Elliptic_integral Knowpia

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse.

Modern mathematics defines an "elliptic integral" as any function $f$ which can be expressed in the form

f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,dt,

where $R$ is a rational function of its two arguments, $P$ is a polynomial of degree 3 or 4 with no repeated roots, and $c$ is a constant.

In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when $P$ has repeated roots, or when $R (x, y)$ contains no odd powers of $y$ or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.

Argument notationEdit

Incomplete elliptic integrals are functions of two arguments; complete elliptic integrals are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral). Most texts adhere to a canonical naming scheme, using the following naming conventions.

For expressing one argument:

$α$ , the modular angle
$k = sin α$ , the elliptic modulus or eccentricity
$m = k 2 = sin 2 α$ , the parameter

Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably.

The other argument can likewise be expressed as $φ$ , the amplitude, or as $x$ or $u$ , where $x = sin φ = sn u$ and $sn$ is one of the Jacobian elliptic functions.

Specifying the value of any one of these quantities determines the others. Note that $u$ also depends on $m$ . Some additional relationships involving $u$ include

\cos \varphi =\operatorname {cn} u,\quad {\textrm {and}}\quad {\sqrt {1-m\sin ^{2}\varphi }}=\operatorname {dn} u.

The latter is sometimes called the delta amplitude and written as $Δ(φ) = dn u$ . Sometimes the literature also refers to the complementary parameter, the complementary modulus, or the complementary modular angle. These are further defined in the article on quarter periods.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude:

F(\varphi ,\sin \alpha )=F\left(\varphi \mid \sin ^{2}\alpha \right)=F(\varphi \setminus \alpha )=F(\sin \varphi ;\sin \alpha ).

This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, $F (k, φ)$ , is often encountered; and similarly $E (k, φ)$ for the integral of the second kind. Abramowitz and Stegun substitute the integral of the first kind, $F (φ, k)$ , for the argument $φ$ in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. $E (F (φ, k) | k 2)$ for $E (φ | k 2)$ . Moreover, their complete integrals employ the parameter $k 2$ as argument in place of the modulus $k$ , i.e. $K (k 2)$ rather than $K (k)$ . And the integral of the third kind defined by Gradshteyn and Ryzhik, $Π(φ, n, k)$ , puts the amplitude $φ$ first and not the "characteristic" $n$ .

Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, Wolfram's Mathematica software and Wolfram Alpha define the complete elliptic integral of the first kind in terms of the parameter $m$ , instead of the elliptic modulus $k$ .

Incomplete elliptic integral of the first kindEdit

The incomplete elliptic integral of the first kind $F$ is defined as

F(\varphi ,k)=F\left(\varphi \mid k^{2}\right)=F(\sin \varphi ;k)=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}.

This is the trigonometric form of the integral; substituting $t = sin θ$ and $x = sin φ$ , one obtains the Legendre normal form:

F(x;k)=\int _{0}^{x}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}}.

Equivalently, in terms of the amplitude and modular angle one has:

F(\varphi \setminus \alpha )=F(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}.

With $x = sn(u, k)$ one has:

F(x;k)=u;

demonstrating that this Jacobian elliptic function is a simple inverse of the incomplete elliptic integral of the first kind.

The incomplete elliptic integral of the first kind has following addition theorem:

F{\bigl [}\arctan(x),k{\bigr ]}+F{\bigl [}\arctan(y),k{\bigr ]}=F\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]

The elliptic modulus can be transformed that way:

F{\bigl [}\arcsin(x),k{\bigr ]}={\frac {2}{1+{\sqrt {1-k^{2}}}}}F\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]

Incomplete elliptic integral of the second kindEdit

The incomplete elliptic integral of the second kind $E$ in trigonometric form is

E(\varphi ,k)=E\left(\varphi \,|\,k^{2}\right)=E(\sin \varphi ;k)=\int _{0}^{\varphi }{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta .

Substituting $t = sin θ$ and $x = sin φ$ , one obtains the Legendre normal form:

E(x;k)=\int _{0}^{x}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,dt.

Equivalently, in terms of the amplitude and modular angle:

E(\varphi \setminus \alpha )=E(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}\,d\theta .

