text:
Talk:Variation of the field

(First comments)
The problem about the landscape coats and coat with field "the sea proper" would be better dealt with if there were a separate article on the field. --Daniel C. Boyer 14:31, 18 Dec 2003 (UTC)

How to deal with fretty, masculy &c.? --Daniel C. Boyer 15:30, 19 Dec 2003 (UTC)

bad links
—Tamfang 06:57, 10 March 2007 (UTC)
 * http://scotshistoryonline.co.uk/HSSforum/viewtopic.php?t=166 — requires login
 * http://www.cundinamarca.gov.co/cundinamarca/municipios/frm_iniciomunicipio.asp?codigo=32 — does not show a CoA

Semé
The image shown under the heading Semé seems a bit misleading. It seems a poor example of a "semé" of crowns, and I think it would be better described as "crowns Or in orle within a bordure Gules." I should think a more exemplary specimen would be more appropriate, in order to clarify graphically, rather than muddle, a relatively obscure term for the common WP user, who is surely unfamiliar with how to imagine what a "semé" is when reading a blazon. Many better examples can be found here. Wilhelm meis (talk) 01:35, 11 July 2008 (UTC)

 * This is absolutely false. Items on a bordure are never blazoned as being "in orle" on the bordure, and semy has a different meaning on a bordure than on the field or on another charge.  --Daniel C. Boyer (talk) 14:40, 11 July 2008 (UTC)

 * Is it absolutely false? Am I wrong in thinking that any of the images I indicated on WM Commons would better exemplify the concept of a semy? If what you say is true, perhaps this could be better explained in the article (with proper references, of course). I study heraldry, and I have an admittedly shaky understanding of this particular term. It is for this reason that I think it is important to clarify this term for the sake of WP users who have no clue what a semy is. Wilhelm meis (talk) 22:55, 11 July 2008 (UTC)

 * Confusingly, semy is almost invariably interpreted two different ways, the first on fields and every charge except the bordure, the second on the bordure. When shown on a field or other charge the charges of the semy are shown in a large number, ideally with some of them running off the edges of the shield, to emphasise that they are part of the semy and not a discrete number of charges ordered in some eccentric manner.  When on a bordure the bordure is shown as if charged with a larger number of the charges of the semy than it would be reasonable to blazon, althought the eight of Jesus College is quite low for this phenomenon.  The charges are then interpreted as a bordure semy and not, say, seventeen bezants.  --Daniel C. Boyer (talk) 20:26, 14 July 2008 (UTC)

 * If there were no bordure and the device had the same eight crowns in the same position, it would undoubtedly be described as having crowns in orle - which, incidentally, has eight of the same charge by default. Don't get me wrong, I'm not insisting that the description is wrong; I'm just saying that this shouldn't be the one and only image WP has to represent the term "semy" when there are so many images available on the Commons that more clearly illustrate the concept. If there are two distinct concepts of a "semy", both should be graphically represented and adequately described, and if a semy on a bordure is easily confused with an "orle" or a charge "in orle", this disambiguation also needs to be adequately explained in the article (and pictures can help this, too). I would like to see this section expanded a bit, and the "orle" concept mentioned as well. I'm not trying to be a pain, just trying to help get this article a little closer to ideal. Wilhelm meis (talk) 01:34, 15 July 2008 (UTC)

 * You're far from being a pain, and I'd wholeheartedly agree. Semy should be represented by two pictures, one normal, one for a bordure, and there must be better examples of the latter than Jesus College.  "In orle" really isn't much of an issue beyond, perhaps, that it should be noted that when charges are on a bordure either as a specific number of charges or functioning as semy they are never so described, it's only when they are in the position that would be occupied by a bordure but there's no bordure there that they are.  --Daniel C. Boyer (talk) 14:43, 15 July 2008 (UTC)

 * Nodding in agreement here. There must be a good France Ancient around somewhere?  And for bordures, how about the first Earl of Cornwall (argent a lion gules, a bordure sable bezanty, if memory serves); or Portugal? — And is it worth mentioning (if the article doesn't already) somewhere that in Continental blazon the color-variants of ermine are considered semy of tails? —Tamfang (talk) 08:34, 17 July 2008 (UTC)

good examples

 * bendy: Halten
 * chevronny: Avry

Widely known facts ≠ Original research
The sentence:
 * The most notable coat of arms of an UN member state that uses this pattern is Croatia.

and similar statements have been repeatedly inserted into the article without any references. While this may be true, Wikipedia requires that verifiability be established by citing at least one reliable source. Information that is added to Wikipedia's articles without citing sources is considered original research, which does not belong here, as Wikipedia is not a publisher of original research. Wilhelm Meis (Quatsch!) 23:06, 29 September 2009 (UTC)

 * Sources:
 * In the book: Carter, David E.; Stephens, Suzanna M. W. The Big Book of Logos 5, Collins Design, 2008, ISBN 0061255742 ─ the Croatian squares are represented within the work of Boris Ljubičić, on as many as 18 pages.
 * In the book: Busch, Akiko (Editor) Design for Sports: The Cult of Performance, 1st ed., Princeton Architectural Press, 1998, ISBN 1568981457 ─ the Croatia Football Shirt was portrayed, which design is also based on squares.
 * In the book (which I do not have in my possesion): Stephenson, Keith; Hampshire, Mark. Squares, Checks, and Grids, Communicating With Pattern, RotoVision, 2008, ISBN 978-2-940361-82-3 ─ a total of six pages is dedicated to Croatian designers which often use the Croatian visual code, the chequered shield, as a visual code for businesses and public administration as well. The Coat of arms of Croatia appears also on the Hungarian (two) pages, as an element from the times of the Austro-Hungarian Empire. Here you can see two pages from the before mentioned book.
 * Here are a number of sources on the CoA.  —Preceding unsigned comment added by Bugoslav (talk • contribs) 22:50, 2 October 2009 (UTC)
 * The Croatian visual code: (minimalist version) can be represented with:
 * Only one check (one red square in a variety of fields of light colours, or one white in a variety of fields of dark colours)
 * Two checks (both red) ─ the Roundel of the Croatian Air Force within the Croatian Armed Forces.
 * Four checks (two by two) (upper red, lower blue) ─ this desing (used on the Flag of the Chairman of the Joint Chiefs of Staff of the Armed Forces of Croatia)
 * Maximally (as far I know):
 * The 29 × 29 check' frame of the Flag of the President of the Republic of Croatia (The Presidential flag of Croatia)
 * Historically the number of checks varied, from 1527 when it was a matrix of 8 by 8 checks, with the first check in white (on the heraldic left), and in times the number of checks contained in a particular row was not equal as the number of checks in a particular column (the matrix was uneven).
 * History (of the chequered (chequy) coat of arms of Croatia:
 * The longest uninterrupted usage of the chequered coat-of-arms, from 1527 to this day, in all of the joint venture state projects in which Croatia participated ─ the Croatian coat of arms was always part of the Coat of arms of that union (of kingdoms, historical states, socialist republics, independently, etc.)
 * Croatia is the only United Nations member state in the World that uses the chequy pattern in the greatest part of its coat-of-arms. Czech Republic implores the pattern also but only in the one quarter of its coat-of-arms, which represents Moravia, not Bohemia (the Czech proper). The main symbol of the Czech Republic is the white lion on a red shield. The main symbol of Croatia (the Croatia proper) and Croats is the checkerboard.
 * Conclusion: The sentence can be rephrased, but Croatia is widely known by its chequy coat-of-arms

 * Bugoslav (talk) 21:48, 2 October 2009 (UTC)

when is a fur not a fur

 * The heraldic furs of the ermine family appear to be semé of the "ermine dots," but they are not counted as such; but fields semy of ermine spots are, as in the coat of Wrexham County Borough Council.

So, ah, ermine is not semy except when it is semy? Perhaps that could be rephrased. In English, a field like Wrexham's would more usually be blazoned vert ermined argent, but in French the ermine mutants are semy, de sinople semé de mouchetures d'hermeline d'argent (possibly misspelled, my Rietstap is in a box). —Tamfang (talk) 04:06, 13 June 2010 (UTC)

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Dubious
I've labelled this statement "The arms of Eyfelsberg zum Wehr provide a perhaps unique example of barry of four different tinctures that do not repeat" as dubious; I intended to link "Eyfelsberg zum Wehr" but  but couldn't find any trace of it outside of this page. Also, the reference given is a bit vague; if it refers to this book it only goes up to p500. The statement and ref was added with this edit (10 years ago!) and the editor then is unavailable for comment now; can anyone shed any light on this? Swanny18 (talk) 23:52, 3 June 2018 (UTC)
 * Turns out it was misspelled. 'Weyr', not 'Wehr'. Number 59 on this page. --Morlark (talk) 07:20, 15 January 2019 (UTC)

Confused newbie
I’ve read this sentence under "Patterning with ordinaries and subordinaries; Barry, paly, bendy" too many times:

A field having the appearance of a number of narrow piles throughout issuing from the dexter of sinister flanks is barry pily.

It might make more sense if "of" was changed to "or", but I don’t know enough about this subject to make that change.

Also, what is the point of the italics at the beginning of the sentence? For that matter, would the sentence still be correct if it read:

"A field of a number of narrow piles throughout issuing from the dexter or sinister flanks is barry pily."

If so, it would be a little clearer to a newbie like me. Pollifax (talk) 23:46, 31 August 2018 (UTC)

Under "Blazoning of French adjectives" this sentence: "Variations of the field of present a particular problem concerning consistent spelling of adjectival endings in English blazons." Is the "of" before "present" superfluous?

Under "Blazoning of French adjectives": "spell all French adjectives in the feminine singular form, for example: a chief undée and a saltire undée, even though the French nouns chef and sautoir are in fact masculine."

I looked this one up, and it looks like the noun "chief" changed into the noon "chef" over the years, but is masculine in both forms. So is there a misspelling here?

Under "Patterning with ordinaries and subordinaries; Barry, paly, bendy": "...and there are even more exceptional examples of barry of a single tincture, as in the arms of Kempten on the Zurich roll." I looked this up, and this is their description (that I don’t understand): "Per fess gules and azure, a chief barry wavy argent." The photo is on this page, as well as others: http://www.vikinganswerlady.com/ZurichRoll/index.html. There are three tinctures, as near as I can tell, and certainly not just one. Or maybe I just plain don’t understand.

Under "Patterning with ordinaries and subordinaries; Gyronny": "...arms of the ancient Scottish family of Campbell: Gyronny of eight or and sable...". In this sentence "or" is not capitalized as it is in other parts of this article. I’ve seen that there is some controversy about capitalization of this word, but is there an effort to keep this article consistent with itself?

Edited—Pollifax (talk) 01:13, 1 September 2018 (UTC)

title: Talk:Variation of the field


text:
Complete Fermi–Dirac integral

In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j  is defined by

 * $$F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1}\,dt, \qquad (j > -1)$$

This equals
 * $$-\operatorname{Li}_{j+1}(-e^x),$$

where $$\operatorname{Li}_{s}(z)$$ is the polylogarithm.

Its derivative is
 * $$\frac{dF_{j}(x)}{dx} = F_{j-1}(x), $$

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for $$F_j$$ appears in the literature, for instance some authors omit the factor $$1/\Gamma(j+1)$$. The definition used here matches that in the NIST DLMF.

Special values
The closed form of the function exists for j = 0:

 * $$F_0(x) = \ln(1+\exp(x)).$$

For x = 0, the result reduces to

$$ F_j(0) = \eta(j+1), $$

where $$\eta$$ is the Dirichlet eta function.

