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Fubini's theorem

In mathematical analysis, the Fubini theorem gives the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. It states that if a function is Lebesgue integrable on a rectangle $$X\times Y$$, then one can evaluate the double integral as an iterated integral:$$\, \iint\limits_{X\times Y} f(x,y)\,\text{d}(x,y) = \int_X\left(\int_Y f(x,y)\,\text{d}y\right)\text{d}x=\int_Y\left(\int_X f(x,y) \, \text{d}x \right) \text{d}y.$$ The formula is not true in general for Riemann integrals, but it is true if the function is continuous on the rectangle. In multivariable calculus, this weaker result is sometimes also called Fubini's theorem, although it was already known by Leonhard Euler.

The Tonelli theorem, introduced by Leonida Tonelli in 1909, is similar but applied to a non-negative measurable function rather than an integrable function over its domain. The Fubini and Tonelli theorems are usually combined and form the Fubini-Tonelli theorem, which gives the conditions under which it is possible to switch the order of integration in an iterated integral.

A related theorem is often called Fubini's theorem for infinite series, which states that: if $\{a_{m,n}\}_{m=1,n=1}^{\infty}$ is a double-indexed sequence of real numbers, and if $\displaystyle \sum_{(m,n)\in \N\times \N} a_{m,n} $  is absolutely convergent, then

 * $$ \sum_{(m,n)\in\N\times \N}a_{m,n} = \sum_{m=1}^\infty\sum_{n=1}^{\infty} a_{m,n} = \sum_{n=1}^\infty \sum_{m=1}^\infty a_{m,n} $$

Although the Fubini theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize the former as a logical consequence of the latter. This is because some properties of measures, in particular subadditivity, are often proved using the Fubini theorem for infinite series. In this case, the Fubini theorem for integrals is a logical consequence of the Fubini's theorem for infinite series.

History
The special case of the Fubini theorem for continuous functions on a product of closed bounded subsets of real vector spaces was known to Leonhard Euler in the 18th century. In 1904, Henri Lebesgue extended this result to bounded, measurable functions on a product of intervals. Levi conjectured that the theorem could be extended to functions that are integrable rather than bounded and this was proven by Fubini in 1907. In 1909, Leonida Tonelli gave a variation of the Fubini theorem that applies to non-negative functions rather than integrable functions.

Product measures
If $$X$$ and $$Y$$ are measure spaces, there are several natural ways to define a product measure on the product $$X\times Y$$.

In the sense of category theory, measurable sets in the product $$X\times Y$$ of measure spaces are the elements of the σ-algebra generated by the products $$A\times B$$, where $$A$$ is measurable in $$X$$ and $$B$$ is measurable in $$Y$$.

A measure μ on X × Y is called a product measure if μ(A × B) = μ1(A)μ2(B) for measurable subsets A ⊂ X and B ⊂ Y and measures μ1 on X and μ2 on Y. In general, there may be many different product measures on X × Y. Fubini's theorem and Tonelli's theorem both need technical conditions to avoid this complication; the most common way is to assume all measure spaces are σ-finite, in which case there is a unique product measure on X×Y. There is always a unique maximal product measure on X × Y, where the measure of a measurable set is the inf of the measures of sets containing it that are countable unions of products of measurable sets. The maximal product measure can be constructed by applying Carathéodory's extension theorem to the additive function μ such that μ(A × B) = μ1(A)μ2(B) on the ring of sets generated by products of measurable sets. (Carathéodory's extension theorem gives a measure on a measure space that in general contains more measurable sets than the measure space X × Y, so strictly speaking, the measure should be restricted to the σ-algebra generated by the products A × B of measurable subsets of X and Y.)

The product of two complete measure spaces is not usually complete. For example, the product of the Lebesgue measure on the unit interval I with itself is not the Lebesgue measure on the square I × I. There is a variation of Fubini's theorem for complete measures, which uses the completion of the product of measures rather than the uncompleted product.

For integrable functions
Suppose X and Y are σ-finite measure spaces and suppose that X × Y is given the product measure (which is unique as X and Y are σ-finite). Fubini's theorem states that if f is X × Y integrable, meaning that f is a measurable function and $$\int_{X\times Y} |f(x,y)|\,\text{d}(x,y) < \infty,$$ then $$\int_X\left(\int_Y f(x,y)\,\text{d}y\right)\,\text{d}x = \int_Y\left(\int_X f(x,y)\,\text{d}x\right)\,\text{d}y = \int_{X\times Y} f(x,y)\,\text{d}(x,y).$$

The first two integrals are iterated integrals with respect to two measures, respectively, and the third is an integral with respect to the product measure. The partial integrals $\int_Y f(x,y)\,\text{d}y$ and $\int_X f(x,y)\,\text{d}x$  need not be defined everywhere, but this does not matter as the points where they are not defined form a set of measure 0.

If the above integral of the absolute value is not finite, then the two iterated integrals may have different values. See below for an illustration of this possibility.

The condition that X and Y are σ-finite is usually harmless because, in practice almost all measure spaces for which one wishes to use Fubini's theorem are σ-finite. Fubini's theorem has some rather technical extensions to the case when X and Y are not assumed to be σ-finite. The main extra complication in this case is that there may be more than one product measure on X×Y. Fubini's theorem continues to hold for the maximal product measure but can fail for other product measures. For example, there is a product measure and a non-negative measurable function f for which the double integral of |f| is zero but the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli theorem (stated below) can fail on non σ-finite spaces, even for the maximal product measure.

Tonelli's theorem for non-negative measurable functions
, named after Leonida Tonelli, is a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumption that $$|f|$$ has a finite integral is replaced by the assumption that $$f$$ is a non-negative measurable function.

Tonelli's theorem states that if $$ (X, A, \mu) $$ and $$ (Y, B, \nu) $$ are σ-finite measure spaces, while $$ f:X\times Y \to [0,\infty] $$ is non-negative measurable function, then $$\int_X\left(\int_Y f(x,y)\,\text{d}y\right)\,\text{d}x = \int_Y\left(\int_X f(x,y)\,\text{d}x\right)\,\text{d}y = \int_{X\times Y} f(x,y)\,\text{d}(x,y).$$

A special case of Tonelli's theorem is in the interchange of the summations, as in $\sum_x \sum_y a_{xy} = \sum_y \sum_x a_{xy}$, where $$a_{xy}$$ are non-negative for all x and y. The crux of the theorem is that the interchange of order of summation holds even if the series diverges. In effect, the only way a change in order of summation can change the sum is when there exist some subsequences that diverge to $$+\infty$$ and others diverging to $$-\infty$$. With all elements non-negative, this does not happen in the stated example.

Without the condition that the measure spaces are σ-finite, all three of these integrals can have different values. Some authors give generalizations of Tonelli's theorem to some measure spaces that are not σ-finite, but these generalizations often add conditions that immediately reduce the problem to the σ-finite case. For example, one could take the σ-algebra on A×B to be that generated by the product of subsets of finite measure, rather than that generated by all products of measurable subsets, though this has the undesirable consequence that the projections from the product to its factors A and B are not measurable. Another way is to add the condition that the support of f is contained in a countable union of products of sets of finite measures. gives some rather technical extensions of Tonelli's theorem to some non σ-finite spaces. None of these generalizations have found any significant applications outside of abstract measure theory, largely because almost all measure spaces of practical interest are σ-finite.

Fubini–Tonelli theorem
Combining Fubini's theorem with Tonelli's theorem gives the Fubini–Tonelli theorem. Often just called Fubini's theorem, it states that if $$X$$ and $$Y$$ are σ-finite measure spaces, and if $$f$$ is a measurable function, then $$\int_X\left(\int_Y |f(x,y)|\,\text{d}y\right)\,\text{d}x=\int_Y\left(\int_X |f(x,y)|\,\text{d}x\right)\,\text{d}y=\int_{X\times Y} |f(x,y)|\,\text{d}(x,y)$$ Furthermore, if any one of these integrals is finite, then $$\int_X\left(\int_Y f(x,y)\,\text{d}y\right)\,\text{d}x=\int_Y\left(\int_X f(x,y)\,\text{d}x\right)\,\text{d}y=\int_{X\times Y} f(x,y)\,\text{d}(x,y).$$

The absolute value of $$f$$ in the conditions above can be replaced by either the positive or the negative part of $$f$$; these forms include Tonelli's theorem as a special case as the negative part of a non-negative function is zero and so has finite integral. Informally, all these conditions say that the double integral of $$f$$ is well defined, though possibly infinite.

The advantage of the Fubini–Tonelli over Fubini's theorem is that the repeated integrals of $$|f|$$ may be easier to study than the double integral. As in Fubini's theorem, the single integrals may fail to be defined on a measure 0 set.

For complete measures
The versions of Fubini's and Tonelli's theorems above do not apply to integration on the product of the real line $$\R$$ with itself with Lebesgue measure. The problem is that Lebesgue measure on $$\R\times\R$$ is not the product of Lebesgue measure on $$\R$$ with itself, but rather the completion of this: a product of two complete measure spaces $$X$$ and $$Y$$ is not in general complete. For this reason, one sometimes uses versions of Fubini's theorem for complete measures: roughly speaking, one replaces all measures with their completions. The various versions of Fubini's theorem are similar to the versions above, with the following minor differences:
 * Instead of taking a product $$X\times Y$$ of two measure spaces, one takes the completion of some product.
 * If $$f$$ is measurable on the completion of $$X\times Y$$ then its restrictions to vertical or horizontal lines may be non-measurable for a measure zero subset of lines, so one has to allow for the possibility that the vertical or horizontal integrals are undefined on a set of measure 0 because they involve integrating non-measurable functions. This makes little difference, because they can already be undefined due to the functions not being integrable.
 * One generally also assumes that the measures on $$X$$ and $$Y$$ are complete, otherwise the two partial integrals along vertical or horizontal lines may be well-defined but not measurable. For example, if $$f$$ is the characteristic function of a product of a measurable set and a non-measurable set contained in a measure 0 set then its single integral is well defined everywhere but non-measurable.

Proofs
Proofs of the Fubini and Tonelli theorems are necessarily somewhat technical, as they have to use a hypothesis related to σ-finiteness. Most proofs involve building up to the full theorems by proving them for increasingly complicated functions, with the steps as follows.
 * 1) Use the fact that the measure on the product is a product measure to prove the theorems for the characteristic functions of rectangles.
 * 2) Use the condition that the spaces are σ-finite (or some related condition) to prove the theorem for the characteristic functions of measurable sets. This also covers the case of simple measurable functions (measurable functions taking only a finite number of values).
 * 3) Use the condition that the functions are measurable to prove the theorems for positive measurable functions by approximating them by simple measurable functions.  This proves Tonelli's theorem.
 * 4) Use the condition that the functions are integrable to write them as the difference of two positive integrable functions and apply Tonelli's theorem to each of these. This proves Fubini's theorem.

Riemann integrals
For Riemann integrals, Fubini's theorem is proven by refining the partitions along the x-axis and y-axis as to create a joint partition of the form $$[x_i,x_{i+1}] \times [y_j,y_{j+1}]$$, which is a partition over $$X\times Y$$. This is used to show that the double integrals of either order are equal to the integral over $$X\times Y$$.

Counterexamples
The following examples show how Fubini's theorem and Tonelli's theorem can fail if any of their hypotheses are omitted.

Failure of Tonelli's theorem for non σ-finite spaces
Suppose that X is the unit interval with the Lebesgue measurable sets and Lebesgue measure, and Y is the unit interval with all subsets measurable and the counting measure, so that Y is not σ-finite. If f is the characteristic function of the diagonal of X×Y, then integrating f along X gives the 0 function on Y, but integrating f along Y gives the function 1 on X. So, the two iterated integrals are different. This shows that Tonelli's theorem can fail for spaces that are not σ-finite no matter what product measure is chosen. The measures are both decomposable, showing that Tonelli's theorem fails for decomposable measures (which are slightly more general than σ-finite measures).

Failure of Fubini's theorem for non-maximal product measures
Fubini's theorem holds for spaces even if they are not assumed to be σ-finite provided one uses the maximal product measure. In the example above, for the maximal product measure, the diagonal has infinite measure so the double integral of |f| is infinite, and Fubini's theorem holds vacuously. However, if we give X×Y the product measure such that the measure of a set is the sum of the Lebesgue measures of its horizontal sections, then the double integral of |f| is zero, but the two iterated integrals still have different values. This gives an example of a product measure where Fubini's theorem fails.

