Elliptical integral

An elliptic integral is an integral of type

${\displaystyle \int R\left(x,{\sqrt {P(x)}}\right)\mathrm {d} x,}$

where is a rational function in two variables and a third or fourth degree polynomial with no multiple zeros . The integral is called elliptic because integrals of this form appear when calculating the perimeter of ellipses and the surface of ellipsoids . There are also far-reaching applications in physics. ${\displaystyle R}$ ${\displaystyle P(x)}$

In general, elliptic integrals cannot be represented by elementary functions , but they can be converted into a sum of elementary functions and integrals of the form described below by transformations. These integrals are called elliptic integrals of the first, second and third kind.

I. Type: ${\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {(1-x^{2})(1-k^{2}x^{2})}}}}$

II. Type: ${\displaystyle \int {\sqrt {\frac {1-k^{2}x^{2}}{1-x^{2}}}}\,\mathrm {d} x}$

III. Type: ${\displaystyle \int {\frac {\mathrm {d} x}{(1-nx^{2}){\sqrt {(1-x^{2})(1-k^{2}x^{2})}}}}}$

In the literature, the parameter is sometimes used instead of in the function call and the definition range is extended to. ${\displaystyle 0<k<1.}$ ${\displaystyle m=k^{2}}$ ${\displaystyle k}$ ${\displaystyle m<0}$

Complete elliptic integrals

Graph of the complete elliptic integrals and ${\displaystyle K(m{=}k^{2})}$ ${\displaystyle E(m{=}k^{2})}$

Definition of complete elliptic integrals

The integrals with the lower integral limit 0 are called incomplete elliptic integrals. If, in addition, the upper integral limit , one speaks of complete elliptic integrals in the case of the I. and II. Types . The complete elliptical integrals I and II. Kind are directly related to the Gaussian hypergeometric function . ${\displaystyle \pi /2}$ ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$

${\displaystyle {\begin{aligned}K(k)&:=F({\tfrac {\pi }{2}},k)={\tfrac {\pi }{2}}\cdot {}_{2}F_{1}({\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2})\\E(k)&:=E({\tfrac {\pi }{2}},k)={\tfrac {\pi }{2}}\cdot {}_{2}F_{1}(-{\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2})\\\Pi (n,k)&:=\Pi ({\tfrac {\pi }{2}},n,k)\end{aligned}}}$

The following table shows the complete elliptic integrals in the integral representation with the parameters and . The Jacobi form can be converted into the Legendre normal form with the substitution . The parameter is used in the function libraries of Matlab , Wolfram-Alpha , Mathematica , Python ( SciPy ) and GNU Octave . ${\displaystyle k}$ ${\displaystyle m}$ ${\displaystyle t=\sin \vartheta }$ ${\displaystyle m=k^{2}}$

Definition of the complete elliptic integrals with parameters and ${\displaystyle k}$ ${\displaystyle m}$