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

title: Complete Fermi–Dirac integral


text:
Incomplete Fermi–Dirac integral

In mathematics, the incomplete Fermi-Dirac integral, named after Enrico Fermi and Paul Dirac, for an index $$j$$ and parameter $$b$$ is given by

 * $$\operatorname{F}_j(x,b) \overset{\mathrm{def}}{=} \frac{1}{\Gamma(j+1)} \int_b^\infty\! \frac{t^j}{e^{t-x} + 1}\;\mathrm{d}t$$

Its derivative is
 * $$\frac{\mathrm{d}}{\mathrm{d}x}\operatorname{F}_j(x,b) = \operatorname{F}_{j-1}(x,b) $$

and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices $$j$$.

This is an alternate definition of the incomplete polylogarithm, since:
 * $$\operatorname{F}_j(x,b) = \frac{1}{\Gamma(j+1)} \int_b^\infty\! \frac{t^j}{e^{t-x} + 1}\;\mathrm{d}t = \frac{1}{\Gamma(j+1)} \int_b^\infty\! \frac{t^j}{\displaystyle \frac{e^t}{e^x} + 1}\;\mathrm{d}t = -\frac{1}{\Gamma(j+1)} \int_b^\infty\! \frac{t^j}{\displaystyle \frac{e^t}{-e^x} - 1}\;\mathrm{d}t = -\operatorname{Li}_{j+1}(b,-e^x) $$

Which can be used to prove the identity:

\operatorname{F}_j(x,b) = -\sum_{n=1}^\infty \frac{(-1)^n}{n^{j+1}}\frac{\Gamma(j+1,nb)}{\Gamma(j+1)}e^{nx} $$ where $$\Gamma(s)$$ is the gamma function and $$\Gamma(s,y)$$ is the upper incomplete gamma function. Since $$\Gamma(s,0)=\Gamma(s)$$, it follows that:
 * $$\operatorname{F}_j(x,0) = \operatorname{F}_j(x)$$

where $$\operatorname{F}_j(x)$$ is the complete Fermi-Dirac integral.

Special values
The closed form of the function exists for $$j=0$$:

 * $$\operatorname{F}_0(x,b) = \ln\!\big(1+e^{x-b}\big) - (b - x) $$

title: Incomplete Fermi–Dirac integral


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Desvres

Desvres (Dèfes; Deveren) is a commune in the Pas-de-Calais department in northern France. It is a market town, known for its pottery. In 2018 its population was 4,930 inhabitants. It is the seat of the canton of Desvres.

title: Desvres


text:
Talk:Evolution to 3G

This is a classic case of a well-written article by a well-informed writer who assumes that any reader is just as well-informed. Most people (like me) coming across this article are totally baffled: what subject are we talking about, anyway? You have to work down quite a ways to realize that it involves telecommunications rather than modern art or aerodynamics or something else.

It needs an introductory sentence or two, putting 3g in a simple context so that readers can decide whether they want/need to learn the details. Such introductions are hard to write and they require a good knowledge of the topic - I hope the author will take pity on the less-informed and return to do this.

title: Talk:Evolution to 3G


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Divion

Divion is a commune and in the Pas-de-Calais department in the Hauts-de-France region of France.

Geography
An ex-coalmining town some 5 mi southwest of Béthune and 34 mi southwest of Lille, at the junction of the D341 and the N41 roads, now bypassed by the D301. Since the mid-1960s, work in farming and light industry have replaced coal mining.

Places of interest

 * The church of St.Martin, rebuilt, as was much of the town, after World War I.
 * The Commonwealth War Graves Commission cemetery.
 * A fifteenth century sandstone cross.
 * Remains of an old castle.

title: Divion


text:
Virulence

Virulence is a pathogen's or microorganism's ability to cause damage to a host.

In most, especially in animal systems, virulence refers to the degree of damage caused by a microbe to its host. The pathogenicity of an organism—its ability to cause disease—is determined by its virulence factors. In the specific context of gene for gene systems, often in plants, virulence refers to a pathogen's ability to infect a resistant host.

The noun virulence (Latin noun virulentia) derives from the adjective virulent, meaning disease severity. The word virulent derives from the Latin word virulentus, meaning "a poisoned wound" or "full of poison." The term virulence does not only apply to viruses.

From an ecological standpoint, virulence is the loss of fitness induced by a parasite upon its host. Virulence can be understood in terms of proximate causes—those specific traits of the pathogen that help make the host ill—and ultimate causes—the evolutionary pressures that lead to virulent traits occurring in a pathogen strain.

Virulent bacteria
The ability of bacteria to cause disease is described in terms of the number of infecting bacteria, the route of entry into the body, the effects of host defense mechanisms, and intrinsic characteristics of the bacteria called virulence factors. Many virulence factors are so-called effector proteins that are injected into the host cells by specialized secretion apparati, such as the type three secretion system. Host-mediated pathogenesis is often important because the host can respond aggressively to infection with the result that host defense mechanisms do damage to host tissues while the infection is being countered (e.g., cytokine storm).

The virulence factors of bacteria are typically proteins or other molecules that are synthesized by enzymes. These proteins are coded for by genes in chromosomal DNA, bacteriophage DNA or plasmids. Certain bacteria employ mobile genetic elements and horizontal gene transfer. Therefore, strategies to combat certain bacterial infections by targeting these specific virulence factors and mobile genetic elements have been proposed. Bacteria use quorum sensing to synchronise release of the molecules. These are all proximate causes of morbidity in the host.

Methods by which bacteria cause disease

 * Adhesion: Many bacteria must first bind to host cell surfaces. Many bacterial and host molecules that are involved in the adhesion of bacteria to host cells have been identified. Often, the host cell surface receptors for bacteria are essential proteins for other functions. Due to the presence of mucus lining and of anti-microbial substances around some host cells, it is difficult for certain pathogens to establish direct contact-adhesion.
 * Colonization: Some virulent bacteria produce special proteins that allow them to colonize parts of the host body. Helicobacter pylori is able to survive in the acidic environment of the human stomach by producing the enzyme urease. Colonization of the stomach lining by this bacterium can lead to gastric ulcers and cancer. The virulence of various strains of Helicobacter pylori tends to correlate with the level of production of urease.
 * Invasion: Some virulent bacteria produce proteins that either disrupt host cell membranes or stimulate their own endocytosis or macropinocytosis into host cells. These virulence factors allow the bacteria to enter host cells and facilitate entry into the body across epithelial tissue layers at the body surface.
 * Immune response inhibitors: Many bacteria produce virulence factors that inhibit the host's immune system defenses. For example, a common bacterial strategy is to produce proteins that bind host antibodies. The polysaccharide capsule of Streptococcus pneumoniae inhibits phagocytosis of the bacterium by host immune cells.
 * Toxins: Many virulence factors are proteins made by bacteria that poison host cells and cause tissue damage. For example, there are many food poisoning toxins produced by bacteria that can contaminate human foods. Some of these can remain in "spoiled" food even after cooking and cause illness when the contaminated food is consumed. Other bacterial toxins are chemically altered and inactivated by the heat of cooking.

Virulent viruses
Virus virulence factors allow it to replicate, modify host defenses, and spread within the host, and they are toxic to the host.

They determine whether infection occurs and how severe the resulting viral disease symptoms are. Viruses often require receptor proteins on host cells to which they specifically bind. Typically, these host cell proteins are endocytosed and the bound virus then enters the host cell. Virulent viruses such as HIV, which causes AIDS, have mechanisms for evading host defenses. HIV infects T-helper cells, which leads to a reduction of the adaptive immune response of the host and eventually leads to an immunocompromised state. Death results from opportunistic infections secondary to disruption of the immune system caused by AIDS. Some viral virulence factors confer ability to replicate during the defensive inflammation responses of the host such as during virus-induced fever. Many viruses can exist inside a host for long periods during which little damage is done. Extremely virulent strains can eventually evolve by mutation and natural selection within the virus population inside a host. The term "neurovirulent" is used for viruses such as rabies and herpes simplex which can invade the nervous system and cause disease there.

Extensively studied model organisms of virulent viruses include virus T4 and other T-even bacteriophages which infect Escherichia coli and a number of related bacteria.

The lytic life cycle of virulent bacteriophages is contrasted by the temperate lifecycle of temperate bacteriophages.

title: Virulence


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Carnac Island

Carnac Island (Noongar: Ngoorloormayup) is a 19 ha, A Class, island nature reserve about 10 km south-west of Fremantle and 3.5 km north of Garden Island in Western Australia.

History
Carnac Island is aeolianite limestone remnant of Pleistocene dunes.

In 1803, French explorer Louis de Freycinet, captain of the Casuarina, named the island Île Pelée (Bald Island). It was also known as Île Lévilian and later Île Berthelot. In 1827, James Stirling changed its name to Pulo Carnac Island in honour of John Rivett Carnac, Second Lieutenant on his ship HMS Success (1825). "Pulo" is Malay for "Island"; it is not known why Stirling included the term, and it was soon dropped.

From, the island served as a whaling station. The whalers transported Perth's first church to the island to be used as a storehouse. It was abandoned within a few years.

From October to November 1838, the island was declared by the Swan River Colony colonial government to be a prison for indigenous Australians. The prison consisted of two guards, an overseer named RM Lyon, and three prisoners named Yagan, Danmera, and Ningina. The solitary conditions resulted in the soldiers assisting the prisoners' escape in a stolen government stores boat.

In 1884, the colonial government gazetted the island as a quarantine station for Fremantle, but it appears never to have been used for that purpose.

In 1916, the Australian Federal Government assumed control over the island for defence purposes, and the island was transferred back to the State of Western Australia in 1961.

Fauna
The island is home to Australian sea lions, bottlenose dolphins and a large range of marine bird life. New Zealand fur seals are frequent visitors. Rabbits inhabited the island in abundance from 1827 to 1897, but were eradicated in 1969. It is particularly noted for the abundance and size of snakes, particularly tiger snakes, which live there. The island is densely populated with up to three tiger snakes in every 25 m2. For this reason, very few people venture away from the beach.

There is no permanent fresh water, providing a challenge for the animals that live there. The origins of the tiger snake colony has attracted significant debate. One example of a theory is that in 1929, a man named Lindsay "Rocky" Vane dumped his tiger snake collection on the island, after snake exhibitions were banned in Western Australia, after Vane's wife and his assistant died from snake bite. and research into how that species has adapted to a harsh island habitat. King skinks also inhabit the island, and there is evidence of confrontation between king skinks and tiger snakes.

In November 2006, naturalist David Attenborough visited the island with a BBC film crew to record a reptile documentary, in which Attenborough provided commentary on the blindness of many of the island's tiger snakes. This is caused by birds defending their chicks by pecking at the snakes' eyes. These blind snakes survive and thrive, relying upon scent and eating immobile prey such as seabird chicks. Carnac Island is the only place where this has been observed. Male tiger snakes largely out-number female tiger snakes on the island, which is another curiosity of the island's tiger snake colony.

Carnac Island is classified as an Important Bird Area because it supports a large colony of the vulnerable fairy tern, as well as small numbers of other nesting seabirds.

title: Carnac Island


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Polygamma function

In mathematics, the polygamma function of order $m$ is a meromorphic function on the complex numbers $$\mathbb{C}$$ defined as the $ψ$th derivative of the logarithm of the gamma function:

 * $$\psi^{(m)}(z) := \frac{\mathrm{d}^m}{\mathrm{d}z^m} \psi(z) = \frac{\mathrm{d}^{m+1}}{\mathrm{d}z^{m+1}} \ln\Gamma(z).$$

Thus

 * $$\psi^{(0)}(z) = \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}$$

holds where $ψ^{(1)}$ is the digamma function and $ψ^{(2)}$ is the gamma function. They are holomorphic on $$\mathbb{C} \backslash\mathbb{Z}_{\le0}$$. At all the nonpositive integers these polygamma functions have a pole of order $ψ^{(3)}$. The function $(m + 1)$ is sometimes called the trigamma function.