Relations with the Jacobi elliptic functions include

{\begin{aligned}E{\bigl (}\operatorname {sn} (u;k);k{\bigr )}&=\int _{0}^{u}\operatorname {dn} ^{2}(w;k)\,dw\\&=u-k^{2}\int _{0}^{u}\operatorname {sn} ^{2}(w;k)\,dw\\&=\left(1-k^{2}\right)u+k^{2}\int _{0}^{u}\operatorname {cn} ^{2}(w;k)\,dw.\end{aligned}}

The meridian arc length from the equator to latitude $φ$ is written in terms of $E$ :

m(\varphi )=a\left(E(\varphi ,e)+{\frac {d^{2}}{d\varphi ^{2}}}E(\varphi ,e)\right),

where

a

is the semi-major axis, and

e

is the eccentricity.

The incomplete elliptic integral of the second kind has following addition theorem:

E{\bigl [}\arctan(x),k{\bigr ]}+E{\bigl [}\arctan(y),k{\bigr ]}=E\left[\arctan \left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}\right)+\arctan \left({\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right),k\right]+{\frac {k^{2}xy}{k'^{2}x^{2}y^{2}+x^{2}+y^{2}+1}}\left({\frac {x{\sqrt {k'^{2}y^{2}+1}}}{\sqrt {y^{2}+1}}}+{\frac {y{\sqrt {k'^{2}x^{2}+1}}}{\sqrt {x^{2}+1}}}\right)

The elliptic modulus can be transformed that way:

E{\bigl [}\arcsin(x),k{\bigr ]}=\left(1+{\sqrt {1-k^{2}}}\right)E\left[\arcsin \left({\frac {\left(1+{\sqrt {1-k^{2}}}\right)x}{1+{\sqrt {1-k^{2}x^{2}}}}}\right),{\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right]-{\sqrt {1-k^{2}}}F{\bigl [}\arcsin(x),k{\bigr ]}+{\frac {k^{2}x{\sqrt {1-x^{2}}}}{1+{\sqrt {1-k^{2}x^{2}}}}}

Incomplete elliptic integral of the third kindEdit

The incomplete elliptic integral of the third kind $Π$ is

\Pi (n;\varphi \setminus \alpha )=\int _{0}^{\varphi }{\frac {1}{1-n\sin ^{2}\theta }}{\frac {d\theta }{\sqrt {1-\left(\sin \theta \sin \alpha \right)^{2}}}}

or

\Pi (n;\varphi \,|\,m)=\int _{0}^{\sin \varphi }{\frac {1}{1-nt^{2}}}{\frac {dt}{\sqrt {\left(1-mt^{2}\right)\left(1-t^{2}\right)}}}.

The number $n$ is called the characteristic and can take on any value, independently of the other arguments. Note though that the value $Π(1; .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}π/2 | m)$ is infinite, for any $m$ .

A relation with the Jacobian elliptic functions is

\Pi {\bigl (}n;\,\operatorname {am} (u;k);\,k{\bigr )}=\int _{0}^{u}{\frac {dw}{1-n\,\operatorname {sn} ^{2}(w;k)}}.

The meridian arc length from the equator to latitude $φ$ is also related to a special case of $Π$ :

m(\varphi )=a\left(1-e^{2}\right)\Pi \left(e^{2};\varphi \,|\,e^{2}\right).

Complete elliptic integral of the first kindEdit

Plot of the complete elliptic integral of the first kind

K (k)

Elliptic Integrals are said to be 'complete' when the amplitude $φ = π / 2$ and therefore $x = 1$ . The complete elliptic integral of the first kind $K$ may thus be defined as

K(k)=\int _{0}^{\tfrac {\pi }{2}}{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}=\int _{0}^{1}{\frac {dt}{\sqrt {\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}},

or more compactly in terms of the incomplete integral of the first kind as

K(k)=F\left({\tfrac {\pi }{2}},k\right)=F\left({\tfrac {\pi }{2}}\,|\,k^{2}\right)=F(1;k).