This gives an example of two different product measures on the same product of two measure spaces. For products of two σ-finite measure spaces, there is only one product measure.

Failure of Tonelli's theorem for non-measurable functions
Suppose that X is the first uncountable ordinal, with the finite measure where the measurable sets are either countable (with measure 0) or the sets of countable complement (with measure 1). The (non-measurable) subset E of X×X given by pairs (x, y) with x<y is countable on every horizontal line and has countable complement on every vertical line. If f is the characteristic function of E then the two iterated integrals of f are defined and have different values 1 and 0. The function f is not measurable. This shows that Tonelli's theorem can fail for non-measurable functions.

Failure of Fubini's theorem for non-measurable functions
A variation of the example above shows that Fubini's theorem can fail for non-measurable functions even if |f| is integrable and both repeated integrals are well defined: if we take f to be 1 on E and –1 on the complement of E, then |f| is integrable on the product with integral 1, and both repeated integrals are well defined, but have different values 1 and –1.

Assuming the continuum hypothesis, one can identify X with the unit interval I, so there is a bounded non-negative function on I×I whose two iterated integrals (using Lebesgue measure) are both defined but unequal. This example was found by. The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory. The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while showed that it is consistent with ZFC that a strong Fubini-type theorem for [0,1] does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.

Failure of Fubini's theorem for non-integrable functions
Fubini's theorem tells us that (for measurable functions on a product of σ-finite measure spaces) if the integral of the absolute value is finite, then the order of integration does not matter; if we integrate first with respect to x and then with respect to y, we get the same result as if we integrate first with respect to y and then with respect to x. The assumption that the integral of the absolute value is finite is "Lebesgue integrability", and without it the two repeated integrals can have different values.

A simple example to show that the repeated integrals can be different in general is to take the two measure spaces to be the positive integers, and to take the function f(x,y) to be 1 if x = y, −1 if x = y + 1, and 0 otherwise. Then the two repeated integrals have different values 0 and 1.

Another example is as follows for the function $$\frac{x^2-y^2}{(x^2+y^2)^2} = -\frac{\partial^2}{\partial x\,\partial y} \arctan(y/x).$$ The iterated integrals

$$\int_{x=0}^1\left(\int_{y=0}^1\frac{x^2-y^2}{(x^2+y^2)^2}\,\text{d}y\right)\,\text{d}x = \frac{\pi}{4}$$ and $$\int_{y=0}^1\left(\int_{x=0}^1\frac{x^2-y^2}{(x^2+y^2)^2}\,\text{d}x\right)\,\text{d}y=-\frac{\pi}{4}$$ have different values. The corresponding double integral does not converge absolutely (in other words the integral of the absolute value is not finite): $$\int_0^1\int_0^1 \left|\frac{x^2-y^2}{\left(x^2 + y^2\right)^2}\right|\,\text{d}y\,\text{d}x=\infty.$$

Product of two integrals
For the product of two integrals with lower limit zero and a common upper limit we have the following formula:
 * {| class = "wikitable"

 * $$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr] = \int_{0}^{1} \int_{0}^{u} x\,v(xy) \,w(x) + x\,v(x) \,w(xy) \,\mathrm{d}x \,\mathrm{d}y $$
 * }

Proof
Let $$ V(x) $$ and $$ W(x) $$ are primitive functions of the functions $$ v(x) $$ and $$ w(x ) $$ respectively, which pass through the origin:

 * $$\int_{0}^{u} v(x) \,\mathrm{d}x = V(u),

\quad \quad \int_{0}^{u} w(x) \,\mathrm{d}x = W(u)$$

Therefore, we have

$$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr] = V(u) W(u) $$

By the product rule, the derivative of the right-hand side is

$$\frac{\mathrm{d}}{\mathrm{d}x} \bigl[V(x) W(x)\bigr] = V(x)w(x) + v(x)W(x)$$

and by integrating we have:

$$\int_{0}^{u} V(x)w(x) + v(x)W(x) \,\mathrm{d}x=V(u) W(u) $$

Thus, the equation from the beginning we get:

$$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} V(x)w(x) + v(x)W(x) \,\mathrm{d}x $$

Now, we introduce a second integration parameter $$y$$ for the description of the antiderivatives $$V(x)$$ and $$W(x)$$:

 * $$\int_{0}^{1} x\,v(xy) \,\mathrm{d}y = \biggl[V(xy)\biggr]_{y = 0}^{y = 1} = V(x) $$

 * $$\int_{0}^{1} x\,w(xy) \,\mathrm{d}y = \biggl[W(xy)\biggr]_{y = 0}^{y = 1} = W(x) $$

By insertion, a double integral appears:

$$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} \biggl[\int_{0}^{1} x\,v(xy) \,\mathrm{d}y\biggr]w(x) + v(x)\biggl[\int_{0}^{1} x\,w(xy) \,\mathrm{d}y\biggr] \,\mathrm{d}x $$

Functions that are foreign to the concerned integration parameter can be imported into the inner function as a factor:

$$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} \biggl[\int_{0}^{1} x\,v(xy) \,w(x) \,\mathrm{d}y\biggr] + \biggl[\int_{0}^{1} x\,v(x)\,w(xy) \,\mathrm{d}y\biggr] \,\mathrm{d}x $$

In the next step, the sum rule is applied to the integrals:

$$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{u} \int_{0}^{1} x\,v(xy) \,w(x) + x\,v(x)\,w(xy) \,\mathrm{d}y \,\mathrm{d}x $$

And finally, we use the Fubini theorem

$$\biggl[\int_{0}^{u} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{u} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{1} \int_{0}^{u} x\,v(xy) \,w(x) + x\,v(x)\,w(xy) \,\mathrm{d}x \,\mathrm{d}y $$

Arcsine Integral
The Arcsine Integral, also called the Inverse Sine Integral, is a function that cannot be represented by elementary functions. The Arcsine Integral does, however, have some elementary function values. These values may be determined by integrating the derivative of the arcsine integral, which is the quotient of the Arcsine divided by the Identity Function-the Cardinalized Arcsine. The Arcsine Integral is exactly the original antiderivative of the Cardinalized Arcsine. For the integration of this function, Fubini's theorem serves as a key, which unlocks the integral by exchanging the order of the integration parameters. When applied correctly, Fubini's theorem leads directly to an antiderivative function that can be integrated in an elementary way, which is shown in cyan in the following equation chain:

$$\operatorname{Si}_{2}(1) = \int_{0}^{1} \frac{1}{x}\arcsin(x) \,\mathrm{d}x = {\color{blue}\int_{0}^{1}} {\color{green}\int_{0}^{1}} \frac{\sqrt{1-x^2}\,y}{(1- x^2 y^2)\sqrt{1-y^2}} \,{\color{green}\mathrm{d}y} \,{\color{blue}\mathrm{d}x} =$$ $$= {\color{green}\int_{0}^{1}} {\color{blue}\int_{0}^{1}} \frac{\sqrt{1-x^2}\ ,y}{(1-x^2 y^2)\sqrt{1-y^2}} {\color{blue}\,\mathrm{d}x} {\color{green}\,\mathrm{ d}y} =\int_{0}^{1} \frac{\pi\,y}{2\sqrt{1-y^2}(1+\sqrt{1-y^2}\,)} \,\mathrm{d}y =$$ $$= {\color{RoyalBlue}\biggl\{ \frac{\pi}{2} \ln\bigl[2 \bigl(1 + \sqrt{1 - y^2}\,\bigr)^ {-1}\bigr] \biggr\}_{y = 0}^{y = 1}} = \frac{\pi}{2}\ln(2)$$

Dirichlet Eta Function
The Dirichlet series defines the Dirichlet Eta Function as follows:

$$ \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} = 1 - \frac{1}{2^ s} + \frac{1}{3^s} - \frac{1}{4^s} + \frac{1}{5^s} - \frac{1}{6^s} \pm \cdots $$

The value η(2) is equal to π²/12 and this can be proven with Fubini's theorem in this way:

$$ \eta(2) = \sum_{n = 1}^\infty (-1)^{n-1}\frac{1}{n^2} = \sum_{n = 1}^\infty \int_{0}^{1} (-1)^{n-1}\frac{1}{n} {x}^{n-1} \,\mathrm{d}x = \int_{0 }^{1} \sum_{n = 1}^\infty (-1)^{n-1}\frac{1}{n} {x}^{n-1} \,\mathrm{d}x = \int_{0}^{1} \frac{1}{x}\ln(x+1) \,\mathrm{d}x $$

The integral of the product of the Reciprocal Function and the Natural Logarithm of the Successor Function is a Polylogarithmic Integral and it cannot be represented by elementary function expressions. Fubini's theorem again unlocks this integral in a combinatorial way. This works by carrying out double integration on the basis of Fubini's theorem used on an additive combination of fractionally rational functions with fractions of linear and square denominators:

$$ \int_{0}^{1} \frac{1}{x}\ln(x+1) \,\mathrm{d}x = {\color{blue}\int_{0}^{ 1}} {\color{green}\int_{0}^{1}} \frac{4}{3(x^2+2xy+1)} + \frac{2x}{3(x^2y+1 )} - \frac{1}{3(xy+1)} \,{\color{green}\mathrm{d}y} \,{\color{blue}\mathrm{d}x} = $$

$$ = {\color{green}\int_{0}^{1}} {\color{blue}\int_{0}^{1}} \frac{4}{3(x^2+2xy +1)} + \frac{2x}{3(x^2y+1)} - \frac{1}{3(xy+1)} \,{\color{blue}\mathrm{d}x} \,{\color{green}\mathrm{d}y} = \int_{0}^{1} \frac{2\arccos(y)}{3\sqrt{1-y^2}} \,\mathrm {d}y = $$

$$ = {\color{RoyalBlue}\biggl[\frac{\pi^2}{12}-\frac{1}{3}\arccos(y)^2 \biggr]_{y = 0} ^{y = 1}} = \frac{\pi^2}{12} $$

This way of working out the integral of the Cardinalized natural logarithm of the successor function was discovered by James Harper and it is described in his work Another simple proof of 1 + 1/2² + 1/3² + ... = π²/6 accurately.

The original antiderivative, shown here in cyan, leads directly to the value of η(2):

 * $$ \eta(2) = \frac{\pi^2}{12} $$

Integrals of Complete Elliptic Integrals
The improper integral of the Complete Elliptic Integral of first kind K takes the value of twice the Catalan constant accurately. The antiderivative of that K-integral belongs to the so-called Elliptic Polylogarithms. The Catalan constant can only be obtained via the Arctangent Integral, which results from the application of Fubini's theorem:

$$ \int_{0}^{1} K(x) \,\mathrm{d}x ={\color{blue}\int_{0}^{1}} {\color{green}\int_ {0}^{1}} \frac{1}{\sqrt{(1 - x^2 y^2)(1 - y^2)}} \,{\color{green}\mathrm{d}y }\,{\color{blue}\mathrm{d}x} = {\color{green}\int_{0}^{1}}{\color{blue}\int_{0}^{1}} \frac{1}{\sqrt{(1 - x^2 y^2)(1 - y^2)}} \,{\color{blue}\mathrm{d}x} \,{\color{green} \mathrm{d}y}= $$

$$ = \int_{0}^{1} \frac{\arcsin(y)}{y\sqrt{1 - y^2}} \,\mathrm{d}y = {\color{RoyalBlue} \biggl\{ 2\,\mathrm{Ti}_{2} \bigl[ y\bigl(1 + \sqrt{1 - y^2}\,\bigr)^{-1} \bigr] \biggr\}_{y = 0}^{y = 1}} = 2\,\mathrm{Ti}_{2}(1) =2\beta(2) =2\,C $$

This time, the expression now in royal cyan color tone is not elementary, but it leads directly to the equally non-elementary value of the "Catalan constant" using the Arctangent Integral, also called Inverse Tangent Integral.