Integral representation
When $ψ(z)$ and $Γ(z)$, the polygamma function equals

 * $$\begin{align}

\psi^{(m)}(z) &= (-1)^{m+1}\int_0^\infty \frac{t^m e^{-zt}}{1-e^{-t}}\,\mathrm{d}t \\ &= -\int_0^1 \frac{t^{z-1}}{1-t}(\ln t)^m\,\mathrm{d}t\\ &= (-1)^{m+1}m!\zeta(m+1,z) \end{align}$$

where $$\zeta(s,q)$$ is the Hurwitz zeta function.

This expresses the polygamma function as the Laplace transform of $m + 1$. It follows from Bernstein's theorem on monotone functions that, for $ψ^{(1)}(z)$ and $ln Γ(z)$ real and non-negative, $ψ^{(0)}(z)$ is a completely monotone function.

Setting $ψ^{(1)}(z)$ in the above formula does not give an integral representation of the digamma function. The digamma function has an integral representation, due to Gauss, which is similar to the $ψ^{(2)}(z)$ case above but which has an extra term $ψ^{(3)}(z)$.

Recurrence relation
It satisfies the recurrence relation
 * $$\psi^{(m)}(z+1)= \psi^{(m)}(z) + \frac{(-1)^m\,m!}{z^{m+1}}$$

which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:

 * $$\frac{\psi^{(m)}(n)}{(-1)^{m+1}\,m!} = \zeta(1+m) - \sum_{k=1}^{n-1} \frac{1}{k^{m+1}} = \sum_{k=n}^\infty \frac{1}{k^{m+1}} \qquad m \ge 1$$

and

 * $$\psi^{(0)}(n) = -\gamma\ + \sum_{k=1}^{n-1}\frac{1}{k}$$

for all $$n \in \mathbb{N}$$, where $$\gamma$$ is the Euler–Mascheroni constant. Like the log-gamma function, the polygamma functions can be generalized from the domain $ψ^{(4)}(z)$ uniquely to positive real numbers only due to their recurrence relation and one given function-value, say $m > 0$, except in the case $Re z > 0$ where the additional condition of strict monotonicity on $$\mathbb{R}^{+}$$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on $$\mathbb{R}^{+}$$ is demanded additionally. The case $(−1)^{m+1} t^{m}⁄1 − e^{−t}$ must be treated differently because $m > 0$ is not normalizable at infinity (the sum of the reciprocals doesn't converge).

Reflection relation

 * $$(-1)^m \psi^{(m)} (1-z) - \psi^{(m)} (z) = \pi \frac{\mathrm{d}^m}{\mathrm{d} z^m} \cot{\pi z} = \pi^{m+1} \frac{P_m(\cos{\pi z})}{\sin^{m+1}(\pi z)}$$

where $x$ is alternately an odd or even polynomial of degree $(−1)^{m+1} ψ^{(m)}(x)$ with integer coefficients and leading coefficient $m = 0$. They obey the recursion equation

 * $$\begin{align} P_0(x) &= x \\ P_{m+1}(x) &= - \left( (m+1)xP_m(x)+\left(1-x^2\right)P'_m(x)\right).\end{align}$$

Multiplication theorem
The multiplication theorem gives

 * $$k^{m+1} \psi^{(m)}(kz) = \sum_{n=0}^{k-1} \psi^{(m)}\left(z+\frac{n}{k}\right)\qquad m \ge 1$$

and

 * $$k \psi^{(0)}(kz) = k\ln{k} + \sum_{n=0}^{k-1}

\psi^{(0)}\left(z+\frac{n}{k}\right)$$

for the digamma function.

Series representation
The polygamma function has the series representation

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

which holds for integer values of $m = 0$ and any complex $z$ not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as

 * $$\psi^{(m)}(z) = (-1)^{m+1}\, m!\, \zeta (m+1,z).$$

This relation can for example be used to compute the special values

\psi^{(2n-1)}\left(\frac14\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta(2n)\right); $$

\psi^{(2n-1)}\left(\frac34\right) = \frac{4^{2n-1}}{2n}\left(\pi^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta(2n)\right); $$

\psi^{(2n)}\left(\frac14\right) = -2^{2n-1}\left(\pi^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right); $$

\psi^{(2n)}\left(\frac34\right) = 2^{2n-1}\left(\pi^{2n+1}|E_{2n}|-2(2n)!(2^{2n+1}-1)\zeta(2n+1)\right). $$

Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

One more series may be permitted for the polygamma functions. As given by Schlömilch,

 * $$\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}}.$$

This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:

 * $$\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^\frac{z}{n}.$$

Now, the natural logarithm of the gamma function is easily representable:

 * $$\ln \Gamma(z) = -\gamma z - \ln(z) + \sum_{k=1}^\infty \left( \frac{z}{k} - \ln\left(1 + \frac{z}{k}\right) \right).$$

Finally, we arrive at a summation representation for the polygamma function:

 * $$\psi^{(n)}(z) = \frac{\mathrm{d}^{n+1}}{\mathrm{d}z^{n+1}}\ln \Gamma(z) = -\gamma \delta_{n0} - \frac{(-1)^n n!}{z^{n+1}} + \sum_{k=1}^{\infty} \left(\frac{1}{k} \delta_{n0} - \frac{(-1)^n n!}{(k+z)^{n+1}}\right)$$

Where $e^{−t}⁄t$ is the Kronecker delta.

Also the Lerch transcendent
 * $$\Phi(-1, m+1, z) = \sum_{k=0}^\infty \frac{(-1)^k}{(z+k)^{m+1}}$$

can be denoted in terms of polygamma function

 * $$\Phi(-1, m+1, z)=\frac1{(-2)^{m+1}m!}\left(\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right)$$

Taylor series
The Taylor series at $$\mathbb{N}$$ is

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

and

 * $$\psi^{(0)}(z+1)= -\gamma +\sum_{k=1}^\infty (-1)^{k+1}\zeta (k+1) z^k$$

which converges for $ψ^{(m)}(1)$. Here, $ζ$ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

Asymptotic expansion
These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:

 * $$ \psi^{(m)}(z) \sim (-1)^{m+1}\sum_{k=0}^{\infty}\frac{(k+m-1)!}{k!}\frac{B_k}{z^{k+m}} \qquad m \ge 1$$

and
 * $$ \psi^{(0)}(z) \sim \ln(z) - \sum_{k=1}^\infty \frac{B_k}{k z^k}$$

where we have chosen $m = 0$, i.e. the Bernoulli numbers of the second kind.

Inequalities
The hyperbolic cotangent satisfies the inequality
 * $$\frac{t}{2}\operatorname{coth}\frac{t}{2} \ge 1,$$

and this implies that the function
 * $$\frac{t^m}{1 - e^{-t}} - \left(t^{m-1} + \frac{t^m}{2}\right)$$

is non-negative for all $m = 0$ and $ψ^{(0)}$. It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that
 * $$(-1)^{m+1}\psi^{(m)}(x) - \left(\frac{(m-1)!}{x^m} + \frac{m!}{2x^{m+1}}\right)$$

is completely monotone. The convexity inequality $P_{m}$ implies that
 * $$\left(t^{m-1} + t^m\right) - \frac{t^m}{1 - e^{-t}}$$

is non-negative for all $|m − 1|$ and $(−1)^{m}⌈2^{m − 1}⌉$, so a similar Laplace transformation argument yields the complete monotonicity of
 * $$\left(\frac{(m-1)!}{x^m} + \frac{m!}{x^{m+1}}\right) - (-1)^{m+1}\psi^{(m)}(x).$$

Therefore, for all $m > 0$ and $δ_{n0}$,
 * $$\frac{(m-1)!}{x^m} + \frac{m!}{2x^{m+1}} \le (-1)^{m+1}\psi^{(m)}(x) \le \frac{(m-1)!}{x^m} + \frac{m!}{x^{m+1}}.$$

Since both bounds are strictly positive for $$x>0$$, we have: This can be seen in the first plot above.
 * $$\ln\Gamma(x)$$ is strictly convex.
 * For $$m=0$$, the digamma function, $$\psi(x)=\psi^{(0)}(x)$$, is strictly monotonic increasing and strictly concave.
 * For $$m$$ odd, the polygamma functions, $$\psi^{(1)},\psi^{(3)},\psi^{(5)},\ldots$$, are strictly positive, strictly monotonic decreasing and strictly convex.
 * For $$m$$ even the polygamma functions, $$\psi^{(2)},\psi^{(4)},\psi^{(6)},\ldots$$, are strictly negative, strictly monotonic increasing and strictly concave.

Trigamma bounds and asymptote
For the case of the trigamma function ($$m=1$$) the final inequality formula above for $$x>0$$, can be rewritten as:

\frac{x+\frac12}{x^2} \le \psi^{(1)}(x)\le \frac{x+1}{x^2} $$ so that for $$x\gg1$$: $$\psi^{(1)}(x)\approx\frac1x$$.

title: Polygamma function


text:
Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

 * $$\psi(z) = \frac{\mathrm{d}}{\mathrm{d}z}\ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}.$$

It is the first of the polygamma functions. This function is strictly increasing and strictly concave on $$(0,\infty)$$, and it asymptotically behaves as

 * $$\psi(z) \sim \ln{z} - \frac{1}{2z},$$

for complex numbers with large modulus ($$|z|\rightarrow\infty$$) in the sector $$|\arg z|<\pi-\varepsilon$$ with some infinitesimally small positive constant $$\varepsilon$$.

The digamma function is often denoted as $$\psi_0(x), \psi^{(0)}(x) $$ or $Ϝ$ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).

Relation to harmonic numbers
The gamma function obeys the equation

 * $$\Gamma(z+1)=z\Gamma(z). \, $$

Taking the logarithm on both sides gives:

 * $$\ln(\Gamma(z+1))=\ln(z)+\ln(\Gamma(z)), $$

Differentiating both sides with respect to $z$ gives:

 * $$\psi(z+1)=\psi(z)+\frac{1}{z}$$

Since the harmonic numbers are defined for positive integers $n$ as

 * $$H_n=\sum_{k=1}^n \frac 1 k, $$

the digamma function is related to them by

 * $$\psi(n)=H_{n-1}-\gamma,$$

where $H_{0} = 0,$ and $γ$ is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values

 * $$ \psi \left(n+\tfrac12\right)=-\gamma-2\ln 2 +\sum_{k=1}^n \frac 2 {2k-1}.$$

Integral representations
If the real part of $z$ is positive then the digamma function has the following integral representation due to Gauss:
 * $$\psi(z) = \int_0^\infty \left(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}}\right)\,dt.$$

Combining this expression with an integral identity for the Euler–Mascheroni constant $$\gamma$$ gives:
 * $$\psi(z + 1) = -\gamma + \int_0^1 \left(\frac{1-t^z}{1-t}\right)\,dt.$$

The integral is Euler's harmonic number $$H_z$$, so the previous formula may also be written
 * $$\psi(z + 1) = \psi(1) + H_z.$$

A consequence is the following generalization of the recurrence relation:
 * $$\psi(w + 1) - \psi(z + 1) = H_w - H_z.$$

An integral representation due to Dirichlet is:
 * $$\psi(z) = \int_0^\infty \left(e^{-t} - \frac{1}{(1 + t)^z}\right)\,\frac{dt}{t}.$$

Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of $$\psi$$.
 * $$\psi(z) = \log z - \frac{1}{2z} - \int_0^\infty \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{e^t - 1}\right)e^{-tz}\,dt.$$

This formula is also a consequence of Binet's first integral for the gamma function. The integral may be recognized as a Laplace transform.

Binet's second integral for the gamma function gives a different formula for $$\psi$$ which also gives the first few terms of the asymptotic expansion:
 * $$\psi(z) = \log z - \frac{1}{2z} - 2\int_0^\infty \frac{t\,dt}{(t^2 + z^2)(e^{2\pi t} - 1)}.$$

From the definition of $$\psi$$ and the integral representation of the gamma function, one obtains
 * $$\psi(z) = \frac{1}{\Gamma(z)} \int_0^\infty t^{z-1} \ln (t) e^{-t}\,dt,$$

with $$\Re z > 0$$.