	Convention with parameters ${\displaystyle k}$	Convention with parameters ${\displaystyle m}$
I. Type: Jacobian shape	${\displaystyle K(k):=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}}$	${\displaystyle K(m):=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(1-m\,t^{2})}}}}$
I. Type: Legendre normal form	${\displaystyle K(k):=\int _{0}^{\pi /2}{\frac {{\text{d}}\vartheta }{\sqrt {1-k^{2}\sin ^{2}\vartheta }}}}$	${\displaystyle K(m):=\int _{0}^{\pi /2}{\frac {{\text{d}}\vartheta }{\sqrt {1-m\sin ^{2}\vartheta }}}}$
II. Type: Jacobian form	${\displaystyle E(k):=\int _{0}^{1}{\sqrt {\frac {1-k^{2}t^{2}}{1-t^{2}}}}\,\mathrm {d} t}$	${\displaystyle E(m):=\int _{0}^{1}{\sqrt {\frac {1-m\,t^{2}}{1-t^{2}}}}\,\mathrm {d} t}$
II. Type: Legendre normal form	${\displaystyle E(k):=\int _{0}^{\pi /2}{\sqrt {1-k^{2}\sin ^{2}\vartheta }}\;{\text{d}}\vartheta }$	${\displaystyle E(m):=\int _{0}^{\pi /2}{\sqrt {1-m\sin ^{2}\vartheta }}\;{\text{d}}\vartheta }$
III. Type: Jacobian shape	${\displaystyle \Pi (n,k):=\int _{0}^{1}{\frac {\mathrm {d} t}{(1-n\,t^{2}){\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}}}$	${\displaystyle \Pi (n,m):=\int _{0}^{1}{\frac {\mathrm {d} t}{(1-n\,t^{2}){\sqrt {(1-t^{2})(1-m\,t^{2})}}}}}$
III. Type: Legendre normal form	${\displaystyle \Pi (n,k):=\int _{0}^{\pi /2}{\frac {{\text{d}}\vartheta }{(1-n\sin ^{2}\vartheta ){\sqrt {1-k^{2}\sin ^{2}\vartheta }}}}}$	${\displaystyle \Pi (n,m):=\int _{0}^{\pi /2}{\frac {{\text{d}}\vartheta }{(1-n\sin ^{2}\vartheta ){\sqrt {1-m\sin ^{2}\vartheta }}}}}$

Definition of complementary elliptic integrals

The complementary full elliptic integrals and are defined with the complementary variable as shown below. ${\displaystyle K'}$ ${\displaystyle E'}$ ${\displaystyle k'}$

${\displaystyle {\begin{aligned}k'&:={\sqrt {1-k^{2}}}\\K'(k)&:=K(k')\\E'(k)&:=E(k')\end{aligned}}}$

Representation by power series

The complete elliptic integrals can be represented as a power series. The specified power series can be used for numerical evaluation. However, it should be noted that the convergence depends on the argument . The use of power series is not the most efficient method for numerical evaluation in terms of computing time. If it is clear in a physical application that the argument lies in a suitable range in terms of accuracy, then the power series representation in the sense of linearization offers a useful method for specifying approximate solutions or rules of thumb. ${\displaystyle k}$ ${\displaystyle k}$

${\displaystyle K(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]^{2}k^{2n}={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}k^{2n}={\frac {\pi }{2}}\left(1+\sum _{n=1}^{\infty }\left({\frac {(2n-1)!}{2^{2n-1}n!(n-1)!}}\right)^{2}k^{2n}\right)}$

${\displaystyle E(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]^{2}{\frac {k^{2n}}{1-2n}}={\frac {\pi }{2}}\left(1-\sum _{n=0}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{2n-1}}\right)={\frac {\pi }{2}}\left(1-\sum _{n=1}^{\infty }\left({\frac {(2n-1)!}{2^{2n-1}n!(n-1)!}}\right)^{2}{\frac {k^{2n}}{2n-1}}\right)}$

Representation as an infinite product

The following table gives product representations of the complete first type elliptic integral and the complementary first type elliptic integral. Often the complementary variable is also used for a more compact representation. The interchanging of and with regard to the two product formulas when compared to the complementary is noticeable . ${\displaystyle k'={\sqrt {1-k^{2}}}}$ ${\displaystyle k_{n}}$ ${\displaystyle k_{n}'={\sqrt {1-k_{n}^{2}}}}$

Product representation of the complete elliptical integral I. Art

	Complete elliptical integral I. Art	Complementary elliptical integral I. Art
Initial value	${\displaystyle k_{0}:=k}$	${\displaystyle k_{0}:=k}$
Recursion equation	${\displaystyle k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}}$	${\displaystyle k_{n+1}={\frac {2{\sqrt {k_{n}}}}{1+k_{n}}}}$
Product formulas	${\displaystyle K(k)={\frac {\pi }{2}}\prod _{n=0}^{\infty }{\frac {2}{1+{\sqrt {1-k_{n}^{2}}}}}={\frac {\pi }{2}}\prod _{n=1}^{\infty }(1+k_{n})}$	${\displaystyle K'(k_{0})={\frac {\pi }{2}}\prod _{n=0}^{\infty }{\frac {2}{1+k_{n}}}={\frac {\pi }{2}}\prod _{n=1}^{\infty }(1+{\sqrt {1-k_{n}^{2}}})}$

