \right) -2\pi \log \Gamma\left(\frac{1}{8}\right)+\frac{\pi}{4}\log \left( \frac{2\pi}{\sqrt{2-\sqrt{2}}} \right)$$
 * $$\operatorname{Cl}_2\left(\frac{3\pi}{4}\right)=

2\pi\log \left( \frac{G\left(\frac{5}{8}\right)}{G\left(\frac{3}{8}\right)} \right) -2\pi \log \Gamma\left(\frac{3}{8}\right)+\frac{3\pi}{4}\log \left( \frac{2\pi}{\sqrt{2+\sqrt{2}}} \right)$$
 * $$\operatorname{Cl}_2\left(\frac{\pi}{6}\right)=

2\pi\log \left( \frac{G\left(\frac{11}{12}\right)}{G\left(\frac{1}{12}\right)} \right) -2\pi \log \Gamma\left(\frac{1}{12}\right)+\frac{\pi}{6}\log \left( \frac{2\pi \sqrt{2} }{\sqrt{3}-1} \right)$$
 * $$\operatorname{Cl}_2\left(\frac{5\pi}{6}\right)=

2\pi\log \left( \frac{G\left(\frac{7}{12}\right)}{G\left(\frac{5}{12}\right)} \right) -2\pi \log \Gamma\left(\frac{5}{12}\right)+\frac{5\pi}{6}\log \left( \frac{2\pi \sqrt{2} }{\sqrt{3}+1} \right)$$

In general, from the Barnes G-function reflection formula,

 * $$ \operatorname{Cl}_2(2\pi z)=2\pi\log\left( \frac{G(1-z)}{G(z)} \right)-2\pi\log\Gamma(z)+2\pi z\log\left(\frac{\pi}{\sin\pi z}\right) $$

Equivalently, using Euler's reflection formula for the gamma function, then,

 * $$ \operatorname{Cl}_2(2\pi z)=2\pi\log\left( \frac{G(1-z)}{G(z)} \right)-2\pi\log\Gamma(z)+2\pi z\log\big(\Gamma(z)\Gamma(1 - z)\big) $$

Generalized special values
Some special values for higher order Clausen functions include

 * $$\operatorname{Cl}_{2m}(0)=\operatorname{Cl}_{2m}(\pi) = \operatorname{Cl}_{2m}(2\pi)=0$$
 * $$\operatorname{Cl}_{2m}\left(\frac{\pi}{2}\right)=\beta(2m)$$
 * $$\operatorname{Cl}_{2m+1}(0)=\operatorname{Cl}_{2m+1}(2\pi)=\zeta(2m+1)$$
 * $$\operatorname{Cl}_{2m+1}(\pi)=-\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^{2m}}\right) \zeta(2m+1)$$
 * $$\operatorname{Cl}_{2m+1}\left(\frac{\pi}{2}\right)=-\frac{1}{2^{2m+1}}\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^{4m+1}}\right)\zeta(2m+1)$$

where $$\beta(x)$$ is the Dirichlet beta function, $$\eta(x)$$ is the Dirichlet eta function (also called the alternating zeta function), and $$\zeta(x)$$ is the Riemann zeta function.

Integrals of the direct function
The following integrals are easily proven from the series representations of the Clausen function:

 * $$\int_0^\theta \operatorname{Cl}_{2m}(x)\,dx=\zeta(2m+1)-\operatorname{Cl}_{2m+1}(\theta)$$
 * $$\int_0^\theta \operatorname{Cl}_{2m+1}(x)\,dx=\operatorname{Cl}_{2m+2}(\theta)$$
 * $$\int_0^\theta \operatorname{Sl}_{2m}(x)\,dx=\operatorname{Sl}_{2m+1}(\theta)$$
 * $$\int_0^\theta \operatorname{Sl}_{2m+1}(x)\,dx=\zeta(2m+2)-\operatorname{Cl}_{2m+2}(\theta)$$

Fourier-analytic methods can be used to find the first moments of the square of the function $$\operatorname{Cl}_2(x)$$ on the interval $$[0,\pi]$$:

 * $$\int_0^\pi \operatorname{Cl}_2^2(x)\,dx=\zeta(4),$$
 * $$\int_0^\pi t\operatorname{Cl}_2^2(x)\,dx=\frac{221}{90720} \pi^{6}-4 \zeta(\overline{5}, 1)-2 \zeta(\overline{4}, 2),$$
 * $$\int_0^\pi t^2\operatorname{Cl}_2^2(x)\,dx=-\frac{2}{3} \pi\left[12 \zeta(\overline{5}, 1)+6 \zeta(\overline{4}, 2)-\frac{23}{10080} \pi^{6}\right].$$

Here $$\zeta$$ denotes the multiple zeta function.