Infinite product representation
The function $$\psi(z)/\Gamma(z)$$ is an entire function, and it can be represented by the infinite product

\frac{\psi(z)}{\Gamma(z)}=-e^{2\gamma z}\prod_{k=0}^\infty\left(1-\frac{z}{x_k} \right)e^{\frac{z}{x_k}}. $$

Here $$x_k$$ is the kth zero of $$\psi$$ (see below), and $$\gamma$$ is the Euler–Mascheroni constant.

Note: This is also equal to $$-\frac{d}{dz}\frac{1}{\Gamma(z)}$$ due to the definition of the digamma function: $$\frac{\Gamma'(z)}{\Gamma(z)}=\psi(z)$$.

Series formula
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16):
 * $$\begin{align}

\psi(z + 1) &= -\gamma + \sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n + z}\right), \qquad z \neq -1, -2, -3, \ldots, \\ &= -\gamma + \sum_{n=1}^\infty \left(\frac{z}{n(n + z)}\right), \qquad z \neq -1, -2, -3, \ldots. \end{align}$$ Equivalently,
 * $$\begin{align}

\psi(z) &= -\gamma + \sum_{n=0}^\infty \left(\frac{1}{n + 1} - \frac{1}{n + z}\right), \qquad z \neq 0, -1, -2, \ldots, \\ &= -\gamma + \sum_{n=0}^\infty \frac{z-1}{(n + 1)(n + z)}, \qquad z \neq 0, -1, -2, \ldots. \end{align}$$

Evaluation of sums of rational functions
The above identity can be used to evaluate sums of the form
 * $$\sum_{n=0}^\infty u_n=\sum_{n=0}^\infty \frac{p(n)}{q(n)},$$

where $p(n)$ and $q(n)$ are polynomials of $n$.

Performing partial fraction on $u_{n}$ in the complex field, in the case when all roots of $q(n)$ are simple roots,

 * $$u_n=\frac{p(n)}{q(n)}=\sum_{k=1}^m \frac{a_k}{n+b_k}.$$

For the series to converge,

 * $$\lim_{n\to\infty} nu_n=0,$$

otherwise the series will be greater than the harmonic series and thus diverge. Hence

 * $$\sum_{k=1}^m a_k=0,$$

and

 * $$\begin{align}

\sum_{n=0}^\infty u_n &= \sum_{n=0}^\infty\sum_{k=1}^m\frac{a_k}{n+b_k} \\ &=\sum_{n=0}^\infty\sum_{k=1}^m a_k\left(\frac{1}{n+b_k}-\frac{1}{n+1}\right) \\ &=\sum_{k=1}^m\left(a_k\sum_{n=0}^\infty\left(\frac{1}{n+b_k}-\frac{1}{n+1}\right)\right)\\ &=-\sum_{k=1}^m a_k\big(\psi(b_k)+\gamma\big) \\ &=-\sum_{k=1}^m a_k\psi(b_k). \end{align}$$

With the series expansion of higher rank polygamma function a generalized formula can be given as

 * $$\sum_{n=0}^\infty u_n=\sum_{n=0}^\infty\sum_{k=1}^m \frac{a_k}{(n+b_k)^{r_k}}=\sum_{k=1}^m \frac{(-1)^{r_k}}{(r_k-1)!}a_k\psi^{(r_k-1)}(b_k),$$

provided the series on the left converges.

Taylor series
The digamma has a rational zeta series, given by the Taylor series at $z = 1$. This is

 * $$\psi(z+1)= -\gamma -\sum_{k=1}^\infty (-1)^k\,\zeta (k+1) \, z^k,$$

which converges for $|z| < 1$. Here, $ζ(n)$ is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function.

Newton series
The Newton series for the digamma, sometimes referred to as Stern series, derived by Moritz Abraham Stern in 1847, reads

 * $$\begin{align}

\psi(s) &= -\gamma + (s-1) - \frac{(s-1)(s-2)}{2\cdot2!} + \frac{(s-1)(s-2)(s-3)}{3\cdot3!}\cdots,\quad\Re(s)> 0, \\ &= -\gamma - \sum_{k=1}^\infty \frac{(-1)^k}{k} \binom{s-1}{k}\cdots,\quad\Re(s)> 0. \end{align}$$

where $( s k )$ is the binomial coefficient. It may also be generalized to
 * $$\psi(s+1) = -\gamma - \frac{1}{m} \sum_{k=1}^{m-1}\frac{m-k}{s+k} - \frac{1}{m}\sum_{k=1}^\infty\frac{(-1)^k}{k}\left\{\binom{s+m}{k+1}-\binom{s}{k+1}\right\},\qquad \Re(s)>-1,$$

where $m = 2, 3, 4, ...$

Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind
There exist various series for the digamma containing rational coefficients only for the rational arguments. In particular, the series with Gregory's coefficients $G_{n}$ is

\psi(v) =\ln v- \sum_{n=1}^\infty\frac{\big| G_{n}\big|(n-1)!}{(v)_{n}},\qquad \Re (v) >0, $$

\psi(v) =2\ln\Gamma(v) - 2v\ln v + 2v +2\ln v -\ln2\pi - 2\sum_{n=1}^\infty\frac{\big|G_{n}(2)\big|}{(v)_{n}}\,(n-1)! ,\qquad \Re (v) >0, $$

\psi(v) =3\ln\Gamma(v) - 6\zeta'(-1,v) + 3v^2\ln{v} - \frac32 v^2 - 6v\ln(v)+  3 v+3\ln{v} - \frac32\ln2\pi + \frac12 - 3\sum_{n=1}^\infty\frac{\big| G_{n}(3) \big|}{(v)_{n}}\,(n-1)! ,\qquad \Re (v) >0, $$ where $(v)_{n}$ is the rising factorial $(v)_{n} = v(v+1)(v+2) ... (v+n-1)$, $G_{n}(k)$ are the Gregory coefficients of higher order with $G_{n}(1) = G_{n}$, $Γ$ is the gamma function and $ζ$ is the Hurwitz zeta function. Similar series with the Cauchy numbers of the second kind $π^{−1}$ reads

\psi(v)=\ln(v-1) + \sum_{n=1}^\infty\frac{C_{n}(n-1)!}{(v)_{n}},\qquad \Re(v) >1, $$ A series with the Bernoulli polynomials of the second kind has the following form

\psi(v)=\ln(v+a) + \sum_{n=1}^\infty\frac{(-1)^n\psi_{n}(a)\,(n-1)!}{(v)_{n}},\qquad \Re(v)>-a, $$ where $C_{n}$ are the Bernoulli polynomials of the second kind defined by the generating equation

\frac{z(1+z)^a}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(a) \,,\qquad |z|<1\,, $$ It may be generalized to

\psi(v)= \frac{1}{r}\sum_{l=0}^{r-1}\ln(v+a+l) + \frac{1}{r}\sum_{n=1}^\infty\frac{(-1)^n N_{n,r}(a)(n-1)!}{(v)_{n}}, \qquad \Re(v)>-a, \quad r=1,2,3,\ldots $$ where the polynomials $ψ_{n}(a)$ are given by the following generating equation

\frac{(1+z)^{a+m}-(1+z)^{a}}{\ln(1+z)}=\sum_{n=0}^\infty N_{n,m}(a) z^n, \qquad |z|<1, $$ so that $N_{n,r}(a)$. Similar expressions with the logarithm of the gamma function involve these formulas

\psi(v)= \frac{1}{v+a-\tfrac12}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{(v)_{n}}(n-1)!\right\},\qquad \Re(v)>-a, $$ and

\psi(v)= \frac{1}{\tfrac{1}{2}r+v+a-1}\left\{\ln\Gamma(v+a) + v - \frac12\ln2\pi - \frac12 + \frac{1}{r}\sum_{n=0}^{r-2} (r-n-1)\ln(v+a+n) +\frac{1}{r}\sum_{n=1}^\infty\frac{(-1)^n N_{n+1,r}(a)}{(v)_{n}}(n-1)!\right\}, $$ where $$\Re(v)>-a$$ and $$r=2,3,4,\ldots$$.

Reflection formula
The digamma and polygamma functions satisfy reflection formulas similar to that of the gamma function:

 * $$\psi(1-x)-\psi(x)=\pi \cot \pi x$$.
 * $$\psi'(-x)+\psi'(x) = \frac{\pi^2}{\sin^2(\pi x)}+\frac{1}{x^2}$$.

Recurrence formula and characterization
The digamma function satisfies the recurrence relation

 * $$\psi(x+1)=\psi(x)+\frac{1}{x}.$$

Thus, it can be said to "telescope" $N_{n,1}(a) = ψ_{n}(a)$, for one has

 * $$\Delta [\psi](x)=\frac{1}{x}$$

where $1⁄x$ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula

 * $$\psi(n)=H_{n-1}-\gamma$$

where $γ$ is the Euler–Mascheroni constant.

Actually, $ψ$ is the only solution of the functional equation

 * $$F(x+1)=F(x)+\frac{1}{x}$$

that is monotonic on $Δ$ and satisfies $R^{+}$. This fact follows immediately from the uniqueness of the $F(1) = −γ$ function given its recurrence equation and convexity restriction. This implies the useful difference equation:

 * $$ \psi(x+N)-\psi(x)=\sum_{k=0}^{N-1} \frac{1}{x+k}$$

Some finite sums involving the digamma function
There are numerous finite summation formulas for the digamma function. Basic summation formulas, such as

 * $$\sum_{r=1}^m \psi\left(\frac{r}{m}\right)=-m(\gamma+\ln m),$$
 * $$\sum_{r=1}^m \psi\left(\frac{r}{m}\right)\cdot\exp\dfrac{2\pi rki}{m} = m\ln \left(1-\exp\frac{2\pi ki}{m}\right), \qquad k\in\Z,\quad m\in\N,\ k\ne m$$
 * $$\sum_{r=1}^{m-1} \psi\left(\frac{r}{m}\right)\cdot\cos\dfrac{2\pi rk}{m} = m \ln \left(2\sin\frac{k\pi}{m}\right)+\gamma, \qquad k=1, 2,\ldots, m-1 $$
 * $$\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\sin\frac{2\pi rk}{m} =\frac{\pi}{2} (2k-m), \qquad k=1, 2,\ldots, m-1 $$

are due to Gauss. More complicated formulas, such as

 * $$\sum_{r=0}^{m-1} \psi \left(\frac{2r+1}{2m}\right)\cdot\cos\frac{(2r+1)k\pi }{m} = m\ln\left(\tan\frac{\pi k}{2m}\right) ,\qquad k=1, 2,\ldots, m-1$$
 * $$\sum_{r=0}^{m-1} \psi \left(\frac{2r+1}{2m}\right)\cdot\sin\dfrac{(2r+1)k\pi }{m} = -\frac{\pi m}{2}, \qquad k=1, 2,\ldots, m-1$$
 * $$\sum_{r=1}^{m-1} \psi\left(\frac{r}{m}\right)\cdot\cot\frac{\pi r}{m}= -\frac{\pi(m-1)(m-2)}{6}$$
 * $$\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right)\cdot \frac{r}{m}=-\frac{\gamma}{2}(m-1)-\frac{m}{2}\ln m -\frac{\pi}{2}\sum_{r=1}^{m-1} \frac{r}{m}\cdot\cot\frac{\pi r}{m} $$
 * $$\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\cos\dfrac{(2\ell+1)\pi r}{m}= -\frac{\pi}{m}\sum_{r=1}^{m-1} \frac{r \cdot\sin\dfrac{2\pi r}{m}}{\cos\dfrac{2\pi r}{m} -\cos\dfrac{(2\ell+1)\pi }{m} }, \qquad \ell\in\mathbb{Z} $$
 * $$\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\sin\dfrac{(2\ell+1)\pi r}{m}=-(\gamma+\ln2m)\cot\frac{(2\ell+1)\pi}{2m} + \sin\dfrac{(2\ell+1)\pi }{m}\sum_{r=1}^{m-1} \frac{\ln\sin\dfrac{\pi r}{m}} {\cos\dfrac{2\pi r}{m} -\cos\dfrac{(2\ell+1)\pi }{m} }, \qquad \ell\in\mathbb{Z}$$
 * $$\sum_{r=1}^{m-1} \psi^2\left(\frac{r}{m}\right)= (m-1)\gamma^2 + m(2\gamma+\ln4m)\ln{m} -m(m-1)\ln^2 2 +\frac{\pi^2 (m^2-3m+2)}{12} +m\sum_{\ell=1}^{ m-1 } \ln^2 \sin\frac{\pi\ell}{m}$$

are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014) ).