The same procedure also works for the Complete Elliptic Integral of the second kind E in the following way:

$$ \int_{0}^{1} E(x) \,\mathrm{d}x ={\color{blue}\int_{0}^{1}} {\color{green}\int_ {0}^{1}} \frac{\sqrt{1 - x^2 y^2}}{\sqrt{1 - y^2}} \,{\color{green}\mathrm{d}y} \,{\color{blue}\mathrm{d}x} = {\color{green}\int_{0}^{1}}{\color{blue}\int_{0}^{1}} \frac {\sqrt{1 - x^2 y^2}}{\sqrt{1 - y^2}} \,{\color{blue}\mathrm{d}x} \,{\color{green}\mathrm {d}y}= $$ $$ = \int_{0}^{1} \biggl[\frac{\arcsin(y)}{2y\sqrt{1 - y^2}} + \frac{1}{2}\biggr] \,\mathrm{d}y = {\color{RoyalBlue}\biggl\{ \mathrm{Ti}_{2} \bigl[ y\bigl(1 + \sqrt{1 - y^2}\,\bigr )^{-1} \bigr] + \frac{1}{2} y \biggr\}_{y = 0}^{y = 1}} = \mathrm{Ti}_{2}(1) + \frac{1}{2} =\beta(2) + \frac{1}{2} =C + \frac{1}{2} $$

Double execution for the Exponential Integral Function
The Mascheroni constant emerges as the Improper Integral from zero to infinity at the integration on the product of negative Natural Logarithm and the Exponential reciprocal. But it is also the improper integral within the same limits on the Cardinalized Difference of the reciprocal of the Successor Function and the Exponential Reciprocal:

$$\gamma = {\color{WildStrawberry}\int_0^\infty \frac{-\ln(x)}{\exp(x)}\,\mathrm{d}x} = {\color{cornflowerblue }\int_{0}^{\infty} \frac{1}{x}\biggl[\frac{1}{x + 1}-\exp(-x)\biggr] \,\mathrm{d}x } $$

The concord of these two integrals can be shown by successively executing the Fubini's Theorem twice and by leading this double execution of that theorem over the identity to an integral of the complementary Exponential Integral Function:

This is how the complementary integral exponential function is defined:

 * $$\mathrm{E}_{1}(x) = \exp(-x)\int_{0}^{\infty} \frac{\exp(-xy)}{y+1} \, \mathrm{d}y $$

This is the derivative of that function:

 * $$\frac{\mathrm{d}}{\mathrm{d}x} \,\mathrm{E}_{1}(x) = -\frac{1}{x}\exp(-x) $$

First implementation of Fubini's theorem:

This integral from a construction of the integral exponential function leads to the integral from the negative Natural Logarithm and the Exponential Reciprocal:

$$ \gamma= {\color{WildStrawberry}\int_{0}^{\infty} -\exp(-y)\ln(y) \,\mathrm{d}y} = {\color{green}\int_{0}^{\infty}} {\color{blue}\int_{0}^{\infty}} \exp(-y)\bigl(\frac{1}{x + y} - \frac{1}{x + 1}\bigr) \,{\color{blue}\mathrm{d}x} \,{\color{green}\mathrm{d}y} = $$ $$ = {\color{blue}\int_{0}^{\infty}}{\color{green}\int_{0}^{\infty}} \exp(-y)\bigl(\frac{1}{x + y} - \frac{1}{x + 1}\bigr) \,{\color{green}\mathrm{d}y} \,{\color{blue}\mathrm{d}x} = {\color{Dandelion}\int_{0}^{\infty} \biggl[\exp(x)\,\mathrm{E}_{1}(x) - \frac{1}{x + 1}\biggr] \,\mathrm{d}x} $$

Second implementation of Fubini's theorem:

The previously described integral from the described cardinalized difference leads to the previously mentioned integral from the Exponential Integral function:

$$ \gamma = {\color{Dandelion}\int_{0}^{\infty} \biggl[\exp(x)\,\mathrm{E}_{1}(x) - \frac{1}{x + 1}\biggr] \,\mathrm{d}x} = {\color{blue}\int_{0}^{\infty}}{\color{blueviolet}\int_{0}^{\infty}} \exp(-xz)\biggl[\frac{1}{z + 1}-\exp(-z)\biggr] \,{\color{blueviolet}\mathrm{d}z} \,{\color{blue}\mathrm{d}x} = $$ $$ = {\color{blueviolet}\int_{0}^{\infty}}{\color{blue}\int_{0}^{\infty}} \exp(-xz)\biggl[\frac{1}{z + 1}-\exp(-z)\biggr] \,{\color{blue}\mathrm{d}x} \,{\color{blueviolet}\mathrm{d}z} ={\color{cornflowerblue}\int_{0}^{\infty} \frac{1}{z}\biggl[\frac{1}{z + 1}-\exp(-z)\biggr] \,\mathrm{d}z} $$

In principle, products from exponential functions and fractionally rational functions can be integrated like this:

$$\frac{\exp(-ax)}{bx + c} = \frac{\mathrm{d}}{\mathrm{d}x} \biggl\{-\frac{1}{b} \exp\bigl(\frac{ac}{b}\bigr)\,\mathrm{E}_{1}\bigl[\frac{a}{b}\bigl(bx + c\bigr)\bigr] \biggr\} $$ $$\int_0^{\infty} \frac{\exp(-ax)}{bx + c} \,\mathrm{d}x = \biggl\{-\frac{1}{b}\exp \bigl(\frac{ac}{b}\bigr)\,\mathrm{E}_{1}\bigl[\frac{a}{b}\bigl(bx + c\bigr)\bigr]\biggr \}_{x = 0}^{x = \infty} = \frac{1}{b}\exp\bigl(\frac{ac}{b}\bigr) \,\mathrm{E}_{1 }\bigl(\frac{ac}{b}\bigr) $$

In this way it is shown accurately by using the Fubini's Theorem twice that these integrals are indeed identical to each other.

Gauss curve integral
Now this formula for the squaring of an integral is set up:

 * {| class = "wikitable"

 * $$\biggl[\int_{0}^{\infty} f(x) \,\mathrm{d}x\biggr]^2 = \int_{0}^{1} \int_{0}^{\infty} 2x\,f(x) \,f(xy)\,\mathrm{d}x \,\mathrm{d}y $$
 * }

This chain of equations can then be generated accordingly:

$$\biggl[\int_{0}^{\infty} \exp(-x^2) \,\mathrm{d}x\biggr]^2 = \int_{0}^{1} \int_{0}^{\infty} 2x\exp(-x^2)\exp(-x^2 y^2) \,\mathrm{d}x\,\mathrm{d}y = \int_{0}^{1} \int_{0}^{\infty} 2x\exp\bigl[-x^2 (y^2 + 1)\bigr] \,\mathrm{d}x\,\mathrm{d}y = $$

$$= \int_{0}^{1} \biggl\{\frac{1}{y^2 + 1} - \frac{1}{y^2 + 1}\exp\biggl[-x^2(y^2 + 1)\biggr]\biggr\}_{x = 0}^{x = \infty} \,\mathrm{d}y = \int_{0}^{1} \frac{1}{y^2 + 1} \,\mathrm{d}y = \arctan(1) = \frac{\pi}{4} $$

For the integral of the Gauss curve this value can be generated:

 * $$\int_{0}^{\infty} \exp(-x^2) \,\mathrm{d}x = \frac{1}{2}\sqrt{\pi} $$

Dilogarithm of one
Now another formula for the squaring of an integral is set up again:

 * {| class = "wikitable"

 * $$\biggl[\int_{0}^{1} g(x) \,\mathrm{d}x\biggr]^2 = \int_{0}^{1} \int_{0}^{1} 2x\,g(x) \,g(xy)\,\mathrm{d}x \,\mathrm{d}y $$
 * }

So this chain of equations applies as a new example:

$$\frac{\pi^2}{4} = \arcsin(1)^2 = \biggl[\int_{0}^{1} \frac{1}{\sqrt{1 - x^2}} \,\mathrm{d}x\biggr]^2 = \int_{0}^{1} \int_{0}^{1} \frac{2x}{\sqrt{(1 - x^2)(1 - x^2 y^2)}} \,\mathrm{d}x \,\mathrm{d}y = $$

$$= \int_{0}^{1} \biggl[\frac{2}{y}\operatorname{artanh}(y) - \frac{2}{y}\operatorname{artanh}\biggl(\frac{\sqrt{1 - x^2}\,y}{\sqrt{1 - x^2 y^2}}\biggr)\biggr]_{x = 0}^{x = 1}\,\mathrm{d}y = \int_{0}^{1} \frac{2}{y}\operatorname{artanh}(y) \,\mathrm{d}y = $$

$$= \biggl[2\,\mathrm{Li}_{2}(y) - \frac{1}{2}\,\mathrm{Li}_{2}(y^2)\biggr]_{y = 0}^{y = 1} = \frac{3}{2}\,\mathrm{Li}_{2}(1) $$

For the Dilogarithm of one this value appears:

 * $$\mathrm{Li}_{2}(1) = \frac{\pi^2}{6} $$

In this way the Basel problem can be solved.

Legendre's relation
In this next example, the more generalized form of the equation is used again as a mold:

 * {| class = "wikitable"

 * $$\biggl[\int_{0}^{1} v(x) \,\mathrm{d}x\biggr]\biggl[\int_{0}^{1} w(x) \,\mathrm{d}x\biggr]= \int_{0}^{1} \int_{0}^{1} \left(x\,v(xy) \,w(x) + x\,v(x)\,w(xy)\right) \,\mathrm{d}x \,\mathrm{d}y $$
 * }

The following integrals can be computed by using the incomplete Elliptic Integrals of the first and second kind as antiderivatives and these integrals have values that can be represented with Complete Elliptic Integrals:

 * $$ \int_{0}^{1} \frac{1}{\sqrt{1 - x^4}} \,\mathrm{d}x = $$

$$ \biggl\{ - \frac{1}{2}\sqrt{2}\,F\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] \biggr\}_{x = 0}^{x = 1} = \frac{1}{2}\sqrt{2}\,K\bigl(\frac{1}{2}\sqrt{2}\bigr) $$

 * $$\int_{0}^{1} \frac{x^2}{\sqrt{1 - x^4}} \,\mathrm{d}x =$$

$$\biggl\{\frac{1}{2}\sqrt{2}\,F\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] - \sqrt{2}\,E\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] \biggr\}_{x = 0}^{x = 1} = \frac{1}{2}\sqrt{2}\biggl[2\,E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac{1}{2}\sqrt{2}\bigr)\biggr] $$

By Insertion of these two integrals into the mentioned mold, this chain of equations results:

$$\biggl[\int_{0}^{1} \frac{1}{\sqrt{1 - x^4}} \,\mathrm{d}x\biggr]\biggl[\int_{0}^{1} \frac{x^2}{\sqrt{1 - x^4}} \,\mathrm{d}x\biggr] =\int_{0}^{1} \int_{0}^{1} \frac{x^3 (y^2+1)}{\sqrt{(1 - x^4)(1 - x^4 y^4)}} \,\mathrm{d}x\,\mathrm{d}y = $$

$$= \int_{0}^{1} \biggl\{\frac{y^2 + 1}{2\,y^2}\biggl[\text{artanh}\bigl(y^2\bigr) - \text{artanh}\biggl(\frac{\sqrt{1 - x^4}\,y^2}{\sqrt{1 - x^4 y^4}}\biggr)\biggr]\biggr\}_{x = 0}^{x = 1}\,\mathrm{d}y = \int_{0}^{1} \frac{y^2 + 1}{2\,y^2}\,\mathrm{artanh}\bigl(y^2\bigr) \,\mathrm{d}y= $$

$$= \biggl[\arctan(y) - \frac{1 - y^2}{2\,y}\,\mathrm{artanh}\bigl(y^2\bigr)\biggr]_{y = 0}^{y = 1} = \arctan(1) = \frac{\pi}{4} $$

For the Lemniscatic special case of Legendre's relation, this result emerges:

 * $$K\bigl(\frac{1}{2}\sqrt{2}\bigr)\biggl[2\,E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac{1}{2}\sqrt{2}\bigr)\biggr] = \frac{\pi}{2} $$

title: Fubini's theorem


text:
Ingelheim am Rhein

Ingelheim, officially Ingelheim am Rhein (Ingelheim upon Rhine), is a town in the Mainz-Bingen district in the Rhineland-Palatinate state of Germany. The town sprawls along the Rhine's left bank. It has been Mainz-Bingen's district seat since 1996.