It can be expressed as a power series

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 }{\bigl (}P_{2n}(0){\bigr )}^{2}k^{2n},

where $P n$ is the Legendre polynomials, which is equivalent to

K(k)={\frac {\pi }{2}}\left(1+\left({\frac {1}{2}}\right)^{2}k^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}k^{4}+\cdots +\left({\frac {\left(2n-1\right)!!}{\left(2n\right)!!}}\right)^{2}k^{2n}+\cdots \right),

where $n!!$ denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

K(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2}\right).

The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed very efficiently in terms of the arithmetic–geometric mean:^[1]

K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}.

Therefore the modulus can be transformed that way:

{\begin{aligned}K(k)&={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{2\operatorname {agm} \left({\frac {1}{2}}+{\frac {\sqrt {1-k^{2}}}{2}},{\sqrt[{4}]{1-k^{2}}}\right)}}\\[4pt]&={\frac {\pi }{\left(1+{\sqrt {1-k^{2}}}\right)\operatorname {agm} \left(1,{\frac {2{\sqrt[{4}]{1-k^{2}}}}{\left(1+{\sqrt {1-k^{2}}}\right)}}\right)}}\\[4pt]&={\frac {2}{1+{\sqrt {1-k^{2}}}}}K\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)\end{aligned}}

This expression is valid for all $n\in \mathbb {N}$ and $0 \leq k \leq 1$ :

K(k)=n\left[\sum _{a=1}^{n}\operatorname {dn} \left({\frac {2a}{n}}K(k);k\right)\right]^{-1}K\left[k^{n}\prod _{a=1}^{n}\operatorname {sn} \left({\frac {2a-1}{n}}K(k);k\right)^{2}\right]

Relation to the gamma functionEdit

If $k 2 = λ (i \sqrt r)$ and $r\in \mathbb {Q} ^{+}$ (where $λ$ is the modular lambda function), then $K (k)$ is expressible in closed form in terms of the gamma function.^[2] For example, $r = 2$ and $r = 7$ give, respectively,^[3]

K\left({\sqrt {2}}-1\right)={\frac {\Gamma \left({\frac {1}{8}}\right)\Gamma \left({\frac {3}{8}}\right){\sqrt {{\sqrt {2}}+1}}}{8{\sqrt[{4}]{2}}{\sqrt {\pi }}}},

and

K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {\Gamma \left({\frac {1}{7}}\right)\Gamma \left({\frac {2}{7}}\right)\Gamma \left({\frac {4}{7}}\right)}{4{\sqrt[{4}]{7}}\pi }}.

More generally, the condition that

{\frac {iK'}{K}}={\frac {iK\left({\sqrt {1-k^{2}}}\right)}{K(k)}}

be in an imaginary quadratic field^{[note 1]} is sufficient.^[4]^[5] For instance, if

k = e 5 πi /6

, then

iK' / K = e 2 πi /3

and^[6]

K\left(e^{5\pi i/6}\right)={\frac {e^{-\pi i/12}\Gamma ^{3}\left({\frac {1}{3}}\right){\sqrt[{4}]{3}}}{4{\sqrt[{3}]{2}}\pi }}.

Relation to Jacobi theta functionEdit

The relation to Jacobi's theta function is given by

K(k)={\frac {\pi }{2}}\theta _{3}^{2}(q),

where the nome

q

is

q(k)=\exp \left(-\pi {\frac {K\left({\sqrt {1-k^{2}}}\right)}{K(k)}}\right).

Asymptotic expressionsEdit

K\left(k\right)\approx {\frac {\pi }{2}}+{\frac {\pi }{8}}{\frac {k^{2}}{1-k^{2}}}-{\frac {\pi }{16}}{\frac {k^{4}}{1-k^{2}}}

This approximation has a relative precision better than 3×10⁻⁴ for

k < 1 / 2

. Keeping only the first two terms is correct to 0.01 precision for

k < 1 / 2

.^{[citation needed]}

Differential equationEdit

The differential equation for the elliptic integral of the first kind is

{\frac {d}{dk}}\left(k\left(1-k^{2}\right){\frac {dK(k)}{dk}}\right)=k\,K(k)

A second solution to this equation is $K (\sqrt 1 - k 2)$ . This solution satisfies the relation

{\frac {d}{dk}}K(k)={\frac {E(k)}{k\left(1-k^{2}\right)}}-{\frac {K(k)}{k}}.