We also have
 * $$ 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{k-1}-\gamma=\frac{1}{k}\sum_{n=0}^{k-1}\psi\left(1+\frac{n}{k}\right), k=2,3, ...$$

Gauss's digamma theorem
For positive integers $r$ and $m$ ($Γ$), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions

 * $$\psi\left(\frac{r}{m}\right) = -\gamma -\ln(2m) -\frac{\pi}{2}\cot\left(\frac{r\pi}{m}\right) +2\sum_{n=1}^{\left\lfloor \frac{m-1}{2} \right\rfloor} \cos\left(\frac{2\pi nr}{m} \right) \ln\sin\left(\frac{\pi n}{m}\right) $$

which holds, because of its recurrence equation, for all rational arguments.

Multiplication theorem
The multiplication theorem of the $$\Gamma$$-function is equivalent to
 * $$\psi(nz)=\frac{1}{n}\sum_{k=0}^{n-1} \psi(z+\frac{k}{n}) +\ln n .$$

Asymptotic expansion
The digamma function has the asymptotic expansion
 * $$\psi(z) \sim \ln z + \sum_{n=1}^\infty \frac{\zeta(1-n)}{z^n} = \ln z - \sum_{n=1}^\infty \frac{B_n}{nz^n},$$

where $B_{k}$ is the $k$th Bernoulli number and $ζ$ is the Riemann zeta function. The first few terms of this expansion are:
 * $$\psi(z) \sim \ln z - \frac{1}{2z} - \frac{1}{12z^2} + \frac{1}{120z^4} - \frac{1}{252z^6} + \frac{1}{240z^8} - \frac{1}{132z^{10}} + \frac{691}{32760z^{12}} - \frac{1}{12z^{14}} + \cdots.$$

Although the infinite sum does not converge for any $z$, any finite partial sum becomes increasingly accurate as $z$ increases.

The expansion can be found by applying the Euler–Maclaurin formula to the sum
 * $$\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{z + n}\right)$$

The expansion can also be derived from the integral representation coming from Binet's second integral formula for the gamma function. Expanding $$t / (t^2 + z^2)$$ as a geometric series and substituting an integral representation of the Bernoulli numbers leads to the same asymptotic series as above. Furthermore, expanding only finitely many terms of the series gives a formula with an explicit error term:
 * $$\psi(z) = \ln z - \frac{1}{2z} - \sum_{n=1}^N \frac{B_{2n}}{2nz^{2n}} + (-1)^{N+1}\frac{2}{z^{2N}} \int_0^\infty \frac{t^{2N+1}\,dt}{(t^2 + z^2)(e^{2\pi t} - 1)}.$$

Inequalities
When $r < m$, the function
 * $$\ln x - \frac{1}{2x} - \psi(x)$$

is completely monotonic and in particular positive. This is a consequence of Bernstein's theorem on monotone functions applied to the integral representation coming from Binet's first integral for the gamma function. Additionally, by the convexity inequality $$1 + t \le e^t$$, the integrand in this representation is bounded above by $$e^{-tz}/2$$. Consequently
 * $$\frac{1}{x} - \ln x + \psi(x)$$

is also completely monotonic. It follows that, for all $x > 0$,
 * $$\ln x - \frac{1}{x} \le \psi(x) \le \ln x - \frac{1}{2x}.$$

This recovers a theorem of Horst Alzer. Alzer also proved that, for $x > 0$,
 * $$\frac{1 - s}{x + s} < \psi(x + 1) - \psi(x + s),$$

Related bounds were obtained by Elezovic, Giordano, and Pecaric, who proved that, for $s ∈ (0, 1)$,
 * $$\ln(x + \tfrac{1}{2}) - \frac{1}{x} < \psi(x) < \ln(x + e^{-\gamma}) - \frac{1}{x},$$

where $$\gamma=-\psi(1)$$ is the Euler–Mascheroni constant. The constants ($$0.5$$ and $$e^{-\gamma}\approx0.56$$) appearing in these bounds are the best possible.

The mean value theorem implies the following analog of Gautschi's inequality: If $x > 0$, where $x > c$ is the unique positive real root of the digamma function, and if $c ≈ 1.461$, then
 * $$\exp\left((1 - s)\frac{\psi'(x + 1)}{\psi(x + 1)}\right) \le \frac{\psi(x + 1)}{\psi(x + s)} \le \exp\left((1 - s)\frac{\psi'(x + s)}{\psi(x + s)}\right).$$

Moreover, equality holds if and only if $s > 0$.

Inspired by the harmonic mean value inequality for the classical gamma function, Horzt Alzer and Graham Jameson proved, among other things, a harmonic mean-value inequality for the digamma function:

$$ -\gamma \leq \frac{2 \psi(x) \psi(\frac{1}{x})}{\psi(x)+\psi(\frac{1}{x})} $$ for $$x>0$$

Equality holds if and only if $$x=1$$.

Computation and approximation
The asymptotic expansion gives an easy way to compute $s = 1$ when the real part of $x$ is large. To compute $ψ(x)$ for small $x$, the recurrence relation

 * $$ \psi(x+1) = \frac{1}{x} + \psi(x)$$

can be used to shift the value of $x$ to a higher value. Beal suggests using the above recurrence to shift $x$ to a value greater than 6 and then applying the above expansion with terms above $ψ(x)$ cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).

As $x$ goes to infinity, $x^{14}$ gets arbitrarily close to both $ψ(x)$ and $ln(x − 1⁄2)$. Going down from $ln x$ to $x$, $ψ$ decreases by $x + 1$, $1⁄x$ decreases by $ln(x − 1⁄2)$, which is more than $ln(x + 1⁄2) / (x − 1⁄2)$, and $1⁄x$ decreases by $ln x$, which is less than $ln(1 + 1⁄x)$. From this we see that for any positive $x$ greater than $1⁄x$,

 * $$\psi(x)\in \left(\ln\left(x-\tfrac12\right), \ln x\right)$$

or, for any positive $x$,

 * $$\exp \psi(x)\in\left(x-\tfrac12,x\right).$$

The exponential $1⁄2$ is approximately $exp ψ(x)$ for large $x$, but gets closer to $x$ at small $x$, approaching 0 at $x − 1⁄2$.

For $x = 0$, we can calculate limits based on the fact that between 1 and 2, $x < 1$, so

 * $$\psi(x)\in\left(-\frac{1}{x}-\gamma, 1-\frac{1}{x}-\gamma\right),\quad x\in(0, 1)$$

or
 * $$\exp \psi(x)\in\left(\exp\left(-\frac{1}{x}-\gamma\right),e\exp\left(-\frac{1}{x}-\gamma\right)\right).$$

From the above asymptotic series for $ψ$, one can derive an asymptotic series for $ψ(x) ∈ [−γ, 1 − γ]$. The series matches the overall behaviour well, that is, it behaves asymptotically as it should for large arguments, and has a zero of unbounded multiplicity at the origin too.

 * $$ \frac{1}{\exp \psi(x)} \sim \frac{1}{x}+\frac{1}{2\cdot x^2}+\frac{5}{4\cdot3!\cdot x^3}+\frac{3}{2\cdot4!\cdot x^4}+\frac{47}{48\cdot5!\cdot x^5} - \frac{5}{16\cdot6!\cdot x^6} + \cdots$$

This is similar to a Taylor expansion of $exp(−ψ(x))$ at $exp(−ψ(1 / y))$, but it does not converge. (The function is not analytic at infinity.) A similar series exists for $y = 0$ which starts with $$\exp \psi(x) \sim x- \frac 12.$$

If one calculates the asymptotic series for $f(y)$ it turns out that there are no odd powers of $x$ (there is no $x$−1 term). This leads to the following asymptotic expansion, which saves computing terms of even order.

 * $$ \exp \psi\left(x+\tfrac{1}{2}\right) \sim x + \frac{1}{4!\cdot x} - \frac{37}{8\cdot6!\cdot x^3} + \frac{10313}{72\cdot8!\cdot x^5} - \frac{5509121}{384\cdot10!\cdot x^7} + \cdots$$

Similar in spirit to the Lanczos approximation of the $$\Gamma$$-function is Spouge's approximation.

Another alternative is to use the recurrence relation or the multiplication formula to shift the argument of $$\psi(x)$$ into the range $$1\le x\le 3$$ and to evaluate the Chebyshev series there.

Special values
The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. Some are listed below:

 * $$\begin{align}

\psi(1) &= -\gamma \\ \psi\left(\tfrac{1}{2}\right) &= -2\ln{2} - \gamma \\ \psi\left(\tfrac{1}{3}\right) &= -\frac{\pi}{2\sqrt{3}} -\frac{3\ln{3}}{2} - \gamma \\ \psi\left(\tfrac{1}{4}\right) &= -\frac{\pi}{2} - 3\ln{2} - \gamma \\ \psi\left(\tfrac{1}{6}\right) &= -\frac{\pi\sqrt{3}}{2} -2\ln{2} -\frac{3\ln{3}}{2} - \gamma \\ \psi\left(\tfrac{1}{8}\right) &= -\frac{\pi}{2} - 4\ln{2} - \frac {\pi + \ln \left (\sqrt{2} + 1 \right ) - \ln \left (\sqrt{2} - 1 \right ) }{\sqrt{2}} - \gamma. \end{align}$$

Moreover, by taking the logarithmic derivative of $$|\Gamma (bi)|^2$$ or $$|\Gamma (\tfrac{1}{2}+bi)|^2$$ where $$b$$ is real-valued, it can easily be deduced that

 * $$\operatorname{Im} \psi(bi) = \frac{1}{2b}+\frac{\pi}{2}\coth (\pi b),$$
 * $$\operatorname{Im} \psi(\tfrac{1}{2}+bi) = \frac{\pi}{2}\tanh (\pi b).$$

Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the imaginary unit the numerical approximation
 * $$\operatorname{Re} \psi(i) = -\gamma-\sum_{n=0}^\infty\frac{n-1}{n^3+n^2+n+1} \approx 0.09465.$$

Roots of the digamma function
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on $ln(f(y) / y)$ at $ln(1 / y) − φ(1 / y)$. All others occur single between the poles on the negative axis:

 * $$\vdots$$
 * $$\vdots$$
 * $$\vdots$$
 * $$\vdots$$
 * $$\vdots$$

Already in 1881, Charles Hermite observed that

 * $$x_n = -n + \frac{1}{\ln n} + O\left(\frac{1}{(\ln n)^2}\right)$$

holds asymptotically. A better approximation of the location of the roots is given by

 * $$x_n \approx -n + \frac{1}{\pi}\arctan\left(\frac{\pi}{\ln n}\right)\qquad n \ge 2$$

and using a further term it becomes still better

 * $$x_n \approx -n + \frac{1}{\pi}\arctan\left(\frac{\pi}{\ln n + \frac{1}{8n}}\right)\qquad n \ge 1$$

which both spring off the reflection formula via

 * $$0 = \psi(1-x_n) = \psi(x_n) + \frac{\pi}{\tan \pi x_n}$$

and substituting $φ(x)$ by its not convergent asymptotic expansion. The correct second term of this expansion is $exp(ψ(x))$, where the given one works well to approximate roots with small $n$.