Integral evaluations involving the direct function
A large number of trigonometric and logarithmo-trigonometric integrals can be evaluated in terms of the Clausen function, and various common mathematical constants like $$\, K \,$$ (Catalan's constant), $$\, \log 2 \,$$, and the special cases of the zeta function, $$\, \zeta(2) \,$$ and $$\, \zeta(3) \,$$.

The examples listed below follow directly from the integral representation of the Clausen function, and the proofs require little more than basic trigonometry, integration by parts, and occasional term-by-term integration of the Fourier series definitions of the Clausen functions.

 * $$\int_0^\theta \log(\sin x)\,dx=-\tfrac{1}{2}\operatorname{Cl}_2(2\theta)-\theta\log 2$$
 * $$\int_0^\theta \log(\cos x)\,dx=\tfrac{1}{2}\operatorname{Cl}_2(\pi-2\theta)-\theta\log 2$$
 * $$\int_0^\theta \log(\tan x)\,dx=-\tfrac{1}{2}\operatorname{Cl}_2(2\theta)-\tfrac{1}{2} \operatorname{Cl}_2(\pi-2\theta)$$
 * $$\int_0^\theta \log(1+\cos x)\,dx=2\operatorname{Cl}_2(\pi-\theta)-\theta\log 2$$
 * $$\int_0^\theta \log(1-\cos x)\,dx=-2\operatorname{Cl}_2(\theta)-\theta\log 2$$
 * $$\int_0^\theta \log(1+\sin x)\,dx=2K-2\operatorname{Cl}_2\left(\frac{\pi}{2}+\theta\right) -\theta\log 2$$
 * $$\int_0^\theta \log(1-\sin x)\,dx=-2K+2\operatorname{Cl}_2\left(\frac{\pi}{2}-\theta\right)-\theta\log 2$$

title: Clausen function


text: User talk:213.106.248.77

title: User talk:213.106.248.77


text:
Geometrized unit system

A geometrized unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

 * $$ c = 1 \ $$
 * $$ G = 1 \ $$

The geometrized unit system is not a completely defined system. Some systems are geometrized unit systems in the sense that they set these, in addition to other constants, to unity, for example Stoney units and Planck units.

This system is useful in physics, especially in the special and general theories of relativity. All physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.

Many equations in relativistic physics appear simpler when expressed in geometric units, because all occurrences of G and of c drop out. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes r = 2m. For this reason, many books and papers on relativistic physics use geometric units. An alternative system of geometrized units is often used in particle physics and cosmology, in which 8πG = 1 instead. This introduces an additional factor of 8π into Newton's law of universal gravitation but simplifies the Einstein field equations, the Einstein–Hilbert action, the Friedmann equations and the Newtonian Poisson equation by removing the corresponding factor.

Definition
Geometrized units were defined in the book Gravitation by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler with the speed of light, $$c$$, the gravitational constant, $$G$$, and Boltzmann constant, $$k_b$$ all set to $$1.0$$. Some authors refer to these units as geometrodynamic units.

In geometric units, every time interval is interpreted as the distance travelled by light during that given time interval. That is, one second is interpreted as one light-second, so time has the geometric units of length. This is dimensionally consistent with the notion that, according to the kinematical laws of special relativity, time and distance are on an equal footing.

Energy and momentum are interpreted as components of the four-momentum vector, and mass is the magnitude of this vector, so in geometric units these must all have the dimension of length. We can convert a mass expressed in kilograms to the equivalent mass expressed in metres by multiplying by the conversion factor G/c2. For example, the Sun's mass of $2 kg$ in SI units is equivalent to $1.5 km$. This is half the Schwarzschild radius of a one solar mass black hole. All other conversion factors can be worked out by combining these two.

The small numerical size of the few conversion factors reflects the fact that relativistic effects are only noticeable when large masses or high speeds are considered.

Conversions
Listed below are all conversion factors that are useful to convert between all combinations of the SI base units, and if not possible, between them and their unique elements, because ampere is a dimensionless ratio of two lengths such as [C/s], and candela (1/683 [W/sr]) is a dimensionless ratio of two dimensionless ratios such as ratio of two volumes [kg⋅m2/s3] = [W] and ratio of two areas [m2/m2] = [sr], while mole is only a dimensionless Avogadro number of entities such as atoms or particles:

title: Geometrized unit system