Continued fractionEdit

A continued fraction expansion is:^[7]

{\frac {K(k)}{2\pi }}=-{\frac {1}{4}}+\sum _{n=0}^{\infty }{\frac {q^{n}}{1+q^{2n}}}=-{\frac {1}{4}}+{\cfrac {1}{1-q+{\cfrac {\left(1-q\right)^{2}}{1-q^{3}+{\cfrac {q\left(1-q^{2}\right)^{2}}{1-q^{5}+{\cfrac {q^{2}\left(1-q^{3}\right)^{2}}{1-q^{7}+{\cfrac {q^{3}\left(1-q^{4}\right)^{2}}{1-q^{9}+\cdots }}}}}}}}}},

where the nome is

q = q (k)

.

Complete elliptic integral of the second kindEdit

Plot of the complete elliptic integral of the second kind

E (k)

The complete elliptic integral of the second kind $E$ is defined as

E(k)=\int _{0}^{\tfrac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\,d\theta =\int _{0}^{1}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\,dt,

or more compactly in terms of the incomplete integral of the second kind $E (φ, k)$ as

E(k)=E\left({\tfrac {\pi }{2}},k\right)=E(1;k).

For an ellipse with semi-major axis $a$ and semi-minor axis $b$ and eccentricity $e = \sqrt 1 - b 2 / a 2$ , the complete elliptic integral of the second kind $E (e)$ is equal to one quarter of the circumference $C$ of the ellipse measured in units of the semi-major axis $a$ . In other words:

C=4aE(e).

The complete elliptic integral of the second kind can be expressed as a power series^[8]

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

which is equivalent to

E(k)={\frac {\pi }{2}}\left(1-\left({\frac {1}{2}}\right)^{2}{\frac {k^{2}}{1}}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {k^{4}}{3}}-\cdots -\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{2n-1}}-\cdots \right).

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

E(k)={\tfrac {\pi }{2}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},-{\tfrac {1}{2}};1;k^{2}\right).

The modulus can be transformed that way:

E(k)=\left(1+{\sqrt {1-k^{2}}}\right)\,E\left({\frac {1-{\sqrt {1-k^{2}}}}{1+{\sqrt {1-k^{2}}}}}\right)-{\sqrt {1-k^{2}}}\,K(k)

ComputationEdit

Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean.^[1]

Define sequences $a n$ and $g n$ , where $a 0 = 1$ , $g 0 = \sqrt 1 - k 2 = k'$ and the recurrence relations $a n + 1 = a n + g n / 2$ , $g n + 1 = \sqrt a n g n$ hold. Furthermore, define

c_{n}={\sqrt {\left|a_{n}^{2}-g_{n}^{2}\right|}}.

By definition,

a_{\infty }=\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }g_{n}=\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right).

Also

\lim _{n\to \infty }c_{n}=0.

Then

E(k)={\frac {\pi }{2a_{\infty }}}\left(1-\sum _{n=0}^{\infty }2^{n-1}c_{n}^{2}\right).

In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all $| k | \leq 1$ . To speed up computation further, the relation $c n + 1 = c n 2 / 4 a n + 1$ can be used.

Furthermore, if $k 2 = λ (i \sqrt r)$ and $r\in \mathbb {Q} ^{+}$ (where $λ$ is the modular lambda function), then $E (k)$ is expressible in closed form in terms of

K(k)={\frac {\pi }{2\operatorname {agm} \left(1,{\sqrt {1-k^{2}}}\right)}}

and hence can be computed without the need for the infinite summation term. For example,

r = 1

and

r = 7

give, respectively,^[9]

E\left({\frac {1}{\sqrt {2}}}\right)=K\left({\frac {1}{\sqrt {2}}}\right)+{\frac {\pi }{4K\left({\frac {1}{\sqrt {2}}}\right)}},

and

E\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)={\frac {7+2{\sqrt {7}}}{14}}K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)+{\frac {\pi {\sqrt {7}}}{28K\left({\frac {3-{\sqrt {7}}}{4{\sqrt {2}}}}\right)}}.