Another improvement of Hermite's formula can be given:

x_n=-n+\frac1{\log n}-\frac1{2n(\log n)^2}+O\left(\frac1{n^2(\log n)^2}\right). $$

Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman
 * $$\begin{align}

\sum_{n=0}^\infty\frac{1}{x_n^2}&=\gamma^2+\frac{\pi^2}{2}, \\ \sum_{n=0}^\infty\frac{1}{x_n^3}&=-4\zeta(3)-\gamma^3-\frac{\gamma\pi^2}{2}, \\ \sum_{n=0}^\infty\frac{1}{x_n^4}&=\gamma^4+\frac{\pi^4}{9} + \frac23 \gamma^2 \pi^2 + 4\gamma\zeta(3). \end{align}$$

In general, the function

Z(k)=\sum_{n=0}^\infty\frac{1}{x_n^k} $$ can be determined and it is studied in detail by the cited authors.

The following results
 * $$\begin{align}

\sum_{n=0}^\infty\frac{1}{x_n^2+x_n}&=-2, \\ \sum_{n=0}^\infty\frac{1}{x_n^2-x_n}&=\gamma+\frac{\pi^2}{6\gamma} \end{align}$$ also hold true.

Regularization
The digamma function appears in the regularization of divergent integrals

 * $$ \int_0^\infty \frac{dx}{x+a},$$

this integral can be approximated by a divergent general Harmonic series, but the following value can be attached to the series

 * $$ \sum_{n=0}^\infty \frac{1}{n+a}= - \psi (a).$$

title: Digamma function


text:
User:Phanly

Main interests are:
 * 1) Global warming and the sub topics
 * 2) Dinghy sailing particularly Heron (dinghy)
 * 3) Parramatta River
 * 4) Remediation
 * 5) Climate change and Australia. I am working on ensuring WP covers this issue: see comment on Australia Project page
 * 6) Stock Markets and investment

I have started (24 Feb 07)a sandbox for development of pages prior to publishing in Wikipedia. My pages under development include: (None yet but Effects of global warming on Australia may well be the first.)

/Timeline of environmental history of Australia is a subpage for preparation of an article on the Timeline of environmental history of Australia

/List of Australian environmentalists is a subpage for preparation of a List of Australian environmentalists

title: User:Phanly


text:
Transport function

In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by

 * $$J(n,x) = \int_0^x t^n \frac{e^t}{(e^t - 1)^2}\,dt.$$

Note that

 * $$\frac{e^t}{(e^t - 1)^2} = \sum_{k=0}^\infty k\,e^{kt}.$$

title: Transport function


text:
User talk:Zombie~enwiki

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title: User talk:Zombie~enwiki


text:
Robert Crippen

Robert Laurel Crippen (born September 11, 1937) is an American retired naval officer and aviator, test pilot, aerospace engineer, and retired astronaut. He traveled into space four times: as pilot of STS-1 in April 1981, the first Space Shuttle mission; and as commander of STS-7 in June 1983, STS-41-C in April 1984, and STS-41-G in October 1984. He was also a part of the Manned Orbiting Laboratory (MOL), Skylab Medical Experiment Altitude Test (SMEAT), ASTP support crew member, and the Approach and Landing Tests (ALT) for the Space Shuttle.

In 1986, Crippen participated in the recovery operations for the remains of crew members after the Space Shuttle Challenger disaster. He was also on the commission responsible for determining the cause of the accident.

After retiring as an astronaut, Crippen worked his way through management at NASA, namely as Director, Space Shuttle, at NASA Headquarters, then Director of the Kennedy Space Center. He also went to Lockheed Martin and Thiokol Propulsion before retiring to private life in Florida.

Crippen has received several awards and honors, including the Congressional Space Medal of Honor in 2006, and having an elementary school named after him in Porter, Texas. He is also a fellow of several organizations, including the American Institute of Aeronautics and Astronautics (AIAA) and the Society of Experimental Test Pilots (SETP).

Early life and education
Robert Crippen was born in Beaumont, Texas, on September 11, 1937. After graduating from New Caney High School in New Caney, Texas in 1955, Crippen went to the University of Texas at Austin to major in Aerospace Engineering. In 1960, he graduated with his Bachelor of Science degree. He was selected as a member of the Texas Alpha chapter of Tau Beta Pi and Sigma Gamma Tau.

Crippen became interested in flying and computers at a very early age. He attended the first computer programming class held at the University of Texas. Throughout his career in the military and at NASA, he worked on computer programming, including programs such as the Manned Orbiting Laboratory, Skylab and the Space Shuttle.

Military career
Crippen was commissioned through the United States Navy's Aviation Officer Candidate School (AOCS) Program at Naval Air Station Pensacola, Florida. He also spent time at the Naval Air Station Whiting Field in Florida and the Naval Air Station Chase Field in Beeville, Texas. That is where he earned his wings. As a Naval Aviator from the summer of 1962 to the end of 1964, he made two deployments aboard the USS Independence, flying the A-4 Skyhawk in Attack Squadron 72 (VA-72). He later went to Edwards Air Force Base in California to attend the U.S. Air Force Aerospace Research Pilot School. After graduation, Crippen worked as an instructor at Edwards Air Force Base until October 1966 when he was picked for the U.S. Air Force's Manned Orbiting Laboratory (MOL) program. He has spent more than 6,500 hours in the air as either a pilot or an astronaut.

Manned Orbiting Laboratory

The Manned Orbiting Laboratory (MOL) program was a follow-on to the X-20 Dyna-Soar program. A joint program between the United States Air Force and the National Reconnaissance Office (NRO), the MOL program would send humans into space.

Publicly, the MOL program was designed to determine the usefulness of man in space. Crews would be in orbit for about a month and they would be able to freely move about the laboratory. The secret and primary mission was to perform reconnaissance missions on Soviet Union and China. The pilots were not told of this true mission, though they were later informed.

When Crippen was selected for astronaut training in October 1966, he had to choose between the military and NASA, deciding to stay in the military to work on the MOL program. He felt that he would get lost at NASA due to the number of astronauts already in programs at the agency. There was an uncrewed flight on November 3, 1966. There were no crewed flights.

The MOL program was canceled in June 1969, after which Crippen transferred to NASA to continue his career. The cancellation was due to differing priorities within the military and space programs, with the government deciding to de-emphasize military involvement in the space program. However, developments like the space suit, waste management, and the high resolution camera were later used in various space programs going forward.

NASA career
After the MOL program was canceled, Crippen became part of NASA Astronaut Group 7 in September 1969. He served as support for the Skylab 2, Skylab 3, and Skylab 4, as well as for the Apollo-Soyuz Test Project.

Skylab Medical Experiment Altitude Test (SMEAT)
The Skylab Medical Experiment Altitude Test (SMEAT) was a 56-day mission (July 26 through September 19, 1972), where astronauts Crippen, Karol J. Bobko and William E. Thornton were housed in a vacuum chamber to conduct medical experiments. The goal was to ensure that crews in space could handle minor medical emergencies, including dentistry. SMEAT also was able to discover and fix the design flaws in the urine handling system for Skylab. Crippen and the team worked to enlarge the original collection system design, which was too small, the collection system of which had burst at one point.

Apollo-Soyuz Test Project
The Apollo-Soyuz Test Project (ASTP) was a joint effort between the United States and the Soviet Union, designed to test rendezvous and docking capabilities if there were ever an emergency in space. Crippen was the capsule communicator (CAPCOM) for the Apollo-Soyuz Test Project (as well as Skylab), during which time he developed a close association with Mission Operations and Flight Control. On July 15, 1975, the United States launched a Saturn IB launch vehicle in an Apollo configuration into space. Two days later, it docked with a Soyuz spacecraft.

Approach and Landing Tests (ALT)
The Approach and Landing tests for the Space Shuttle were critical to the overall program, in which a crew tested the maneuverability and landing capabilities of the spacecraft. Launched from a 747 aircraft, the prototype Space Shuttle Enterprise was first tested on August 12, 1977. Astronauts Fred W. Haise Jr. and C. Gordon Fullerton piloted the Enterprise from about 7300 m to the ground and landed at Edwards Air Force Base. A second crew of astronauts Joe H. Engle and Richard H. Truly alternated with Haise and Fullerton to test the Enterprise's capabilities. Crippen was first assigned to family support with the testing, then progressed to being a chase pilot in a T-38. The ALT program lasted about a year, testing all aspects of approach and touchdown from aerodynamics to maneuverability, gliding and landing characteristics.

STS-1

Crippen was the pilot of the Space Shuttle Columbia on STS-1, the first test flight of the Space Shuttle in orbit. His job involved working with the Shuttle computers, electrical systems, and auxiliary power units. He was also responsible for operating the payload bay doors. STS-1 had several firsts: a crewed vehicle to be flown into Earth orbit without any prior non-crewed orbital testing; a crewed, winged vehicle to launch with solid rocket boosters; a reentry vehicle to land on a conventional runway. It launched April 12, 1981, and landed April 14, 1981.

STS-7
Crippen served as the commander of STS-7, the second launch of the Space Shuttle Challenger, and headed a crew of five people. During flight, the team deployed the Canadian Anik-C2 satellite, as well as the Palapa-B1 satellite from Indonesia. They also used the Canadian Remote Manipulator System (Canadarm) to deploy and retrieve the Shuttle Pallet Satellite (SPAS-01). They conducted formation flying with an untethered satellite (SPAS-02), operated a joint U.S.-German materials experiment (OSTA-2) and activated seven Getaway Special (GAS) experiments. Finally, they worked with the Continuous Flow Electrophoresis System (CFES) and the Monodisperse Latex Reactor (MLR). It launched June 18, 1983, and landed on June 24, 1983.

STS-41-C
Crippen was the commander of Space Shuttle Challenger on STS-41-C, a seven-day mission during which the crew deployed the Long Duration Exposure Facility (LDEF). The crew also retrieved, repaired and redeployed Solar Maximum Mission (SMM) satellite using the Canadarm. The flight also tested the Manned Maneuvering Units (MMU) through two extravehicular activities (EVAs). The team also operated the Cinema 360 and IMAX Camera Systems. Finally, the crew ran a student experiment on honey bees. It launched April 6, 1984, and landed April 13, 1984.

STS-41-G
On his final spaceflight, Crippen served as the commander of STS-41-G. During the eight-day flight, the Space Shuttle Challenger crew deployed the Earth Radiation Budget Satellite (ERBS). They also performed scientific observations of the Earth with a Large Format Camera, and demonstrated potential satellite refueling with the Orbital Refueling System (ORS). Finally, the Office of Space and Terrestrial Applications-3 (OSTA-3) also had experiments for the crew to perform. It launched October 5, 1984, and landed October 13, 1984, at Kennedy Space Center, Florida.

Post-spaceflight career
After STS-41-G, Crippen became deputy director of flight crew operations (1984-1986) and was also named commander of the STS-62-A mission which would have launched from the new SLC-6 facility at Vandenberg Air Force Base, California. That mission was canceled after the Space Shuttle Challenger disaster in 1986, and SLC-6 was mothballed in 1989.

Crippen was personally involved in the recovery efforts of the Challenger disaster, brooking no interference in recovering all seven bodies of the Challenger crew for their families. Crippen was also a part of the STS-51-L Interim Mishap Review Board to examine the cause of the Challenger accident. The Board became a sub-team for the Rogers Commission Report, the team put together by President Ronald Reagan to investigate the accident.