Representation using the AGM algorithm

In addition to the power series, there is a representation as a limit value of the iterated arithmetic-geometric mean value (AGM algorithm). The following shows the arithmetic mean , the geometric mean and an auxiliary variable. The initial values are defined as indicated by the argument . It should be noted that for the complete elliptic integral of type I runs into infinity. Therefore it can not be calculated. However, this is not a problem, as this value is exactly too known. A case distinction is therefore required for an implementation. The parameter convention can also be calculated using the AGM algorithm. Only substitution is required . Practice shows that when using double-precision ( decimal places after the decimal point ), a choice of recursion steps delivers the best results. With the accuracy decreases due to rounding errors . This small number of recursion steps shows the efficiency of the AGM algorithm. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a_{0},b_{0},c_{0}}$ ${\displaystyle k}$ ${\displaystyle k=1}$ ${\displaystyle E(1)}$ ${\displaystyle E(1)=1}$ ${\displaystyle E(m),\;K(m)}$ ${\displaystyle m=k^{2}}$ ${\displaystyle \approx 16}$ ${\displaystyle n=4}$ ${\displaystyle n>4}$

AGM algorithm for calculating elliptic integrals

Initial values	Recursion equations	Elliptic integrals
${\displaystyle a_{0}:=1}$	${\displaystyle a_{n+1}={\frac {a_{n}+b_{n}}{2}}}$	${\displaystyle K(k)={\frac {\pi }{2\displaystyle \lim _{n\rightarrow \infty }a_{n}}}\equiv {\frac {\pi }{2\displaystyle \lim _{n\rightarrow \infty }b_{n}}}}$
${\displaystyle b_{0}:={\sqrt {1-k^{2}}}}$	${\displaystyle b_{n+1}={\sqrt {a_{n}b_{n}}}}$	${\displaystyle E(k)=\left(1-\sum _{n=0}^{\infty }2^{n-1}c_{n}^{2}\right)\cdot K(k)}$
${\displaystyle c_{0}:=k}$	${\displaystyle c_{n+1}={\frac {a_{n}-b_{n}}{2}}}$	${\displaystyle \Pi (n,k)={\frac {1}{2}}\cdot \left(2+{\frac {n}{1-n}}\sum _{n=0}^{\infty }Q_{n}\right)\cdot K(k)}$
${\displaystyle p_{0}:={\sqrt {1-n}}}$	${\displaystyle p_{n+1}={\frac {p_{n}^{2}+a_{n}b_{n}}{2p_{n}}}}$
${\displaystyle Q_{0}:=1}$	${\displaystyle Q_{n+1}={\frac {Q_{n}}{2}}\cdot {\frac {p_{n}^{2}-a_{n}b_{n}}{p_{n}^{2}+a_{n}b_{n}}}}$

The so-called Quartic AGM algorithm, the iteration rule of which is shown in the following table, is also found by substitution according to . The term "quartic" refers to the convergence of the algorithm. The convergence order of the algorithm in the table above is quadratic. ${\displaystyle \alpha _{n}={\sqrt {a_{2n}}}\;,\;\beta _{n}={\sqrt {b_{2n}}}}$

Quartic AGM algorithm for computing elliptic integrals

Initial values	Recursion equations	Elliptic integrals
${\displaystyle \alpha _{0}:=1}$	${\displaystyle \alpha _{n+1}={\frac {\alpha _{n}+\beta _{n}}{2}}}$	${\displaystyle K(k)={\frac {\pi }{2\displaystyle \lim _{n\rightarrow \infty }\alpha _{n}^{2}}}}$
${\displaystyle \beta _{0}:=\left(1-k^{2}\right)^{1/4}}$	${\displaystyle \beta _{n+1}=\left({\frac {\alpha _{n}^{3}\beta _{n}+\beta _{n}^{3}\alpha _{n}}{2}}\right)^{1/4}}$	${\displaystyle E(k)=\left(1-\sum _{n=0}^{\infty }4^{n}\left(\alpha _{n}^{4}-\left({\frac {\alpha _{n}^{2}+\beta _{n}^{2}}{2}}\right)^{2}\right)\right)\cdot K(k)}$