From the later half of the 8th century, the Ingelheim Imperial Palace, which served emperors and kings as a lodging and a ruling seat until the 11th century, was to be found here.

Etymology
The typically Rhenish-Hessian placename ending —heim might well go back to Frankish times, that is to say, likely as far back as the 5th or 6th century. Settlements or estates then took their lords’ names and were given this suffix, which means "home" in German. The name is recorded in later documents as Ingilinhaim, Ingilinheim (782), Ingilenhaim, Engelheim, Hengilonheim, Engilonheim (822), Engilinheim (826), Hingilinheim (855), Ingilunheim (874), Ingulinheim (889), Ingelesheim (891), Ingelenheim (940), Anglia sedes (1051), Ingilheim and Ingelnheim (1286), among other forms.

Since 1269, a distinction has been made between Nieder-Ingelheim and Ober-Ingelheim (Lower and Upper Ingelheim).

Location
Ingelheim am Rhein lies in the north of Rhein Hessen on the so-called Rhein Knee, west of the state capital, Mainz. The Rhein forms the town's northern limit. Southwards, the town stretches into the valley of the river Selz, which empties into the Rhein in the constituent community of Frei-Weinheim or Ingelheim-Nord ("North").

The constituent communities of Ingelheim-Mitte and Ingelheim-Süd ("Middle" and "South") are nestled against the corner of the so-called Mainzer Berg (Ingelheim) ("Mainz Mountain").

The municipal area's lowest point is the harbour on the Rhein at 80.8 m above sea level. The two highest points are the Mainzer Berg at 247.8 m above sea level and the Westerberg (Rheinhessen) at 247.5 m above sea level.

An obelisk on the south side of the village in direction Wackernheim, marks the road begun by Charlemagne, and completed by Napoleon. From this point a fine prospect of the entire Rheingau could be obtained.

Municipal area’s extent
The municipal area's north-south extent is 7.9 km, while the east-west extent is 5 km.

Neighbouring municipalities
Clockwise from the north, these are Geisenheim, Oestrich-Winkel on the Rhine's right bank, and on the left bank Budenheim, Finthen, the Verbandsgemeinde of Nieder-Olm, Schwabenheim, Gau-Algesheim (both belonging to the Verbandsgemeinde of Gau-Algesheim) and Bingen am Rhein. Since 1 July 2019 Wackernheim and Heidesheim are incorporated into the city of Ingelheim.

Constituent communities
Ingelheim is currently divided into six Stadtteile: Ingelheim-Mitte, Ingelheim-Nord, Ingelheim-Süd, Sporkenheim, Groß-Winternheim and Ingelheim-West. Before Ingelheim became a town in 1939, the first three centres bore the names Nieder-Ingelheim, Frei-Weinheim and Ober-Ingelheim. Official changes notwithstanding, the old names are still quite often used.

Climate
The town lies in the temperate zone. The average yearly temperature in Ingelheim is 9.8 °C. The warmest months are July and August with average temperatures of 18.0 and 18.5 °C respectively, and the coldest month is January at 1.0 °C on average. The most precipitation falls in June and August with an average of 64 mm, and the least in March with an average of 31 mm. Like all Rhenish Hesse, Ingelheim, too, is sheltered from the weather by the Hunsrück, the Taunus, the Odenwald and the Donnersberg, thereby limiting the yearly precipitation to only 560 mm.

History
The Ingelheim area was already settled in prehistoric times. The place first earned itself particular importance, though, only under Charlemagne and his successors. Charlemagne had built the Ingelheim Imperial Palace (Ingelheimer Kaiserpfalz) here, where synods and Imperial diets were held in the time that followed. His son and successor, Emperor Louis the Pious, died on 20 June 840 in Ingelheim.

In the High and Late Middle Ages, the Palatinate's, and thereby also Ingelheim's, importance shrank.

For German justice history, the Ingelheimer Oberhof ("Ingelheim Upper Court") is of particular importance, as a unique collection of judgments from the 15th and 16th centuries that it handed down has been preserved.

Late 19th century Ingelheim was the residence of the Dutch writer Multatuli (Eduard Douwes Dekker).

In 1939, the formerly self-administering municipalities of Nieder-Ingelheim, Ober-Ingelheim and Frei-Weinheim were merged into the Town of Ingelheim am Rhein. From the Second World War, Ingelheim emerged as the only unscathed town between Mainz and Koblenz. Today, Ingelheim is a middle centre in Rhineland-Palatinate, a Great District-Bound Town (Große kreisangehörige Stadt – a status deriving from the Rhineland-Palatinate Municipal Order) and the seat of district administration for Mainz-Bingen.

Furthermore, Ingelheim harbours the business Boehringer Ingelheim which is active worldwide.

Religion
In 2004, 36% of Ingelheim's inhabitants belonged to the Lutheran faith, and 34% were Catholic, while 24% were without any religious faith; from 2% of the population, no data were forthcoming.

The six Catholic parishes belong, within the Roman Catholic Diocese of Mainz to the Deanery of Bingen.

The five Protestant parishes of the EKHN belong to the Provostship (Propstei) of Mainz, and within this to the Deanery of Ingelheim.

Besides these, the Baptists, Religious humanists and Muslims each have small communities in Ingelheim, as do the Jehovah's Witnesses and Buddhists. Until 1942 there was a Jewish community, whose beginnings went back to the 16th century. About 1850, roughly 200 Jewish inhabitants lived in Ober-Ingelheim, and by 1933 there were still 134 all together in Oberingelheim and Niederingelheim. In 1840 and 1841, a synagogue that was important to architectural history was built. It was dedicated on 27 August 1841 and destroyed on 9 November 1938 – Kristallnacht. Many Jewish inhabitants lost their lives after being deported to the death camps during the time of the Third Reich.

Amalgamations
On 22 April 1972 the municipality of Groß-Winternheim was amalgamated. The former municipalities Heidesheim am Rhein and Wackernheim were merged into Ingelheim am Rhein on 1 July 2019.

Town council
The municipal election held in 2004 yielded the following results:

Mayor
In the last mayoral elections, held on 26 May 2019, Ralf Claus, mayor of Ingelheim since 2012, was reelected as mayor:

Results of council elections since 1946

 * 1946
 * CDU: 42.2%
 * SPD: 26%
 * KPD: 9.2%
 * Liste Gemünden/Gaul: 22.6%
 * Eligible voters: 6,899
 * Voter turnout: 88.6%

 * 1948
 * CDU: 35.1%
 * SPD: 33.1%
 * DP: 25.3%
 * KP: 6.5%
 * 9 November 1952
 * Freie Bürgerliste Rausch: 40.1%, 2,882 votes — 11 seats
 * SPD: 23.04%, 1,656 votes — 6 seats
 * CDU: 22.43%, 1,612 votes — 6 seats
 * FDP: 10.21%, 734 votes — 2 seats
 * KPD: 4.2%, 303 votes
 * Eligible voters: 9,488
 * Voter turnout: 77.76%, 7378 votes, 7,187 valid votes

 * 1956
 * SPD: 36.79%, 2,611 votes — 9 seats
 * CDU: 27.06%, 1,920 votes — 7 seats
 * Wählergruppe Bambach: 24.45%, 1,735 votes — 6 seats
 * FDP: 11.7%, 830 votes — 3 seats
 * Eligible voters: 9,979
 * Voter turnout: 72.62%, 7,247 votes, 7,096 valid votes

 * 23 October 1960
 * SPD: 42.61%, 3,114 votes — 11 seats
 * CDU: 36.65%, 2,679 votes — 10 seats
 * FDP: 16.92%, 1,237 votes — 2 seats
 * Wählergruppe Kaufmann: 3.82%, 279 votes
 * Eligible voters: 10,695
 * Voter turnout: 70.14%, 7,502 votes, 7,309 valid votes

 * 25 October 1964
 * SPD: 51.7% — 13 seats (absolute majority)
 * CDU: 34.7%, 2,800 votes — 9 seats
 * FDP: 13.6%, 1,098 votes — 3 seats
 * Eligible voters: 11,369 (50a CDU) 11,312 (40a Ing)
 * Voter turnout: 72.77%, 8,231 votes (50a CDU) 8,232 (40a Ing)

 * 8 June 1969
 * CDU: 37.15%, 3,397 votes — 12 seats
 * SPD: 34.45%, 3,150 votes — 11 seats
 * FDP: 10.45%, 956 votes — 3 seats
 * Freie Wählergruppe Kaege: 17.95%, 1,641 votes — 5 seats
 * Eligible voters: 12,295
 * Voter turnout: 75.51%, 9,309 votes, 9,144 valid votes

 * 23 April 1972
 * SPD: 41.99%, 4,263 votes — 14 seats (according to 40a 4,264)
 * CDU: 38.92%, 3,952 votes — 12 seats
 * FDP: 8.79%, 892 votes — 2 seats
 * Wählergruppe Kaege: 10.28%, 1,044 votes — 3 seats
 * Eligible voters: 13,992
 * Voter turnout: 73.46%, 10,280 votes, 10,153 valid votes

 * 17 March 1974
 * CDU: 46.6%, 5,092 votes — 17 seats (40a: 46.40%)
 * SPD: 34.34%, 3,769 votes — 12 seats
 * FDP: 10.26%, 1,126 votes — 3 seats
 * FWG: 8.98%, 986 votes — 3 seats
 * Eligible voters: 14,027
 * Voter turnout: 79.17%, 11,106 votes, 10,973 valid votes

 * 10/11 June 1979
 * SPD: 42.12%, 4,322 votes — 14 seats
 * CDU: 41.52%, 4,261 votes — 13 seats
 * FDP: 8.21%, 842 votes — 2 seats
 * FWG: 8.15%, 837 votes — 2 seats
 * Eligible voters: 14,238
 * Voter turnout: 73.54%, 10,470 votes, 10,262 valid votes

 * 17 June 1984
 * CDU: 40.7%, 4,576 votes — 15 seats
 * SPD: 44.1%, 4,966 votes — 16 seats
 * FDP: 7.8% — 2 seats
 * FWG: 10.6% — 2 seats
 * DKP: 1% — 112 votes
 * Eligible voters: 15,408
 * Voter turnout: 74.9%, 11,252 valid votes

 * 18 June 1989
 * SPD: 41.0% — 15 seats
 * CDU: 31.2% — 11 seats
 * FWG: 10.6% — 4 seats
 * FDP: 7.75% — 3 seats
 * Grüne: 7.38% — 2 seats
 * 12 June 1994
 * SPD: 36.6% — 13 seats
 * CDU: 31.0% — 11 seats
 * FWG: 6 seats
 * Grüne: 4 seats
 * FDP: 2 seats
 * Voter turnout: 70%, 11,781 votes

Mayors before 1939

 * Nieder-Ingelheim
 * Weitzel about 1881
 * Leonard Muntermann (DDP, 1912 - 7 April 1932)
 * Ober-Ingelheim
 * Dr. Georg Rückert (February 1932 - April 1933)
 * Gaul (1933-)

(Chief) Mayors since 1939
Mayors (Bürgermeister) from 1946, Chief Mayors (Oberbürgermeister) from 1972:

 * 1939-1945: Franz Bambach (NSDAP)
 * 15 April 1945 - 23 June 1945: Georg Schick
 * 23 June 1945 - : Dr. iur. Georg Rückert (SP)
 * 22 September 1946 – 1948: Dr. iur. Georg Rückert (SP)
 * 1949 - 1 October 1956: Dr. rer. pol. Heinz Brühne (SPD)
 * 1957-1964: Heinz Kühn
 * 1964-1965: Albert Saalwächter
 * 1966-1972: Hans-Ulrich Oehlschlägel, BM (SPD)
 * 1972-1975: Hans-Ulrich Oehlschlägel, OB (SPD)
 * 1975-1995: Anno Vey (CDU)
 * 1995-2011: Dr. Joachim Gerhard (CDU)
 * 2011-   : Ralf Claus (SPD)

Coat of arms
The town's arms might be described thus: Argent an eagle displayed sable armed and langued gules.

The eagle is the Imperial Eagle. The arms have their roots in the Imperial Freedom enjoyed by the Ingelheimer Grund (Ingelheim area).