Crippen was stationed at Kennedy Space Center (KSC) from July 1987 to December 1989 where he was Deputy Director of Shuttle Operations under Arnie Aldrich, Director of Space Shuttle. He was accountable for Shuttle missions and the return of the craft to KSC after landings at Edwards.

From January 1990 to January 1992, Crippen served as Director of the Space Shuttle program at NASA Headquarters. As such, he was responsible for the all aspects of the Shuttle program, including scheduling, budget and overall program management. Afterwards, he moved to KSC to become center director. This lasted from January 1992 through to January 1995.

Post-NASA career
After leaving NASA, from April 1995 through November 1996, Crippen was the Vice President of Lockheed Martin Information Systems. His main focus was simulation work supporting the military.

Crippen became President of Thiokol Propulsion from December 1996 through to April 2001. They were responsible for producing the Space Shuttle Reusable Solid Rocket Motors. Crippen enjoyed continuing his connection with the Space Shuttle program. In particular, he appreciated his role in working to improve the design of the solid rocket boosters to prevent another Challenger disaster.

Organizations
Crippen is a fellow in the American Institute of Aeronautics and Astronautics (AIAA), the Society of Experimental Test Pilots (SETP), and the American Astronautical Society (AAS). He served as President of the American Institute of Aeronautics and Astronautics in 1999. He was selected to be a member of Naval Aviators Golden Eagles in 2009. He was elected to the National Academy of Engineering in 2012 for leadership in human space flight and development of solid fueled rockets.

Awards and honors

Crippen has earned several awards throughout his career, including the SETP Iven C. Kincheloe Award (1981) and the NASA Exceptional Service Medal (1972). In 1981, after the inaugural Space Shuttle flight, he received the Department of Defense Distinguished Service Award, the American Astronautical Society's Flight Achievement Award, the National Geographic Society's Gardiner Greene Hubbard Medal, and the American Legion's Distinguished Service Medal. He was awarded the Federal Aviation Administration (FAA)'s Award for Distinguished Service (1982), the Dr. Robert H. Goddard Memorial Trophy and the Harmon Trophy. He received the Navy Distinguished Flying Cross (1984) and the Defense Meritorious Service Medal (1984). Finally, Crippen received the Golden Plate Award of the American Academy of Achievement (1986).

He also received NASA Outstanding Leadership Medal, four NASA Space Flight Medals, and three NASA Distinguished Service Medals. In 1996, Crippen became the tenth individual to receive the Rotary National Award for Space Achievement's National Space Trophy. He was inducted into the International Space Hall of Fame in 1991 and the U.S. Astronaut Hall of Fame in 2001. On April 6, 2006, he received the Congressional Space Medal of Honor, the highest award for spaceflight achievement.

On November 18, 2015, at the National Business Aviation Association convention in Las Vegas, Crippen was announced as a 2016 Inductee into the National Aviation Hall of Fame.

On December 15, 2023, Crippen received the Wright Brothers Memorial Trophy from the National Aeronautic Association in Washington, D.C. for his devotion to public service and the advancement of American aerospace; his achievements as an aviator, astronaut, and leader; and his selfless dedication to the future of humankind.

Personal life
Crippen was first married to Virginia Hill on September 8, 1959. They had three daughters together: Ellen Marie (born June 1962), Susan Lynn (born December 1964), and Linda Ruth (born May 1967). On November 7, 1987, he married Pandora Puckett, NASA's first female lead Orbiter Project Engineer, on the and  at the Kennedy Space Center.

title: Robert Crippen


text:
Edward Smith-Stanley, 13th Earl of Derby

Edward Smith-Stanley, 13th Earl of Derby (21 April 1775 – 30 June 1851), KG, of Knowsley Hall in Lancashire (styled Lord Stanley from 1776 to 1832, known as Baron Stanley of Bickerstaffe from 1832-4), was a politician, peer, landowner, builder, farmer, art collector and naturalist. He was the patron of the writer Edward Lear.

Origins
He was the eldest child and only son and heir of Edward Smith-Stanley, 12th Earl of Derby (1752–1834) by his wife Elizabeth Hamilton, a daughter of James Hamilton, 6th Duke of Hamilton.

Career
He was educated at Eton College and Trinity College, Cambridge. On 10 November 1796 he was appointed a Deputy Lieutenant of Lancashire and in the same year he was elected as a Member of Parliament for Preston. He held this seat until 1812 and then represented Lancashire until 1832, when he was ennobled as Baron Stanley of Bickerstaffe, of Bickerstaffe in the County Palatine of Lancaster.

Military career
He was commissioned Colonel of the 1st Royal Lancashire Supplementary Militia on 1 March 1797; this regiment subsequently became the 2nd Royal Lancashire Militia. He was breveted as a colonel in the regular Army with seniority from that date, retaining the rank until his regiment was disembodied, which occurred at the end of 1799. He resigned his commission as colonel on 13 April 1847.

Naturalist
In 1834 he succeeded his father as 13th Earl of Derby and withdrew from politics, instead concentrating on his natural history collection at Knowsley Hall, near Liverpool. He had a large collection of living animals: at his death, there were 1,272 birds and 345 mammals at Knowsley, shipped to England by explorers such as Joseph Burke. From 1828 to 1833 he was President of the Linnean Society. Many of Derby's collections are now housed in Liverpool's World Museum. Several species were named after him, for example the Derbyan parakeet, Psittacula derbiana and an Australian species of parrot named firstly by Nicholas Vigors as Platycercus stanleyii, in 1830 when he was Lord Stanley, and referred to in the vernacular as "The Earl of Derby’s Parrakeet" by the author John Gould in the sixth volume of his magnum opus Birds of Australia. However the latter species was found to have been named previously as Platycercus icterotis, and thus Platycercus stanleyii was found to have been an invalid name due to the pre-existence of a published description for the species, according to "the inviolable laws of precedence in deliberations on biological nomenclature". From the Earl of Derby's Collection, the State Library of NSW purchased six volumes of exquisite Australian natural history drawings dating from the early days of British settlement in NSW and this Library publishes talks and exhibitions of its research on this collection.

He founded in 1851 with his natural history's collection a museum in Liverpool, the Derby Museum, the current World Museum, the oldest of the National Museums Liverpool group.

Marriage and issue
On 30 June 1798 he married his first cousin Charlotte Margaret Hornby (d.1817), second daughter of Rev. Geoffrey Hornby (1750–1812), of Scale Hall, near Lancaster in Lancashire, High Sheriff of Lancashire in 1774 and a Deputy Lieutenant of Lancashire, Colonel of a regiment of Lancashire militia, by his wife Lucy Smith-Stanley (d.1833) a daughter of James Smith-Stanley, Lord Strange (1716–1771), (son and heir apparent of Edward Stanley, 11th Earl of Derby (1689–1776)) and a sister of Edward Smith-Stanley, 12th Earl of Derby (1752–1834). Charlotte's brother was Edmund Hornby (1773–1857) of Dalton Hall, near Burton, Westmorland, a Member of Parliament for Preston, Lancashire, from 1812–1826, who married his first cousin Lady Charlotte Stanley (d.1805), a daughter of Edward Smith-Stanley, 12th Earl of Derby (1752–1834). By Charlotte Hornby, he had issue:
 * Edward George Geoffrey Smith-Stanley, 14th Earl of Derby (1799–1869), thrice Prime Minister (1852, 1858–9, 1866–8)
 * Lady Charlotte Elizabeth Smith-Stanley (1801–1853), married Edward Penrhyn
 * Hon. Henry Thomas Smith-Stanley (1803–1875), MP for Preston (1832–7)
 * Lady Emily Lucy Smith-Stanley (1804), died in infancy
 * Lady Louisa Emily Stanley (1805–1825), married Lt.-Col. Samuel Long
 * Lady Eleanor Mary Smith-Stanley (b. 1807)
 * Colonel Hon. Charles James Fox Stanley (1808–1884)

Death
He died on 30 June 1851.


text:
Synchrotron function

In mathematics the synchrotron functions are defined as follows (for x ≥ 0):

 * First synchrotron function $$F(x) = x \int_x^\infty K_{\frac{5}{3}}(t)\,dt$$
 * Second synchrotron function $$G(x) = x K_{\frac{2}{3}}(x)$$

where Kj is the modified Bessel function of the second kind.

Use in astrophysics
In astrophysics, x is usually a ratio of frequencies, that is, the frequency over a critical frequency (critical frequency is the frequency at which most synchrotron radiation is radiated). This is needed when calculating the spectra for different types of synchrotron emission. It takes a spectrum of electrons (or any charged particle) generated by a separate process (such as a power law distribution of electrons and positrons from a constant injection spectrum) and converts this to the spectrum of photons generated by the input electrons/positrons.

title: Synchrotron function


text:
Slobozia

Slobozia is the capital city of Ialomița County, Muntenia, Romania, with a population of 41,550 in 2021.

Etymology
Its name is from the Romanian "slobozie", which meant a recently colonized village which was free of taxation. The word itself comes from the Slavic word "slobod" which means "free". As it is located in the middle of flat land (Bărăgan Plain), it was very vulnerable to Tatar and Ottoman incursions. To encourage peasants to settle there, they were exempted from some taxes, hence the name.

Geography
Slobozia lies roughly in the middle of the county, on the banks of Ialomița River, at about 120 km east of Bucharest and 150 km west of Constanța, important port at the Black Sea. The city is within 17 km of the Bucharest-Constanța A2 Motorway (Autostrada Soarelui).

The total area of the municipality is 126.72 km2. In the present administrative form, Slobozia consists of Slobozia proper and the neighbourhoods of Bora and Slobozia Nouă.

Economy
The main activity in the area is agriculture, processing of the agricultural products and light industry.

Culture
In 1990 the Cultural Centre was inaugurated, bearing the name of the conductor and composer Ionel Perlea, a city native. The building houses exhibition and performance rooms, bookstores, cultural institutions. In 1999 the Cultural Centre Ionel Perlea entered the UNESCO circuit.

The city is the headquarters of the Romanian Orthodox Diocese of Slobozia and Călărași, established in 1993.

Demographics
At the 2002 census, 97.6% of inhabitants were ethnic Romanians and 2.2% Roma. 98.6% were Romanian Orthodox, 0.4% Seventh-day Adventist and 0.2% Roman Catholic.

Tourism
The main tourist attraction consists of the nearby Lake Amara, situated 5 km away. Amara Resort is also a balneoclimateric resort. Access to Amara is by minibuses that leave every 15 minutes from the Slobozia Train Station. As part of a private tourist complex, there is a small copy of the Eiffel Tower 54 m high.