Special characteristics and identities

Here are , and again, the complementary sizes . ${\displaystyle K'}$ ${\displaystyle E'}$ ${\displaystyle k'}$

Special function values

${\displaystyle {\begin{aligned}K(0)&={\frac {\pi }{2}}&\lim _{k\rightarrow 0}K'(k)&=\infty \\\lim _{k\rightarrow 1}K(k)&=\infty &K'(1)&={\frac {\pi }{2}}\\E(0)&={\frac {\pi }{2}}&E'(0)&=1\\E(1)&=1&E'(1)&={\frac {\pi }{2}}\\\Pi (0,0)&={\frac {\pi }{2}}\\\lim _{k\rightarrow 1}\Pi (0,k)&=\infty \end{aligned}}}$

Special identities

${\displaystyle {\begin{aligned}K(k)={\tfrac {1}{n}}F({\tfrac {n\pi }{2}},k)\\E(k)={\tfrac {1}{n}}E({\tfrac {n\pi }{2}},k)\end{aligned}}}$

${\displaystyle {\begin{aligned}\Pi (0,m)&\equiv K(m)\\\Pi (n,0)&\equiv {\frac {\pi /2}{\sqrt {1-n}}}\\\Pi (m,m)&\equiv {\frac {E(m)}{1-m}}\end{aligned}}}$

${\displaystyle {\begin{aligned}K(k)&\equiv {\frac {1}{1+k}}\;K\left({\frac {2{\sqrt {k}}}{1+k}}\right)&K'(k)&\equiv {\frac {2}{1+k}}\;K'\left({\frac {2{\sqrt {k}}}{1+k}}\right)\\K(k)&\equiv {\frac {2}{1+k'}}\;K\left({\frac {1-k'}{1+k'}}\right)&K'(k)&\equiv {\frac {1}{1+k'}}\;K'\left({\frac {1-k'}{1+k'}}\right)\\E(k)&\equiv {\frac {1+k}{2}}\;E\left({\frac {2{\sqrt {k}}}{1+k}}\right)+{\frac {k'^{2}}{2}}\;K\left(k\right)&E'(k)&\equiv (1+k)\;E'\left({\frac {2{\sqrt {k}}}{1+k}}\right)-k\;K'\left(k\right)\\E(k)&\equiv (1+k')\;E\left({\frac {1-k'}{1+k'}}\right)-k'\;K(k)&E'(k)&\equiv {\frac {1+k'}{2}}\;E'\left({\frac {1-k'}{1+k'}}\right)+{\frac {k^{2}}{2}}\;K'(k)\end{aligned}}}$

Derivatives

${\displaystyle {\begin{aligned}{\frac {{\text{d}}K(k)}{{\text{d}}k}}&={\frac {E(k)-(1-k^{2})\cdot K(k)}{k\cdot (1-k^{2})}}\\{\frac {{\text{d}}E(k)}{{\text{d}}k}}&={\frac {E(k)-K(k)}{k}}\end{aligned}}}$

Inverse functions

Inverse functions or algebraic functions of inverse functions of the elliptic integrals are called elliptic functions . They are related to the trigonometric functions .