Old coats of arms

 * Nieder-Ingelheim: Argent a wall embattled gules masoned sable, issuant therefrom a demi-eagle displayed of the third beaked and langued of the second.

 * Ober-Ingelheim: Argent an eagle displayed sable armed, beaked and langued gules.

Sponsorships

 * Airbus Ingelheim am Rhein D-ABJE, Boeing 737-530, SN 25310/2126
 * Until her decommissioning on 28 June 2001 there was a partnership with S58 Pinguin, a German Navy Fast Attack Craft.

Twin towns – sister cities
Ingelheim am Rhein is twinned with:

 * 🏴 Stevenage, England, United Kingdom (1963)
 * 🇫🇷 Autun, France (1963)
 * 🇮🇹 San Pietro in Cariano, Italy (1984)
 * 🇩🇪 Friedrichshain-Kreuzberg (Berlin), Germany (1971)
 * 🇩🇪 Limbach-Oberfrohna, Germany (1990)
 * 🇵🇱 Nysa, Poland (2002)

On 24 October 1975, the three-way partnership between Ingelheim, Autun and Stevenage was officially sealed.

kING culture centre
Ingelheim has a multi-purpose culture centre, named kING, located close to the station.

Museums
The Museum bei der Kaiserpfalz ("Museum at the Imperial Palace") has an exhibit dedicated to the Imperial Palace built in Ingelheim after 785 by Charlemagne. On show are small archaeological finds, objects from architectural sculpture and a demonstrative model of the once imposing building. Remnants of the Imperial Palace can be seen right near the museum. Of Europe-wide importance is the golden solidus found in 1996, which is hitherto still the only gold coin ever found struck with Charlemagne's effigy.

Ingelheimer Fassenacht
There is in Ingelheim a well-developed carnival culture, which admittedly is very much under the Mainz carnival’s influence. All together, the town counts four Carnival clubs:
 * Carneval-Verein "Wäschbächer" 1885
 * Carnevalverein Frei-Weinheim
 * Ingelheimer Carnevalverein
 * Narrenclub Ingelheim 1987 ("Fools’ Club")

Music

 * Further Education Centre Symphony Orchestra
 * Ingelheimer Konfettis (performing and singing group)
 * Ingelheim church choir
 * Bläserchöre Ingelheim (wind choirs)
 * Carolus Magnus-Ingelheimer Kaiserpfalzbläser (wind ensemble)
 * Telemann-Chor Ingelheim (choir)

Singing clubs

 * GV Liederkranz 1857
 * GV Einigkeit 1885
 * GV Germania 1862
 * MGV 1866
 * Schubert-Quartett 1924 e.V.
 * Boehringer Jazz & Pop Chor 2009

Buildings
The town has at its disposal a range of historical buildings worth seeing:

Others

 * St. Michael with Plague Cross
 * Carolingian aqueduct
 * Heidesheimer Tor (gate)
 * Bismarck Tower
 * Ohrenbrücker Tor (gate)
 * Jewish graveyard
 * Old market hall in Nieder-Ingelheim

Parks

 * Kommerzienrat-Boehringer-Anlage
 * Emmerlingscher Park
 * Rosengärtchen
 * Uffhubtor-Anlage

Natural monuments
In the cadastral areas of Nieder-Ingelheim and Frei-Weinheim, mainly north of the Autobahn along Konrad-Adenauer-Straße, but also south of the Autobahn – even within the Boehringer Ingelheim industrial lands – are found drifting chalk sands. Likewise a deposit is to be found in the area of the Griesmühle (mill).
 * Drifting chalk sands and dunes

These formations are under conservational protection under the Rhineland-Palatinate State Care Law. Damaging them or removing them, among other acts, is considered an incompensable encroachment on nature and the landscape. Municipal building uses in drifting chalk sand areas are therefore routinely excluded or only approved in very special cases. Two such exceptions were the building of Konrad-Adenauer-Straße (from the Autobahn bridge to Rheinstraße) and the building of the daycare centre on Sporkenheimer Straße.

Sport

 * 1. Schwimmsportverein Ingelheim 1966 e.V. (swimming)
 * RV Ingelheim
 * SpVgg Ingelheim
 * TUS Ober-Ingelheim
 * Turngemeinde 1847 Corp. Nieder-Ingelheim (gymnastics)
 * SV Ingelheim 1949 e.V.
 * VfL Frei Weinheim
 * HSC Ingelheim
 * TV Frei-Weinheim (gymnastics)
 * MFG-Ingelheim e.V. - Modellfluggruppe Ingelheim e.V.
 * TSC Ingelheim
 * FSC Ingelheim 07

 * Bridgeclub Ingelheim

Common welfare
The MütZe ("Mothers’ and Families’ Centre", with the abbreviation resembling the word Mütze – "cap") is to be found at the old Gymnasium. The MütZe takes upon itself a generation-spanning exchange for all Ingelheim residents. A babysitter exchange, handicraft classes, breakfast and lunch, housework and holiday support are regularly offered, as well as courses and events covering every family theme from babies to health to creativity.
 * Mütter- und FamilienZentrum e.V. MütZe

In Ingelheim there are also a House of Youth (Haus der Jugend, although this is soon to become a shopping centre and will be replaced with another House of Youth) and a Mehrgenerationshaus.

Regular events

 * Since 1972 there has been a yearly folk music event, the Eurofolkfestival Ingelheim, on the Burgkirche Fairgrounds. It is said to be one of the successor festivals to the famous Waldeck-Festivals. A great number of the visitors are people from the hippie culture and youths from the local area and from throughout Germany. The number of visitors varies from 2,000 to 3,000. It is usually held between mid-June and mid-July and always lasts from Friday to Sunday. Out of the Eurofolkfestival grew the OpenOhr Festival (a youth cultural festival) in Mainz in 1974 and 1975.
 * Hafenfest auf der Jungau ("Harbour Festival on the Jungau"), each year in early August.
 * Ingelheimer Rotweinfest ("Ingelheim Red Wine Festival") on the Burgkirche Fairgrounds, is held each year from the last weekend in September to the first weekend in October.
 * Kerb in Groß-Winternheim ("kermis, or church consecration festival"), second weekend in September
 * Internationale Tage ("International Days"), each year since 1959. Organized for Boehringer Ingelheim by François Lachenal till 1997.till 2000 curated by Patricia Rochat and since then by Ulrich Luckhardt.
 * Umsonst-und-drinnen, international music festival for new blood groups.
 * Kinderfest der DPSG Ingelheim ("Ingelheim DPSG Children’s Festival"), each year on Ascension Day since 1969 on the Jungau in Frei-Weinheim.
 * Entekerb ("Harvest Kermis"), in October in Frei-Weinheim.
 * Altstadtfest ("Old Town Festival"), second weekend in August, staged by NCI
 * Fest der Generationen, second Saturday in September around the old Gymnasium, staged by the MütZe

Culinary specialities
Regional Rhenish-Hessian specialities are asparagus and morello cherries (a cultivar of sour cherries).

Transport
The Autobahn A 60 runs through the municipal area and has two interchanges there. Bundesstraße 41 ends in Ingelheim. The Autobahnen A 61 and A 63 lie right nearby. Frankfurt Airport can be reached by Autobahn in roughly 30 minutes. Frankfurt-Hahn Airport can be reached in roughly 50 minutes by Autobahnen A 60 and A 61 or Bundesstraße 50. A Bus to Hahn can be caught in Mainz

Ingelheim lies on the Mainz-Bingen-Cologne (West Rhine Railway) and Saarbrücken-Mainz-Frankfurt railway lines. Between Ingelheim-Nord and Oestrich-Winkel runs a Rhine ferry. The constituent communities and the surrounding municipalities are served by city and regional bus routes of Omnibusverkehr Rhein-Nahe GmbH. The local rail transport is served by the Rhein-Nahe-Nahverkehrsverbund.

Established businesses

 * Boehringer Ingelheim, pharmaceutical enterprise
 * Envision Entertainment GmbH, formerly EA Phenomic, a video game developer
 * Goldener Engel, brewery
 * Karl Gemünden, building company
 * Rheinhessische Energie- und Wasserversorgungs-GmbH, energy and water supply
 * Vereinigte Obst- und Gemüsemärkte (VOG), Europe's biggest transshipment centre for sour cherries
 * WetterKontor, supplier of weather information

Agricultural produce
Of the 4,987-hectare municipal area, 641 ha is used for winegrowing and 1 373 ha is used for crops. The main agricultural produce is sour cherries, white asparagus and Wine. Although the town lies in a region dominated by white wine, 54.9% of the vineyard area in Ingelheim am Rhein is used for growing red wine varieties. With 641 ha in vineyards, the town is moreover one of Rhenish Hesse’s biggest winegrowing centres after Worms, (1,490 ha), Nierstein (783 ha), Alzey (769 ha), Westhofen (764 ha), Alsheim (704 ha) and Bechtheim (654 ha), and one of the biggest in the whole state of Rhineland-Palatinate.

"The red wines of Ingelheim and Heidesheim (…) opposite to Eltville (…) enjoy a high reputation."

The Geisenheim Grape Breeding Institute’s vegetable farming department runs an experimental asparagus field in Ingelheim. The research results can be viewed on the Internet.

Media
Local daily newspaper: Allgemeine Zeitung Ingelheim within the Rhein Main Presse, published by the Verlagsgruppe Rhein Main, Mainz.

Municipal television: "Blickpunkt Ingelheim", which is broadcast every Monday and Thursday on regional channel K3.

Public institutions
Since 1996, Ingelheim has been the seat of district administration for Mainz-Bingen.

Education
Ingelheim is home to:

 * three primary schools (Präsident-Mohr-Grundschule, Theodor-Heuss-Grundschule, Brüder-Grimm-Grundschule)
 * a combination primary school and Hauptschule (Pestalozzi-Grund- u. Hauptschule)
 * a professional college, die BBS Ingelheim
 * a school for those with learning difficulties (Albert-Schweitzer-Sonderschule)
 * a Realschule (Kaiserpfalz-Realschule)
 * an integrated comprehensive school (Kurt Schumacher)
 * a Gymnasium (Sebastian-Münster-Gymnasium)

Under the umbrella of the Ingelheim Further Education Centre Weiterbildungszentrum Ingelheim the following institutions work:
 * Volkshochschule (folk high school)
 * Fridtjof-Nansen-Akademie für politische Bildung (political education)
 * Music school
 * Jugendbildungswerk (youth education)

Honorary citizens

 * Christian Rauch (1878–1976), archaeologist, named an honorary citizen of Ingelheim 16 December 1974
 * Robert Boehringer (1884–1974), entrepreneur and lyricist, named an honorary citizen of Ingelheim in 1974

Sons and daughters of the town

 * Sebastian Münster (1488–1552), scientist (cosmographer) and a Hebraist.
 * Johannes Anspach (1752–1823), pastelist, draftsman, painter, owner of an artist's studio in the Netherlands
 * Klaus Knopper (born 1968), developer of Linux distribution Knoppix
 * Markus Kreuz (born 1977), footballer

Other celebrities

 * Charlemagne (747–814), held court in Ingelheim in 807.
 * Louis the Pious (778–840), died in a summer tent on an island in the Rhine off Ingelheim.
 * Pope Joan Johanna von Ingelheim, according to legend she reigned as Pope from 855 to 857.
 * Anselm Franz von Ingelheim (1634–1695), Archbishop-Elector of Mainz from 1679.
 * Jean-Baptiste Kléber (1753–1800), during the Siege of Mainz in 1793 headquartered in Ingelheim
 * Eduard Douwes Dekker, known as Multatuli (1820–1887 in Ingelheim am Rhein), Dutch writer
 * Richard von Weizsäcker (1920–2015), noble and politician, from 1962 to 1966 managing partner of Boehringer Ingelheim

Documents

 * picture of Oberingelheim from J.F. Dielmann, A. Fay, J. Becker (draughtsman): F.C. Vogels Panorama des Rheins, Bilder des rechten und linken Rheinufers, Lithographische Anstalt F.C. Vogel, Frankfurt 1833

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

title: Ingelheim am Rhein


text:
Broussaisia

Broussaisia arguta, the kanawao, is a species of perennial flowering plant in the Hydrangea family, Hydrangeaceae, that is endemic to Hawaii. It is the only species in the monotypic genus Broussaisia.