Twin towns – sister cities
Slobozia is twinned with:
 * 🇨🇳 Nanyang, China
 * 🇧🇬 Razgrad, Bulgaria
 * 🇧🇬 Silistra, Bulgaria
 * 🇲🇰 Veles, North Macedonia

Notable people

 * Alin Badea (born 1988), sabre fencer
 * Mihaela Bulică (born 1990), sabre fencer
 * Mircea Dinescu (born 1950), poet, journalist, and editor
 * Petre Dumitru (born 1957), weightlifter
 * Petru Filip (born 1955), politician
 * Cristina Gheorghe (born 1986), handballer
 * Adrian Mihalcea (born 1976), footballer
 * Minelli (born 1988), singer, songwriter, and lyricist
 * Doina Spîrcu (born 1970), rower
 * Iulian Teodosiu (born 1994), sabre fencer
 * Adrian Ursea (born 1967), football player and manager
 * Elena Voicu (born 1990), handballer

title: Slobozia


text:
Crag

Crag may refer to:

 * Crag (climbing), a cliff or group of cliffs, in any location, which is or may be suitable for climbing
 * Crag (dice game), a dice game played with three dice
 * Crag, Arizona, US
 * Crag, West Virginia, US
 * Crag and tail, a geological formation caused by the passage of a glacier over an area of hard rock
 * Crag Group, a geological group outcropping in East Anglia, UK
 * Coralline Crag Formation
 * Norwich Crag Formation
 * Red Crag Formation
 * Wroxham Crag Formation, see Cromer Forest Bed
 * Crag Hotel, Penang, Malaysia
 * Crag Jones (born 1962), Welsh climber
 * USS Crag (AM-214), a 1943 US Navy Admirable-class minesweeper
 * The Crag, the final event in the Nickelodeon Guts action sports program
 * Club de Radioaficionados de Guatemala, an amateur radio organization in Guatemala
 * Constitutional Reform and Governance Act 2010, concerning United Kingdom constitutional law
 * CRAG stands for Corrective Retrieval Augmented Generation

title: Crag


text:
Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables $s$ with $Re(s) > 1$ and $a ≠ 0, −1, −2, …$ by

 * $$\zeta(s,a) = \sum_{n=0}^\infty \frac{1}{(n+a)^{s}}.$$

This series is absolutely convergent for the given values of $s$ and $a$ and can be extended to a meromorphic function defined for all $s ≠ 1$. The Riemann zeta function is $ζ(s,1)$. The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.

Integral representation
The Hurwitz zeta function has an integral representation
 * $$\zeta(s,a) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}e^{-ax}}{1-e^{-x}} dx$$

for $$\operatorname{Re}(s)>1$$ and $$\operatorname{Re}(a)>0.$$ (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing
 * $$\zeta(s,a)\Gamma(s)

= \sum_{n=0}^\infty \frac{1}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x} = \sum_{n=0}^\infty \int_0^\infty y^s e^{-(n+a)y} \frac{dy}{y}$$ and then interchanging the sum and integral.

The integral representation above can be converted to a contour integral representation
 * $$\zeta(s,a) = -\Gamma(1-s)\frac{1}{2 \pi i} \int_C \frac{(-z)^{s-1}e^{-az}}{1-e^{-z}} dz$$

where $$C$$ is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation $$(-z)^{s-1}$$. Unlike the previous integral, this integral is valid for all s, and indeed is an entire function of s.

The contour integral representation provides an analytic continuation of $$\zeta(s,a)$$ to all $$s \ne 1$$. At $$s = 1$$, it has a simple pole with residue $$1$$.

Hurwitz's formula
The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function:
 * $$\zeta(1-s,a) = \frac{\Gamma(s)}{(2\pi)^s} \left( e^{-\pi i s/2} \sum_{n=1}^\infty \frac{e^{2\pi ina}}{n^s} + e^{\pi i s/2} \sum_{n=1}^\infty \frac{e^{-2\pi ina}}{n^s} \right),$$

valid for Re(s) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:
 * $$\zeta(1-s) = \frac{2\Gamma(s)}{(2\pi)^{s}} \cos\left(\frac{\pi s}{2}\right) \zeta(s)$$

Hurwitz's formula can also be expressed as
 * $$\zeta(s,a) = \frac{2\Gamma(1-s)}{(2\pi)^{1-s}} \left( \sin\left(\frac{\pi s}{2}\right) \sum_{n=1}^\infty \frac{\cos(2\pi na)}{n^{1-s}} + \cos\left(\frac{\pi s}{2}\right) \sum_{n=1}^\infty \frac{\sin(2\pi na)}{n^{1-s}} \right)$$

(for Re(s) < 0 and 0 < a ≤ 1).

Hurwitz's formula has a variety of different proofs. One proof uses the contour integration representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation. These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper. Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral
 * $$\zeta(s,a) = s \int_{-a}^\infty \frac{\lfloor x \rfloor - x + \frac{1}{2}}{(x+a)^{s+1}} dx$$

(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series.

Functional equation for rational a
When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers $$1\leq m \leq n $$,
 * $$\zeta \left(1-s,\frac{m}{n} \right) =

\frac{2\Gamma(s)}{ (2\pi n)^s } \sum_{k=1}^n \left[\cos \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\; \zeta \left( s,\frac {k}{n} \right)\right] $$ holds for all values of s.

This functional equation can be written as another equivalent form:

$$ \zeta \left(1-s,\frac{m}{n} \right) = \frac{\Gamma(s)}{ (2\pi n)^s} \sum_{k=1}^n \left[e^{\frac{\pi is}{2}}e^{-\frac{2\pi ikm}{n}}\zeta \left( s,\frac {k}{n} \right) + e^{-\frac{\pi is}{2}}e^{\frac{2\pi ikm}{n}}\zeta \left( s,\frac {k}{n} \right) \right] $$.

Some finite sums
Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form

\sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right) \cos\dfrac{2\pi rk}{m} =\frac{m \Gamma(1-s)}{(2\pi m)^{1-s}} \sin\frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) + \zeta\left(1-s,1-\frac{k}{m}\right) \right\} - \zeta(s) $$

\sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right) \sin\dfrac{2\pi rk}{m}= \frac{m \Gamma(1-s)}{(2\pi m)^{1-s}} \cos \frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) - \zeta\left(1-s,1-\frac{k}{m}\right)\right\} $$

\sum_{r=1}^{m-1} \zeta^2\left(s,\frac{r}{m}\right) = \big(m^{2s-1}-1 \big)\zeta^2(s) + \frac{2m\Gamma^2(1-s)}{(2\pi m)^{2-2s}} \sum_{l=1}^{m-1} \left\{\zeta\left(1-s,\frac{l}{m}\right) - \cos\pi s \cdot \zeta\left(1-s,1-\frac{l}{m}\right)\right\} \zeta\left(1-s,\frac{l}{m}\right) $$ where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.

Series representation
A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:

 * $$\zeta(s,a)=\frac{1}{s-1}

\sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (a+k)^{1-s}.$$

This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the nth forward difference of $$a^{1-s}$$; that is,

 * $$\Delta^n a^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (a+k)^{1-s}$$

where Δ is the forward difference operator. Thus, one may write:

 * $$\begin{align}

\zeta(s, a) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n a^{1-s}\\ &= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} a^{1-s} \end{align}$$

Taylor series
The partial derivative of the zeta in the second argument is a shift:

 * $$\frac {\partial} {\partial a} \zeta (s,a) = -s\zeta(s+1,a).$$

Thus, the Taylor series can be written as:

 * $$\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!}

\frac {\partial^k} {\partial x^k} \zeta (s,x) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).$$

Alternatively,

 * $$\zeta(s, q) = \frac{1}{q^s} + \sum_{n=0}^{\infty} (-q)^n {s + n - 1 \choose n} \zeta(s + n),$$

with $$|q| < 1$$.

Closely related is the Stark–Keiper formula:

 * $$\zeta(s,N) =

\sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right] {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N) $$

which holds for integer N and arbitrary s. See also Faulhaber's formula for a similar relation on finite sums of powers of integers.

Laurent series
The Laurent series expansion can be used to define generalized Stieltjes constants that occur in the series
 * $$\zeta(s,a) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) (s-1)^n.$$

In particular, the constant term is given by
 * $$\lim_{s\to 1} \left[ \zeta(s,a) - \frac{1}{s-1}\right] =

\frac{-\Gamma'(a)}{\Gamma(a)} = -\psi(a)$$ where $$\Gamma$$ is the gamma function and $$\psi = \Gamma' / \Gamma$$ is the digamma function. As a special case, $$\gamma_0(1) = -\psi(1) = \gamma_0 = \gamma$$.

Discrete Fourier transform
The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function.

Negative integers
The values of ζ(s, a) at s = 0, −1, −2, ... are related to the Bernoulli polynomials:
 * $$\zeta(-n,a) = -\frac{B_{n+1}(a)}{n+1}.$$

For example, the $$n=0$$ case gives
 * $$\zeta(0,a) = \frac{1}{2} - a.$$

s-derivative
The partial derivative with respect to s at s = 0 is related to the gamma function:
 * $$\left. \frac{\partial}{\partial s} \zeta(s,a) \right|_{s=0} = \log\Gamma(a) - \frac{1}{2} \log(2\pi)$$

In particular, $\zeta'(0) = -\frac{1}{2} \log(2\pi).$ The formula is due to Lerch.

Relation to Jacobi theta function
If $$\vartheta (z,\tau)$$ is the Jacobi theta function, then

 * $$\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}=

\pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]$$

holds for $$\Re s > 0$$ and z complex, but not an integer. For z=n an integer, this simplifies to

 * $$\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}=

2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s) =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).$$

where ζ here is the Riemann zeta function. Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic delta function, or Dirac comb in z as $$t\rightarrow 0$$.

Relation to Dirichlet L-functions
At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ(s) when a = 1, when a = 1/2 it is equal to (2s−1)ζ(s), and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then

 * $$\zeta(s,n/k)=\frac{k^s}{\varphi(k)}\sum_\chi\overline{\chi}(n)L(s,\chi),$$

the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination

 * $$L(s,\chi)=\frac {1}{k^s} \sum_{n=1}^k \chi(n)\; \zeta \left(s,\frac{n}{k}\right).$$

There is also the multiplication theorem

 * $$k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),$$

of which a useful generalization is the distribution relation

 * $$\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).$$

(This last form is valid whenever q a natural number and 1 − qa is not.)

Zeros
If a=1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if a=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<a<1 and a≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn for rational or transcendental irrational a, and by Cassels for algebraic irrational a.

Rational values
The Hurwitz zeta function occurs in a number of striking identities at rational values. In particular, values in terms of the Euler polynomials $$E_n(x)$$:

 * $$E_{2n-1}\left(\frac{p}{q}\right) =

(-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}} \sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right) \cos \frac{(2k-1)\pi p}{q}$$

and

 * $$E_{2n}\left(\frac{p}{q}\right) =

(-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}} \sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right) \sin \frac{(2k-1)\pi p}{q}$$

One also has

 * $$\zeta\left(s,\frac{2p-1}{2q}\right) =

2(2q)^{s-1} \sum_{k=1}^q \left[ C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) + S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right) \right]$$

which holds for $$1\le p \le q$$. Here, the $$C_\nu(x)$$ and $$S_\nu(x)$$ are defined by means of the Legendre chi function $$\chi_\nu$$ as

 * $$C_\nu(x) = \operatorname{Re}\, \chi_\nu (e^{ix})$$

and

 * $$S_\nu(x) = \operatorname{Im}\, \chi_\nu (e^{ix}).$$

For integer values of ν, these may be expressed in terms of the Euler polynomials. These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

Applications
Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theory, where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems. In applied statistics, it occurs in Zipf's law and the Zipf–Mandelbrot law. In particle physics, it occurs in a formula by Julian Schwinger, giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.

Special cases and generalizations
The Hurwitz zeta function with a positive integer m is related to the polygamma function:
 * $$\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z) \ .$$

The Barnes zeta function generalizes the Hurwitz zeta function.

The Lerch transcendent generalizes the Hurwitz zeta:
 * $$\Phi(z, s, q) = \sum_{k=0}^\infty

\frac { z^k} {(k+q)^s}$$ and thus
 * $$\zeta(s,a)=\Phi(1, s, a).\,$$

Hypergeometric function

 * $$\zeta(s,a)=a^{-s}\cdot{}_{s+1}F_s(1,a_1,a_2,\ldots a_s;a_1+1,a_2+1,\ldots a_s+1;1)$$ where $$a_1=a_2=\ldots=a_s=a\text{ and }a\notin\N\text{ and }s\in\N^+.$$

Meijer G-function

 * $$\zeta(s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1 \; \left| \; \begin{matrix}0,1-a,\ldots,1-a\\0,-a,\ldots,-a\end{matrix}\right)\right.\qquad\qquad s\in\N^+.$$

title: Hurwitz zeta function