Incomplete elliptic integrals

Definition of incomplete elliptic integrals

Graph of the elliptic integrals of the first kind in Legendre form for various parameters ${\displaystyle F(\varphi ,m)}$ ${\displaystyle m=k^{2}}$

Graph of the elliptic integrals of the second kind in Legendre form for various parameters ${\displaystyle E(\varphi ,m)}$ ${\displaystyle m=k^{2}}$

In the following table the definitions of the incomplete elliptic integrals are given in Jacobi form and in Legendre normal form. The Jacobi form can be converted into the Legendre normal form with the substitution . Compared to the complete elliptic integrals, the incomplete elliptic integrals have an additional degree of freedom which corresponds to the upper limit of integration. Thus the complete elliptic integrals represent a special case of the incomplete. In the function libraries of Matlab , Wolfram-Alpha , Mathematica , Python ( SciPy ) and GNU Octave the parameter and the Legendre normal form are used. ${\displaystyle t=\sin \vartheta }$ ${\displaystyle m=k^{2}}$

Definition of the incomplete elliptic integrals with parameters and ${\displaystyle k}$ ${\displaystyle m}$

	Convention with parameters ${\displaystyle k}$	Convention with parameters ${\displaystyle m}$
I. Type: Jacobian shape	${\displaystyle F(x,k):=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}}$	${\displaystyle F(x,m):=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(1-m\,t^{2})}}}}$
I. Type: Legendre normal form	${\displaystyle F(\varphi ,k):=\int _{0}^{\varphi }{\frac {\mathrm {d} \vartheta }{\sqrt {1-k^{2}\sin ^{2}\vartheta }}}}$	${\displaystyle F(\varphi ,m):=\int _{0}^{\varphi }{\frac {\mathrm {d} \vartheta }{\sqrt {1-m\sin ^{2}\vartheta }}}}$
II. Type: Jacobian form	${\displaystyle E(x,k):=\int _{0}^{x}{\sqrt {\frac {1-k^{2}t^{2}}{1-t^{2}}}}\,\mathrm {d} t}$	${\displaystyle E(x,m):=\int _{0}^{x}{\sqrt {\frac {1-m\,t^{2}}{1-t^{2}}}}\,\mathrm {d} t}$
II. Type: Legendre normal form	${\displaystyle E(\varphi ,k):=\int _{0}^{\varphi }{\sqrt {1-k^{2}\sin ^{2}\vartheta }}\,\mathrm {d} \vartheta }$	${\displaystyle E(\varphi ,m):=\int _{0}^{\varphi }{\sqrt {1-m\sin ^{2}\vartheta }}\,\mathrm {d} \vartheta }$
III. Type: Jacobian shape	${\displaystyle \Pi (n,x,k):=\int _{0}^{x}{\frac {\mathrm {d} t}{(1-n\,t^{2}){\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}}}$	${\displaystyle \Pi (n,x,m):=\int _{0}^{x}{\frac {\mathrm {d} t}{(1-n\,t^{2}){\sqrt {(1-t^{2})(1-m\,t^{2})}}}}}$
III. Type: Legendre normal form	${\displaystyle \Pi (n,\varphi ,k):=\int _{0}^{\varphi }{\frac {\mathrm {d} \vartheta }{(1-n\sin ^{2}\vartheta ){\sqrt {1-k^{2}\sin ^{2}\vartheta }}}}}$	${\displaystyle \Pi (n,\varphi ,m):=\int _{0}^{\varphi }{\frac {\mathrm {d} \vartheta }{(1-n\sin ^{2}\vartheta ){\sqrt {1-m\sin ^{2}\vartheta }}}}}$

Alternative representations

Symmetrical Carlson shapes

The symmetrical Carlson forms are an alternate set of functions by which the classical elliptical integrals can be expressed. The more modern Carlson forms were not invented until the 1960s, while the Legendre forms had been formulated as early as 1825. The Carlson shapes offer several advantages over the classic elliptical integrals.