Kanawao is a widespread species occurring in mesic and wet forests on all of the main Hawaiian Islands. It is a dioecious, evergreen plant that either grows as a 1.5 – shrub or a 6 m tree. The obovate leaves are 10 - 25 cm long and 4 - 9 cm wide with finely serrated margins.

Relationship to other Organisms

The Orthotylus broussaisia (O. broussaisiae) from Kauai feeds on Broussaisia arguta. Although there has been extensive research on Broussaisia arguta, only the Orthotylus broussaisiae species has been found feeding on it.

title: Broussaisia


text:
Finns Party

The Finns Party,  (Perussuomalaiset, PS; Sannfinländarna, Sannf) formerly known as the True Finns, is a right-wing populist political party in Finland. It was founded in 1995 following the dissolution of the Finnish Rural Party.

The party achieved its electoral breakthrough in the 2011 Finnish parliamentary election, when it won 19.1% of votes, becoming the third largest party in the Parliament of Finland.

In the 2015 election the party got 17.7% of the votes, making it the parliament's second-largest political party. The party was in opposition for the first 20 years of its existence. In 2015, it joined the coalition government formed by Prime Minister Juha Sipilä.

Following a 2017 split, over half of the party's MPs left the parliamentary group and were subsequently expelled from their membership in the party. This defector group, Blue Reform, continued to support the government coalition, while the Finns Party went into opposition. The party, having been reduced to 17 seats after the split, increased its representation to 39 seats in the 2019 Finnish parliamentary election, while Blue Reform failed to win any seats.

During the 2023 Finnish parliamentary election, the Finns Party finished in second place with 46 seats, recording their strongest result since its founding. They then, out of Petteri Orpo’s request, proceeded to form a coalition government with the winning National Coalition Party, the Christian Democrats and Swedish People's Party of Finland. The party currently holds seven ministerial portfolios in the Orpo Cabinet.

Finnish Rural Party
The predecessor of the Finns Party was the Finnish Rural Party (SMP), founded by Agrarian League dissident Veikko Vennamo in 1959. Vennamo ran into serious disagreement with Arvo Korsimo, the Agrarian League's party secretary, and was excluded from the parliamentary group. As a result, Vennamo immediately started building his own organization and founded the Finnish Rural Party. Vennamo was a populist and became a critic of President Urho Kekkonen and of political corruption within the "old parties", particularly the Centre Party (the renamed Agrarian League). The Rural Party achieved two major victories in the elections of 1970 and 1983, winning 18 and 17 seats, respectively.

In the 1970s, Vennamo's personalized leadership style alienated some in the party, which led to a split in the parliamentary group in 1972. After the Rural Party's new rise in 1983 under Vennamo's son Pekka, the party became a partner in two coalition governments. However, the party's support declined steadily in the late-1980s and early-1990s. In 1995, the party won only one seat in the Finnish parliament and soon filed for bankruptcy.

Founding of the Finns Party and its rise in popularity

In the summer of 1995, following the collapse of the Finnish Rural Party, the decision to found the Finns Party was made by Timo Soini, Raimo Vistbacka, Urpo Leppänen and Kari Bärlund. Soini had been the Rural Party's last party secretary and Vistbacka its last chair and MP. The party collected the five thousand signatures needed for registration and was added to the official party register on 13 October 1995. The first party congress was held in November. Vistbacka was elected party chair and Soini the party secretary.

It took some time before the Finns Party gained ground in Finnish elections. At the time of its founding in 1995, the party's sole MP was Vistbacka, who was reelected in the 1999 election. In 2003, the party won three seats: besides Vistbacka, Soini and Tony Halme were elected. In the 2007, the party gained two further seats for a total of five. In the 2008 municipal election, the Finns Party were most successful in those districts where the Social Democrats and the Left Alliance lost most. In the 2011 election, the Centre Party suffered the largest blow from the Finns Party's success.

According to a 2008–2009 study, Finns Party supporters viewed themselves as centrist: on a scale where 1 was extreme left and 10 was extreme right, the average supporter placed themselves at 5.4. According to the same study, the supporters were united by patriotism and social conservatism. A 2011 study indicated that the Finns Party was the most popular party among voters with an annual income of 35,000–50,000 euros, while over a quarter of the party's voters earn over 50,000 per year. The same study also indicated that the party's voters included a higher percentage of blue-collar workers than those of the Social Democrats.

Timo Soini
Timo Soini led the Finns Party for twenty years, from 1997 until 2017. He was first elected to the parliament in 2003. He was the party's candidate in the 2006 presidential election, and was elected to the European Parliament in 2009 with the highest personal vote share in the country. He served as an MEP for two years, returning to the Finnish parliament in the 2011 election. Soini was the party's presidential candidate for a second time in the election of 2012. Jussi Halla-aho succeeded Soini as party chair in 2017.

2011–2017
The Finns Party obtained 39 seats in the 2011 election, making them the third largest party, narrowly behind the National Coalition Party (44) and the Social Democrats (42). Soini received 43,212 personal votes, the highest number of all candidates, leaving behind the Foreign Minister Alexander Stubb and the Finance Minister Jyrki Katainen in their Uusimaa electoral district. The popularity of the party rose from 4.1% to 19.1% in just four years. Helsingin Sanomat wrote in an editorial that the party and Soini had "rewritten the electoral history books". According to political analyst Jan Sundberg, Soini had the ability to appeal to common people and make complicated things look easy. The election result was also referred to as "shocking" and "exceptional".

After the election, the National Coalition Party (NCP) began negotiations aiming to form a cabinet between the NCP, the Social Democrats, and the Finns Party. However, when it became clear that the NCP and the Social Democrats would continue to support EU bailouts, which the Finns Party vehemently opposed during the electoral campaign, the party voluntarily broke from the negotiations to become the leading opposition party. Soini said that the party would not compromise its core principles just to enter the government. According to an opinion poll, most of the party's supporters accepted this decision.

The Finns Party's popularity initially continued to rise after the 2011 election: in one opinion poll from June 2011 gave the party a record popularity of 23 percent. The party's membership rose to over 8,000 members by 2013 (up from circa 5,500 in 2011 and circa 1,000 in 2005 ). Membership in the party's youth organisation rose as well, going from 800 before the 2011 election to over 2,200 in 2013.

The party nominated Soini as its candidate for the 2012 presidential election; Soini finished fourth with 9.4 percent. Soini interpreted the result by saying that half of the party's voters wanted him for president, while the other half wanted to him to remain as party chair. In municipal elections later in 2012, the party got 12.3 percent of votes and 1,195 seats in the municipal councils, up more than 750 from the previous municipal election. However, this result saw the votes for the party shrink significantly from the 2011 parliamentary election result. The party got 12.9 percent of votes in the 2014 European Parliament election and increased its number of MEPs to two.

In the 2015 election, the Finns Party got 17.7% of the votes and 38 seats. This meant that they were the third largest party by votes but the second largest party by seats. The Finns Party subsequently entered into a coalition government with the Centre Party and the NCP, led by Prime Minister Juha Sipilä. The party's participation in the Sipilä Cabinet marked a softening of its Eurosceptic positions. On 22 June 2016, Finns Party MP Maria Tolppanen joined the Social Democrats, after which the Finns Party had 37 seats in the parliament. In March 2017, Soini announced that he would step down as party chair in the next party congress in June.

2017 leadership election and splits
In June 2017, Jussi Halla-aho and Sampo Terho faced off in the leadership election, in which Halla-aho received 949 votes against Terho's 646 votes and thus succeeded Soini as party chair. Sipilä and Minister of Finance Petteri Orpo soon announced that they would not continue their coalition with the Finns Party if it was led by Halla-aho. Subsequently, twenty Finns Party MPs, including Soini and Terho, defected to form a new parliamentary group under the name New Alternative, later renamed to Blue Reform and after that, in 2022, into Finnish Reform Movement. As all cabinet ministers were among the defectors, the then Blue Reform made an agreement with Sipilä to stay in the government.

Following the split, MPs Veera Ruoho and Arja Juvonen left the Finns Party parliamentary group to continue as independents, after which the party's seats were reduced to fifteen. All of the defecting MPs were subsequently expelled from the Finns Party. In the following weeks, MPs Ritva Elomaa and Arja Juvonen regretted their decision and re-joined the party, raising the amount of MPs to seventeen.

The party nominated MP Laura Huhtasaari as its candidate for the 2018 presidential election. In the election, Huhtasaari placed third with 6.9 percent of the votes, while the incumbent president Sauli Niinistö went on to secure his second term with a majority of votes.

2019 general election and revival
At the 2019 Finnish parliamentary election the Finns Party finished in second place and increased its number of MPs to 39 (with its strongest result being in Satakunta) while the breakaway Blue Reform party lost all of its seats.

On 21 June 2021, Jussi Halla-aho announced that he would retire from his position as a party leader in August 2021. He was succeeded by MP Riikka Purra on 14 August.

Since 2020, further minor splits have emerged within the party, forming the Power Belongs to the People party and Blue-and-Black Movement.

During the 2023 Finnish parliamentary election the party finished in second place ahead of the Social Democrats with 20% of the vote and 46 seats, marking the strongest result to date for the party.

In April 2023, National Coalition Party leader Petteri Orpo announced his attention to form a governing coalition with the Finns Party, Swedish People's Party, and the Christian Democrats.

In the Orpo Cabinet, the Finns have seven ministers out of 19. Former party leader Jussi Halla-aho was elected Speaker of the Parliament of Finland.

In the European Parliament
When the Finns Party first gained representation in the European Parliament in 2009, it became a founding member of the Europe of Freedom and Democracy Group (EFD) in the Parliament. After the 2014 election, the party chose to leave the EFD to join the European Conservatives and Reformists Group (ECR). Commenting on the party's choice of group, party secretary Riikka Slunga-Poutsalo said in 2014 that joining a right-wing parliamentary group would not change the party's characteristic of being a "centre-left workers' party". After the 2019 election, the party joined the Identity and Democracy Group; however, this decision was reverted after the 2023 parliamentary election, with the Finns Party rejoining the ECR after a four-year break.

Ideology
Ideologically, the Finns Party has been described as right-wing and far-right. It is a nationalist and national-conservative party that opposes immigration, while on foreign stances it is Eurosceptic. The party combines right-wing economic policies and economic nationalism with socially conservative values, and ethnic nationalism. Several scholars have described them as radically right-wing populist. In the parliament seating order, the party was seated in the centre of the plenary until 2019 when it was moved to the right of the plenary despite the party's opposition to the move. Some of the party's supporters have described themselves as centrists. The party has drawn people from left-wing parties but central aspects of their manifesto have gained support from right-wing voters as well. The Finns Party has been compared by international media to the other Nordic populist parties and other similar nationalist and right-wing populist movements in Europe. It also calls for austerity policies to curb deficit spending.

Policies and platform
In evaluating the Finns Party's 70-page programme for the 2011 election Mikko Lahtinen, political scientist in the University of Tampere, and Markku Hyrkkänen, historian of ideas in the University of Turku, note that nationalism is a theme consistently repeated throughout the programme. According to them the party presents populism as a noble ideology, which seeks to empower the people. Lahtinen describes the rhetoric used in the program as a refreshing change to the politically correct "jargon" of mainstream media, and believes that the Finns Party may have succeeded in gaining supporters from the traditional left-wing parties by presenting a more attractive form of criticism of neoliberalism than those parties.

Ville Pernaa, political scientist, described the party's 2015 electoral program by saying that the Finns Party combines elements of both right-wing and left-wing politics along with populist rhetoric.

Fiscal policies

 * Progressive taxation and the welfare state

The Finns Party has proposed more progressivity to taxes to avoid the establishment of flat taxation. The party has called for the raising of the capital gains tax and the re-institution of the wealth tax. According to the party, the willingness to pay taxes is best guaranteed by a society unified by correct social policies – the electoral program warns against individualist policies, which weaken the solidarity among citizens. "The willingness to pay taxes is guaranteed by having a unified people", the program reads (p. 46).