Incomplete elliptic integrals

Incomplete elliptic integrals can be expressed using the symmetric Carlson forms , and : ${\displaystyle R_{F}}$ ${\displaystyle R_{D}}$ ${\displaystyle R_{J}}$

${\displaystyle F(\varphi ,m)=\sin \varphi \;R_{F}({\cos ^{2}}\varphi ,1-{m\sin ^{2}}\varphi ,1)}$

${\displaystyle E(\varphi ,m)=\sin \varphi \;R_{F}({\cos ^{2}}\varphi ,1-{m\sin ^{2}}\varphi ,1)-{\tfrac {1}{3}}{m\sin ^{3}}\varphi \;R_{D}({\cos ^{2}}\varphi ,1-{m\sin ^{2}}\varphi ,1)}$

${\displaystyle \Pi (n,\varphi ,m)=\sin \varphi \;R_{F}({\cos ^{2}}\varphi ,1-{m\sin ^{2}}\varphi ,1)+{\tfrac {1}{3}}{n\sin ^{3}}\phi \;R_{J}({\cos ^{2}}\varphi ,1-{m\sin ^{2}}\varphi ,1,1-{n\sin ^{2}}\varphi )}$

(for and ) ${\displaystyle 0\leq \varphi \leq 2\pi }$ ${\displaystyle 0\leq {m\sin ^{2}}\varphi \leq 1}$

Complete elliptic integrals

Complete elliptic integrals are obtained by inserting φ = π / 2:

${\displaystyle K(m)=R_{F}\left(0,1-m,1\right)}$

${\displaystyle E(m)=R_{F}\left(0,1-m,1\right)-{\tfrac {1}{3}}mR_{D}\left(0,1-m,1\right)}$

${\displaystyle \Pi (n,m)=R_{F}\left(0,1-m,1\right)+{\tfrac {1}{3}}nR_{J}\left(0,1-m,1,1-n\right)}$

Bulirsch integrals

A generalized version of the complete elliptic integrals is the Bulirsch -Integral

${\displaystyle \operatorname {cel} (k_{c},p,a,b)=\int _{0}^{\pi /2}{\frac {a\cos ^{2}\theta +b\sin ^{2}\theta }{(\cos ^{2}\theta +p\sin ^{2}\theta ){\sqrt {\cos ^{2}\theta +k_{c}^{2}\sin ^{2}\theta }}}}\,\mathrm {d} \theta }$

It applies

${\displaystyle {\begin{aligned}K(m)&=\operatorname {cel} ({\sqrt {1-m}},1,1,1)\\E(m)&=\operatorname {cel} ({\sqrt {1-m}},1,1,1-m)\\\Pi (n,m)&=\operatorname {cel} ({\sqrt {1-m}},1-n,1,1)\end{aligned}}}$

The function cel has the advantage that certain combinations of the normal elliptic integrals occurring in practice can be represented as a common function, and thus numerical instabilities and undefined value ranges can be avoided.

${\displaystyle {\begin{aligned}\lambda \,K(m)+\mu \,E(m)&=\operatorname {cel} ({\sqrt {1-m}},1,\lambda +\mu ,\lambda +\mu (1-m))\\\lambda \,K(m)+\mu \,\Pi (n,m)&=\operatorname {cel} ({\sqrt {1-m}},1-n,\lambda +\mu ,\lambda (1-n)+\mu )\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {K(m)-E(m)}{m}}&=\operatorname {cel} ({\sqrt {1-m}},1,0,1)\\{\frac {(m-1)K(m)+E(m)}{m}}&=\operatorname {cel} ({\sqrt {1-m}},1,1,0)\\{\frac {\Pi (n,m)-K(m)}{n}}&=\operatorname {cel} ({\sqrt {1-m}},1-n,0,1)\end{aligned}}}$

Numerical evaluation

The elliptical integrals can be calculated efficiently with the aid of the above-mentioned arithmetic-geometric mean. They can also be converted into the symmetrical Carlson form for evaluation . An approximation using fractional-rational functions of a higher order is also possible. A direct numerical quadrature e.g. B. with the tanh-sinh method is also possible.

Application examples

A classic application of elliptic integrals is the exact movement of a pendulum .