Some observers have compared the Finns Party's fiscal policies to the old national Social Democratic taxation policy, which has given the left-wing brand to the Finns Party. During the electoral campaign in 2011 Soini stated that he preferred the Social Democrats over the center-right National Coalition Party as a possible coalition partner in a future cabinet. Soini has stated that the Finns Party is a "workers' party without socialism". A researcher for the opinion polling company Taloustutkimus agreed, describing the Finns Party as a "non-socialist workers' party".

 * State support for rural regions, including support for agriculture

The Finns Party's rural policy program suggests state subsidies to relieve the effect of structural changes on the rural areas. This policy is shared by the Centre Party in Finland and originates from the agrarian and rural policies of both parties.

 * Increased state investment in infrastructure and industry

The Finns Party favours state investments in infrastructure and industry as well. A tendency towards favouring old industrial policies have led some political analysts to label the Finns Party as a center-left party.

Energy policies

 * Aspiration to energy self-reliance and support for nuclear energy
 * Pro-industry environmental policy – opposition to green tax reform and to taxpayers' involvement in emission trading funds
 * The Party of the Finns strongly supports the peat industry, which produces massive amounts of greenhouse gases, having even proposed exempting this production from any tax. The party generally scores best in municipalities whose economies are tied to this industry. The party has strongly denounced the Paris Agreement, signed in December 2015, saying it was "catastrophic" for the economy, and demanded that the private sector and taxpayers be spared its "disastrous economic consequences".

Cultural policies

 * Teaching "healthy national pride" in schools, because the unity of citizens is the basis of society.
 * Removal of the obligatory character of the second official language (Swedish in Finnish-language schools and vice versa) in curriculums on all levels of education, freeing up time for the learning of other foreign languages such as English, German, French, Spanish and Russian (especially in the eastern part of the country). Allowance regarding the use of the Swedish language and its teaching will have to be made for those communes where Swedish-speaking populations are in the majority or a large percentage of the population – Swedish is one of Finland's national languages.
 * Support for cultural activities that "promote Finnish identity"

The cultural program of the Finns Party, which proposed subsidizing traditional art over postmodernist art, prompted criticism from outside the party and generated debate within the party as well. Some critics of the policy called it overtly populist or said that the state should not interfere with the content of art. A poll commissioned by Helsingin Sanomat at the time of the controversy found that a majority, 51 percent, of Finns agreed with the party's stance on ending subsidies for postmodern art.

Social policies

 * Rejection of gender identity, opposition to same-sex marriage, same-sex adoption and in vitro fertilisation given to same-sex couples and single women.

Immigration policies
Regarding immigration policy the 2011 manifesto emphasises:

The party also requires that immigrants accept Finnish cultural norms. The only written declaration to the European Parliament made by a True Finn MEP also concerns immigration matters. The party underlines the role of national sovereignty in immigration issues: "[True] Finnish immigration policy should be based on the fact that the Finns should always be able to decide for themselves the conditions under which a foreigner can come to our country and reside in our country."
 * Limiting humanitarian immigration strictly to refugee quotas (which should be adapted to correspond with the economic situation),
 * Limiting family unification to proven direct relatives only, and requiring means of subsistence from the immigrant,
 * Deporting those immigrants guilty of serious or recurrent crimes or those that does not meet the Immigration laws,
 * Welcoming work-based immigration, provided the immigrants pay taxes and abide by Finnish labour laws,
 * Granting Finnish nationality after five years' residence in Finland, provided the immigrant masters Finnish, has no criminal record, and has means of subsistence

In 2015 the party's immigration programme included demands like:

 * Lowering the refugee quota
 * Opposition to the planned burden-sharing mechanisms of the Common European Asylum Policy
 * Opposition to using public funds to advance multiculturalism
 * Tightening the conditions of family unification by migrants
 * Allowing the immigration of workers from outside the EU and EEA countries only if they are found to be necessary in a given field in a means test by the Finnish Labour Office
 * Making sure that migrants living on welfare benefits are not concentrated in the same areas
 * Outlawing begging in public places
 * Ending affirmative action

In their 2019 election manifesto the party called for a prohibition on wearing the burqa and the niqāb in public.

Timo Soini signed a pan-European charter against racism in 1998. However, in 2009, before the European Parliament election, Soini refused to sign an anti-racism appeal, saying that the appeal was an attempt to influence the party's choice of candidates (the appeal was drawn up by another political party). All other Finnish parties signed this appeal against racism. In May 2011, following controversies surrounding the remarks of the Finns Party's MP Teuvo Hakkarainen, the Finns Party's parliamentary group issued a statement condemning all racism and discrimination, including affirmative action. The party invited other parties to sign the statement as well, but no other party did so. In December 2011, an opinion poll revealed 51% of Finns Party voters agreed with the statement, "Joihinkin rotuihin kuuluvat ihmiset eivät kerta kaikkiaan sovi asumaan moderniin yhteiskuntaan;" "People of certain races are unable to live within (fit into) a modern society."

Foreign and defence policies
The party also holds the following positions:

 * Opposition to the integration of the European Union and opposition to Finnish Eurozone membership.
 * Reductions in foreign aid

Timo Soini has been an outspoken critic of both the EU and NATO, but has stated that if a choice had to be made, NATO is a lesser evil than the EU. The Finns Party favors non-alliance or neutrality, as international activities abroad for the Defence Forces would undermine the defence budget's funds for sustaining a large conscript army of war-time personnel (which is 350,000) to guarantee the defence of all of Finland. When the Finnish Parliament voted to ratify the Ottawa Treaty, banning anti-personnel mines, in November 2011, the Finns Party was the only party unified in opposing the treaty.

The party believes in national sovereignty: "[T]he eternal and unlimited right to always decide freely and independently of all of one's affairs lies only and solely with the people, which forms a nation separate of others."

Shortly after the leadership election of Jussi Halla-aho, the party hardened its position towards the European Union. In 2017, Laura Huhtasaari stated that she would support leaving the EU should she win the 2018 Finnish presidential election citing the growth of the Union's power at the expense of the member states. Other party members have supported the idea of Finland withdrawing from both the Schengen Agreement and the Eurozone. In its latest platform, the Finns Party states that it supports a "European policy" based on appreciation for Western and Christian shaped values. The party argues that Finland is needed in the European Parliament to defend Finnish interests in the short-term, but states the "long-term strategic goal" is to take gradual steps to withdraw Finland from the European Union and proposes introducing a parallel currency within Finland to initiate phasing out Finnish membership of the Eurozone.

In 2023, Halla-aho reiterated that the party's long-term goal is strategic preparation for a "Fixit" (Finnish exit) from the European Union and Eurozone, highlighting what he described as a "democratic deficit" within the EU while Foreign Trade Minister Ville Tavio stated the party will vote against joint-EU debt policies in government. However, the party leadership stated that it also wished to strengthen European unity and cooperation in the meantime in response to Russia's invasion of Ukraine.

Initially, the party was opposed to Finnish admission into NATO in its 2011 program, however following the Russian invasion of Ukraine the party signaled a change to this policy and stated it was willing to back NATO membership.

Judicial policies
During the 2011 election the party's judicial programme included the following policies:
 * Tougher punishments for violent crime
 * More resources for police and prosecutors
 * Opposition to any incorporation of Sharia law into judicial practices

Chairpersonship and party secretaries

The party chairpersonship is divided between four persons, elected at party congress biannually. Riikka Purra is the party's chair. The first deputy chair is Leena Meri, the second deputy chair is Mauri Peltokangas and the third deputy chair is Sebastian Tynkkynen.

Raimo Vistbacka chaired the Finns Party from 1995 to 1997. The party secretary Timo Soini succeeded Vistbacka as chairperson in 1997.

Rolf Sormo followed Timo Soini as party secretary and served from 1997 to 1999. The third party secretary, Hannu Purho, served for eight years, from 1999 to 2007. After him, Timo Soini's parliamentary assistant, Ossi Sandvik, was elected party secretary in 2007. He was succeeded by Riikka Slunga-Poutsalo, who was elected as party secretary in 2013. She was succeeded by Simo Grönroos in 2019, who was in turn succeeded by Arto Luukkanen.

Board
The board of the Finns Party has 13 members: the party chairperson, the three deputy chairs, the party secretary, chair of the parliamentary group and seven other members.

Foundations
The foundation Perussuomalaisten tukisäätiö ("The Finns Party support fund") was founded in 1990. It used the name SMP:n tukisäätiö until 2006. The fund borrowed 1.7 million euros from the party in 2012 to buy a 450 m2 commercial property in downtown Helsinki on Yrjönkatu for use as the Party's new headquarters. The Party rented these premises from the fund. Following the split of 2017, this foundation was left in the control the defector group, Blue Reform.

Another foundation, Suomen Perusta ("The Foundation of Finland"), was set up in 2012. Its role is to function as a think tank affiliated with the party.

Current members of the Finnish Parliament
Jani Mäkelä is the current chair of the parliamentary group.

 * Sanna Antikainen (Savonia-Karelia, 2019–)
 * Miko Bergbom (Pirkanmaa, 2023–)
 * Juho Eerola (South-East, 2011–)
 * Kike Elomaa (Finland Proper, 2011–)
 * Kaisa Garedew (Central Finland, 2023–)
 * Jussi Halla-aho (Helsinki, 2011–14, 2019–)
 * Laura Huhtasaari (Satakunta, 2015–2019, 2023–)
 * Petri Huru (Satakunta, 2019–)
 * Tomi Immonen (Central Finland, 2023–)
 * Vilhelm Junnila (Finland Proper, 2019–)
 * Kaisa Juuso (Lapland, 2019–)
 * Arja Juvonen (Uusimaa, 2011–)
 * Antti Kangas (Oulu, 2023–)
 * Teemu Keskisarja (Uusimaa, 2023–)
 * Ari Koponen (Uusimaa, 2019–)
 * Jari Koskela (Satakunta, 2019–)
 * Sheikki Laakso (South-East, 2019–)
 * Rami Lehtinen (Häme, 2023–)
 * Mikko Lundén (Finland Proper, 2019–)
 * Leena Meri (Uusimaa, 2015–)
 * Juha Mäenpää (Vaasa, 2019–)
 * Jani Mäkelä (South-East, 2015–)
 * Veijo Niemi (Pirkanmaa, 2019–)
 * Mira Nieminen (Häme, 2023–)
 * Mauri Peltokangas (Vaasa, 2019–)
 * Jorma Piisinen (Uusimaa, 2023–)
 * Mikko Polvinen (Oulu, 2023–)
 * Sakari Puisto (Pirkanmaa, 2019–)
 * Riikka Purra (Uusimaa, 2019–)
 * Lulu Ranne (Häme, 2019–)
 * Mari Rantanen (Helsinki, 2019–)
 * Minna Reijonen (Savonia-Karelia, 2019–)
 * Anne Rintamäki (Vaasa, 2023–)
 * Jari Ronkainen (Häme, 2015–)
 * Onni Rostila (Uusimaa, 2023–)
 * Wille Rydman (Helsinki, 2023–; NCP MP 2015–2023)
 * Sami Savio (Pirkanmaa, 2015–)
 * Sara Seppänen (Lapland, 2023–)
 * Pia Sillanpää (Vaasa, 2023–)
 * Jenna Simula (Oulu, 2019–)
 * Jaana Strandtman (Southeast Finland, 2023–)
 * Ville Tavio (Finland Proper, 2015–)
 * Sebastian Tynkkynen (Oulu, 2019–)
 * Joakim Vigelius (Pirkanmaa, 2023–)
 * Ville Vähämäki (Oulu, 2011–)