Circumference of an ellipse

A classic application is the calculation of the circumference of an ellipse . Below an ellipsis parameter form is with the semi-axes , indicated. The result is represented by the complete elliptic integral II. Art. The parameter convention is used here. ${\displaystyle {\boldsymbol {x}}(\psi )}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle m=k^{2}}$

${\displaystyle {\begin{aligned}{\boldsymbol {x}}(\psi )&={\begin{pmatrix}a\cos \psi \\b\sin \psi \end{pmatrix}}\\U&=\int _{0}^{2\pi }\left\|{\frac {{\text{d}}{\boldsymbol {x}}}{{\text{d}}\psi }}\right\|\,{\text{d}}\psi =\int _{0}^{2\pi }{\sqrt {a^{2}\sin ^{2}\psi +b^{2}\cos ^{2}\psi }}\;{\text{d}}\psi =4b\int _{0}^{\pi /2}{\sqrt {1-{\frac {b^{2}-a^{2}}{b^{2}}}\sin ^{2}\psi }}\;{\text{d}}\psi =4b\cdot E\left(1-{\frac {a^{2}}{b^{2}}}\right)=4a\cdot E\left(1-{\frac {b^{2}}{a^{2}}}\right)\end{aligned}}}$

The equivalence of the last two expressions can be seen if instead of before . The last expression is for . The associated application of the incomplete elliptic integral of type II results from the upper limit of integration being set as a variable as in the following. This gives the arc length of the ellipse as a function of the parameter . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle m>0}$ ${\displaystyle a>b}$ ${\displaystyle \psi }$ ${\displaystyle s}$ ${\displaystyle \psi }$

${\displaystyle s(\psi )=\int _{0}^{\psi }{\sqrt {a^{2}\sin ^{2}\xi +b^{2}\cos ^{2}\xi }}\;{\text{d}}\xi =b\cdot E\left(\psi ,\;1-{\frac {a^{2}}{b^{2}}}\right)}$

Electrical scalar potential of a homogeneous, continuous, ring-shaped charge distribution

A classic problem from electrostatics is the calculation of the electrical scalar potential for a given spatial charge distribution . In the case of a homogeneous, continuous, ring-shaped charge distribution, the electrical scalar potential can be described with the help of the complete elliptic integral of type I. The result is here with the parameter convention with specified. In the given solution represents the total electrical charge, the radius of the ring and the vacuum permittivity . The scalar potential is also given with the cylindrical coordinates. Since there is no dependency on the azimuth coordinate , it can be seen that the problem is cylinder-symmetrical. ${\displaystyle \varphi _{e}}$ ${\displaystyle K(m)}$ ${\displaystyle m=k^{2}}$ ${\displaystyle Q}$ ${\displaystyle R}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle (\rho ,\varphi ,z)}$ ${\displaystyle \varphi }$

${\displaystyle \varphi _{e}(\rho ,z)={\frac {Q}{4\pi ^{2}\varepsilon _{0}}}\int _{0}^{\pi }{\frac {{\text{d}}\psi }{\sqrt {\rho ^{2}+z^{2}+R^{2}-2R\rho \cos \psi }}}={\frac {Q}{2\pi ^{2}\varepsilon _{0}}}\cdot {\frac {1}{\sqrt {(\rho +R)^{2}+z^{2}}}}\cdot K\left({\frac {4R\rho }{(\rho +R)^{2}+z^{2}}}\right)}$

Electrical scalar potential of a homogeneous, continuous, ring-shaped dipole distribution

In addition to the simple charge distribution, it is also possible to consider an annular distribution of axially aligned dipoles . The solution of the electrical scalar potential is given below. In this case, represents the component of the electric dipole moment , the radius of the ring and the vacuum permittivity. The result is here with the parameter convention with specified. ${\displaystyle p_{z}}$ ${\displaystyle z}$ ${\displaystyle R}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle E(m)}$ ${\displaystyle m=k^{2}}$

${\displaystyle \varphi _{e}(\rho ,z)={\frac {p_{z}\cdot z}{4\pi ^{2}\varepsilon _{0}}}\int _{0}^{\pi }{\frac {{\text{d}}\psi }{{\sqrt {\rho ^{2}+z^{2}+R^{2}-2R\rho \cos \psi }}^{3}}}={\frac {p_{z}}{2\pi ^{2}\varepsilon _{0}}}\cdot {\frac {z}{((\rho -R)^{2}+z^{2})\cdot {\sqrt {(\rho +R)^{2}+z^{2}}}}}\cdot E\left({\frac {4R\rho }{(\rho +R)^{2}+z^{2}}}\right)}$

Magnetic vector potential of a ring-shaped current-carrying conductor

An example from the magnetostatics of stationary currents is the calculation of the magnetic field of a ring conductor through which current flows . It is possible to calculate the magnetic vector potential , from which the magnetic flux density can be determined in further consideration with the help of the rotation . Here represents the electrical current strength , the radius of the ring conductor and the vacuum permeability . Furthermore, the magnetic vector potential is indicated with the cylindrical coordinates and with the unit base vector in the azimuthal direction. The solution is presented by a combination of complete elliptic integral 1st and 2nd Art. The result is here with the parameter convention with specified. ${\displaystyle {\boldsymbol {A}}}$ ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle \mu _{0}}$ ${\displaystyle (\rho ,\varphi ,z)}$ ${\displaystyle {\boldsymbol {e}}_{\varphi }}$ ${\displaystyle E(m),\;K(m)}$ ${\displaystyle m=k^{2}}$

${\displaystyle {\boldsymbol {A}}(\rho ,z)={\boldsymbol {e}}_{\varphi }\cdot {\frac {\mu _{0}IR}{2\pi }}\cdot \int _{0}^{\pi }{\frac {\cos \psi \;{\text{d}}\psi }{\sqrt {\rho ^{2}+z^{2}+R^{2}-2R\rho \cos \psi }}}={\boldsymbol {e}}_{\varphi }\cdot {\frac {\mu _{0}I}{2\pi }}\cdot {\frac {1}{\rho }}\left({\sqrt {(\rho +R)^{2}+z^{2}}}E\left({\frac {4R\rho }{(\rho +R)^{2}+z^{2}}}\right)-{\frac {\rho ^{2}+R^{2}+z^{2}}{\sqrt {(\rho +R)^{2}+z^{2}}}}K\left({\frac {4R\rho }{(\rho +R)^{2}+z^{2}}}\right)\right)}$

literature

• Louis Vessot King: On the direct numerical calculation of elliptic functions and integrals. Cambridge University Press, 1924, archive.org.
• Jonathan M. Borwein, Peter B. Borwein: Pi and the AGM. A study in analytical Number Theory and Computational Complexity. John Wiley & Sons, 1987.
• Harris Hancock: Elliptic Integrals . John Wiley & Sons, 1917.
• PF Byrd, MD Friedman: Handbook of Elliptic Integrals for Engineers and Scientists . Springer-Verlag 1971.
• Viktor Prasolov, Yuri Solovyev: Elliptic Functions and Elliptic Integrals . AMS 1997.

Web links

• Elliptic integrals. At: dlmf.nist.gov. (NIST Digital Library of Mathematical Functions).
• Eric W. Weisstein : Elliptic Integral . In: MathWorld (English).

Individual evidence

1. ↑ See Eric W. Weisstein : Complete Elliptic Integral of the First Kind . In: MathWorld (English). The shape without the !! symbol comes from:
   Bronstein, Semendjajew: Taschenbuch der Mathematik. Frankfurt / Main 1991, p. 223.
2. ↑ NIST Digital Library of Mathematical Functions 19.2: Bulirsch's Integrals .
3. ^ R. Bulirsch: Numerical calculation of elliptic integrals and elliptic functions. III . In: Numerical Mathematics . 13, No. 4, 1969, ISSN  0029-599X , pp. 305-315. doi : 10.1007 / BF02165405 .
4. ^ WH Press, SA Teukolsky, WT Vetterling, BP Flannery: Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions . In: Numerical Recipes: The Art of Scientific Computing , 3rd Edition, Cambridge University Press, New York 2007, ISBN 978-0-521-88068-8 .
5. ↑ Cephes Mathematical Library .