Former members of the Finnish Parliament

 * Simon Elo (2015–17; defected to Blue Reform in 2017)
 * Tiina Elovaara (2015–17; defected to Blue Reform in 2017)
 * Teuvo Hakkarainen (2011–2019)
 * Tony Halme (2003–07)
 * James Hirvisaari (2011–13; expelled from the party in 2013)
 * Reijo Hongisto (2011–17; defected to Blue Reform in 2017)
 * Laila Koskela (2011–14; defected to the Centre Party in 2014)
 * Lauri Heikkilä (2011–15)
 * Olli Immonen (2011–2023)
 * Ari Jalonen (2011–17; defected to Blue Reform in 2017)
 * Anssi Joutsenlahti (2011–15; Rural MP 1979–87)
 * Johanna Jurva (2011–15)
 * Pietari Jääskeläinen (2009–15)
 * Toimi Kankaanniemi (2015–)
 * Pentti Kettunen (2011–15; Rural MP 1983–87, 1989–91)
 * Kimmo Kivelä (2011–17; defected to Blue Reform in 2017)
 * Osmo Kokko (2011–15)
 * Jouni Kotiaho (2019–2023)
 * Kari Kulmala (2015–17; defected to Blue Reform in 2017)
 * Rami Lehto (2015–2023)
 * Jari Lindström (2011–17, Minister for Justice and Employment 2015–19; defected to Blue Reform in 2017)
 * Maria Lohela (2011–17, Speaker of the Parliament 2015–2018; defected to Blue Reform in 2017)
 * Anne Louhelainen (2011–17; defected to Blue Reform in 2017)
 * Pirkko Mattila (2011–17; Minister for Social Affairs and Health 2016–19, defected to Blue Reform in 2017)
 * Jukka Mäkynen (2019–2023)
 * Lea Mäkipää (2011–17; Rural MP 1983–95; defected to Blue Reform in 2017)
 * Hanna Mäntylä (2011–17, Minister for Social Affairs and Health 2015–2016; defected to Blue Reform in 2017)
 * Martti Mölsä (2011–17; defected to Blue Reform in 2017)
 * Mika Niikko (2011–2023)
 * Jussi Niinistö (2011–17, Minister for Defence 2015–19; defected to Blue Reform in 2017)
 * Pentti Oinonen (2007–11; defected to Blue Reform in 2017)
 * Tom Packalén (2011–2023)
 * Mika Raatikainen (2014–19)
 * Veera Ruoho (2015–17; defected to the National Coalition Party in 2017)
 * Pirkko Ruohonen-Lerner (2007–15)
 * Vesa-Matti Saarakkala (2011–17; defected to Blue Reform in 2017)
 * Riikka Slunga-Poutsalo (2019–2023)
 * Timo Soini (2011–17, 2003–09, Minister for Foreign Affairs 2015–19; defected to Blue Reform in 2017)
 * Ismo Soukola (2011–15)
 * Sampo Terho (2015–17, Minister for European Affairs, Culture and Sports 2017–19; defected to Blue Reform in 2017)
 * Maria Tolppanen (2011–16; defected to the SDP in 2016)
 * Reijo Tossavainen (2011–15)
 * Ano Turtiainen (2019–20, expelled from the parliamentary group in 2020 and from the party in 2021)
 * Kaj Turunen (2011–17; defected to Blue Reform in 2017)
 * Kauko Tuupainen (2011–15)
 * Markku Uusipaavalniemi (2010–11; Centre MP 2007–10)
 * Veikko Vallin (2019–2023)
 * Veltto Virtanen (2007–15; Ecological Party MP 1995–99)
 * Raimo Vistbacka (1995–11; Rural MP 1987–95)
 * Juha Väätäinen (2011–15)
 * Jussi Wihonen (2019–2023)
 * Timo Vornanen (Savonia-Karelia, 2023–2024; expelled)

Current members of the European Parliament

 * Sebastian Tynkkynen (2024–)

Former members of the European Parliament

 * Pirkko Ruohonen-Lerner (2015–2019, 2023–2024)
 * Teuvo Hakkarainen (2019–2024)
 * Laura Huhtasaari (2019–2023)
 * Jussi Halla-aho (2014–19)
 * Timo Soini (2009–11)
 * Sampo Terho (2011–15)

Party chairpeople

 * Riikka Purra (2021–)
 * Jussi Halla-aho (2017–2021)
 * Timo Soini (1997–2017)
 * Raimo Vistbacka (1995–97)

Party secretaries

 * Arto Luukkanen (2021–)
 * Simo Grönroos (2019–2021)
 * Riikka Slunga-Poutsalo (2013–19)
 * Ossi Sandvik (2007–13)
 * Hannu Purho (1999–2007)
 * Rolf Sormo (1997–99)
 * Timo Soini (1995–97)

Controversies
Several Finns Party MPs and other party leaders have made public statements which others have interpreted as being racist or otherwise inflammatory. In 2011 Finns party MP James Hirvisaari was fined 1,425 euro by the Kouvola Court of Appeals for comments he made on his blog about Muslims. In 2011 President Tarja Halonen was quoted characterizing some Finns party voters as racist. Her comments were broadly condemned by the Finns party. A 2011 book by Swedish journalist Lisa Bjurwald made a similar characterization, that the party's leaders support racist positions, while publicly denying that they do so.

In 2011 MP Pentti Oinonen declined an invitation to the presidential Independence Day ball, citing his aversion to seeing same-sex couples dance.

Jussi Halla-aho, the previous leader of the party, a former MEP and now the Speaker of the House wrote that Somalis were genetically predisposed to rob passersby, wrote that he would be happy if a gang of immigrants raped a Green party MP, and commented "Violence is an underrated problem-solving tool these days," while pondering if he should shoot a gay man. A number of senior Finns Party politicians, including Halla-aho and Immonen, are also Suomen Sisu members, an organization that Länsiväylä describes as a Nazi group. The website of Suomen Sisu has promoted books by Nazis and neo-Nazis such as Alfred Rosenberg, George Lincoln Rockwell and David Duke.

In a judgement given on 8 June 2012, MP Jussi Halla-aho, then Chairperson of the Administration Committee was found guilty by the Supreme Court of both disturbing religious worship and ethnic agitation for statements he made about Muhammad in his blog.

In October 2013 it was reported that MP James Hirvisaari, had invited far-right activist Seppo Lehto as his guest to the parliament. During his visit, Lehto made several Nazi salutes, including at least one instance where Hirvisaari took a photo of Lehto performing the Nazi salute from the spectator gallery overlooking the Parliament House's Session Hall. Photos and videos of Lehto performing the Nazi salute in the Parliament House were then distributed on Lehto's public Facebook page and on YouTube. After newspapers broke news of the incident, Speaker of the Parliament Eero Heinäluoma issued a notice of censure to Hirvisaari for the incident and the Finns Party leadership unanimously decided to expel Hirvisaari from the party, citing multiple cases of acting against the party's interest.

Hirvisaari then became affiliated with the Change 2011 party as the party's MP, until he was unseated in the parliamentary election of 2015.

The party's Ruovesi municipal chapter's official postal address had Nazi flags hanging in the windows in 2019. The chapter chairperson denied that the flags were his.

In 2020, a number of members of The Finns party in parliament criticised the Research Council of Finland for funding the research on the Holocaust of historian and Green League candidate Oula Silvennoinen at the University of Helsinki.

Finns Party politicians have had contacts with far-right and neo-nazi movements such as the Finnish Defense League, Soldiers of Odin, Nordic Resistance Movement (NRM), Rajat Kiinni (Close the Borders), and Suomi Ensin (Finland First). An anti-mosque demonstration was supported by the youth branch of the PS, whose chair, Jarmo Keto, said that, "Islam as an ideology is responsible for many conflicts and terror attacks. Thus such a mosque project is an irresponsible idea."

Members of the Finns Party have attracted criticism from the other parties and antifascists for attending events organized by or with the NRM. Two municipal-level politicians of the Finns Party have taken part in an event where the participants shot and threw knives at shooting targets, using photos of members of the Rinne Cabinet.

The Finns' General Secretary, Olli Immonen, attended an NRM event commemorating Eugen Schauman, who assassinated Nikolay Bobrikov. Some members have been fired from the party as a result of contacts with the far-right.

In 2020, the party sacked the youth wing's vice leader Toni Jalonen for declaring himself a fascist at a conference in Estonia. The party additionally cut financial support to its youth wing, so it was declared bankrupt as a result. A new organization called Perussuomalainen nuoriso in Finnish was founded with new personnel to replace it.

NRM and other far-right activists attend an annual torch march demonstration in Helsinki on the Finnish independence day which ends at the Hietaniemi cemetery where members visit the tomb of Carl Gustaf Emil Mannerheim and the monument to the Finnish Volunteer Battalion of the Waffen-SS. According to B'nai B'rith's report, "The main organizers and guests of the event have been drawn from either non-party-affiliated far-right-activists or members of the right-wing populist Finns Party (Perussuomalaiset), its youth organization Finns Party Youth (Perussuomalaiset Nuoret)".

Finns Party MP Vilhelm Junnila spoke at an event organized in memory of victims of a terrorist attack in 2019. His participation in the event was criticized in the media due to the event being organized by the NRM and Soldiers of Odin. The matter was especially brought up in the media after the formation of the Orpo Cabinet in 2023, where he was appointed Minister of Economic Affairs. Junnila survived a vote of no-confidence, but announced his resignation from the cabinet afterwards.

He was replaced by Wille Rydman, who gained attention for his multiple nazi-related controversies after his appointment.

Viljam Nyman, a Finns party Lapland area board member became a leader of the Atomwaffen Division's Finnish chapter and a prominent adherent to the satanic Order of Nine Angles (ONA). Nyman's associate, a Finns party member born in 1980 and a fellow accelerationism and ONA adherent had dozens of his writings published by the official Finns party organ, Suomen Uutiset. Both were suspected of planning murders and terrorist attacks, and Nyman has been subsequently convicted of terror offenses.

The person born in 1980 is also suspected of sending a string of letter bombs to the Social Democratic, Green League and Left Alliance parties' offices.

A Finns Party Espoo city council member Jiri Keronen told that he "avows" the teachings of ONA and that he is republishing their works.

On 9 May 2024 the Finns Party council expelled Timo Vornanen alongside Teuvo Hakkarainen from the party, because of the police suspecting that Timo Vornanen pointed at a group of people with a gun and then shot the ground after an argument on the 26 April in the Kamppi subdivision of Helsinki, outside a bar. After the shooting incident, Vornanen has been on sick leave from the parliament.

He announced on May 14 that he'd give up his parliamentary committee seats to the rest of the Finns Party to decide on replacements. He also posted on Facebook that he would leave the Joensuu city council’s Finns Party council group and create his own city council and parliamentary groups.

Former MEP Teuvo Hakkarainen, on the other hand, was expelled from the party because he stood as a candidate for the European Parliament on the Freedom Alliance's list. He claims that he did this because the Finns Party’s leadership had "left the field" and adopted a mentality that if you even slightly questioned the party's leadership or criticized their actions, you'd be kicked out.

title: Finns Party


text:
Carpenteria

Carpenteria, the tree anemone or bush anemone, is a genus of flowering plants in the hydrangea family Hydrangeaceae. It is closely related to the similar genus Philadelphus and is monotypic, being represented by the single species Carpenteria californica which is a flowering evergreen shrub native to the Sierra Nevada foothills in California.

The genus was named in honor of Dr. William Marbury Carpenter, a noted botanist from Louisiana.

Description
Carpenteria californica grows to 1 - 3 m tall, with flaky bark on older stems. The leaves are opposite, lanceolate, 4–10 cm long and 1–2.5 cm broad, glossy green above, blue-green to whitish and downy beneath.

The sweetly-scented flowers are 3–7 cm across with five to eight pure white petals and a cluster of yellow stamens. It flowers from late spring to midsummer. The fruit is a leathery capsule 6–12 mm in diameter, containing numerous seeds.

Distribution and habitat
The bush anemone is a rare species, endemic to only seven sites in Fresno and Madera Counties, where it grows in chaparral and oak woodlands at 340 - 1340 m altitude, between the San Joaquin River and Kings River. It is well adapted to wildfire, reproducing by stump sprouts after burning. Natural seedlings are rare.

Cultivation
Carpenteria californica is cultivated as an ornamental plant, grown for its lush foliage, large scented flowers and drought tolerance. It is used in traditional and wildlife gardens in subtropical and temperate locations in the northern hemisphere. It is hardy down to -10 C in sheltered  locations in full sun.

It has been in cultivation since 1875, and is now much more common in gardens than in its natural habitat. It first flowered in England for the famed plantswoman Gertrude Jekyll at Godalming in 1885.

Cultivars include:
 * 'Bodnant' — cold-tolerant cultivar, hardy to -15 C in the British Isles
 * 'Elizabeth' — masses of smaller white flowers, more compact growth habit
 * 'Ladham's' — large flowers

The cultivars 'Elizabeth' and 'Bodnant' have gained the Royal Horticultural Society's Award of Garden Merit.

title: Carpenteria