Derivative and differential equationEdit

{\frac {dE(k)}{dk}}={\frac {E(k)-K(k)}{k}}

\left(k^{2}-1\right){\frac {d}{dk}}\left(k\;{\frac {dE(k)}{dk}}\right)=kE(k)

A second solution to this equation is $E (\sqrt 1 - k 2) - K (\sqrt 1 - k 2)$ .

Complete elliptic integral of the third kindEdit

Plot of the complete elliptic integral of the third kind

Π(n, k)

with several fixed values of

n

The complete elliptic integral of the third kind $Π$ can be defined as

\Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1-n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic $n$ ,

\Pi '(n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\left(1+n\sin ^{2}\theta \right){\sqrt {1-k^{2}\sin ^{2}\theta }}}}.

Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.^[1]

Partial derivativesEdit

{\begin{aligned}{\frac {\partial \Pi (n,k)}{\partial n}}&={\frac {1}{2\left(k^{2}-n\right)(n-1)}}\left(E(k)+{\frac {1}{n}}\left(k^{2}-n\right)K(k)+{\frac {1}{n}}\left(n^{2}-k^{2}\right)\Pi (n,k)\right)\\[8pt]{\frac {\partial \Pi (n,k)}{\partial k}}&={\frac {k}{n-k^{2}}}\left({\frac {E(k)}{k^{2}-1}}+\Pi (n,k)\right)\end{aligned}}

Jacobi zeta functionEdit

In 1829, Jacobi defined the Jacobi zeta function:

Z(\varphi ,k)=E(\varphi ,k)-{\frac {E(k)}{K(k)}}F(\varphi ,k).

It is periodic in

\varphi

with minimal period

\pi

. It is related to the Jacobi zn function by

Z(\varphi ,k)=\operatorname {zn} (F(\varphi ,k),k)

.

Functional relationsEdit

Legendre's relation:

K(k)E\left({\sqrt {1-k^{2}}}\right)+E(k)K\left({\sqrt {1-k^{2}}}\right)-K(k)K\left({\sqrt {1-k^{2}}}\right)={\frac {\pi }{2}}.

ReferencesEdit

NotesEdit

^ $K$ can be analytically extended to the complex plane.

ReferencesEdit

^ ^a ^b ^c Carlson 2010, 19.8.
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 296
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 298
^ Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. Bibcode:1949PNAS...35..371C. doi:10.1073/PNAS.35.7.371. PMC 1063041. PMID 16588908. S2CID 45071481.
^ Chowla, S.; Selberg, A. (1967). "On Epstein's Zeta-Function". Journal für die Reine und Angewandte Mathematik. 227: 86–110.
^ "Legendre elliptic integrals (Entry 175b7a)".
^ N.Bagis,L.Glasser.(2015)"Evaluations of a Continued fraction of Ramanujan". Rend.Sem.Mat.Univ.Padova, Vol.133 pp 1-10
^ "Complete elliptic integral of the second kind: Series representations (Formula 08.01.06.0002)".
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 26, 161

SourcesEdit

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 17". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 587. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Byrd, P. F.; Friedman, M.D. (1971). Handbook of Elliptic Integrals for Engineers and Scientists (2nd ed.). New York: Springer-Verlag. ISBN 0-387-05318-2.
Carlson, B. C. (1995). "Numerical Computation of Real or Complex Elliptic Integrals". Numerical Algorithms. 10 (1): 13–26. arXiv:math/9409227. Bibcode:1995NuAlg..10...13C. doi:10.1007/BF02198293. S2CID 11580137.
Carlson, B. C. (2010), "Elliptic integral", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953). Higher transcendental functions. Vol II (PDF). McGraw-Hill Book Company, Inc., New York-Toronto-London. MR 0058756.
Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.1.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.
Greenhill, Alfred George (1892). The applications of elliptic functions. New York: Macmillan.
Hancock, Harris (1910). Lectures on the Theory of Elliptic Functions. New York: J. Wiley & sons.
King, Louis V. (1924). On The Direct Numerical Calculation Of Elliptic Functions And Integrals. Cambridge University Press.
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

External linksEdit

"Elliptic integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Eric W. Weisstein, "Elliptic Integral" (Mathworld)
Matlab code for elliptic integrals evaluation by elliptic project
Rational Approximations for Complete Elliptic Integrals (Exstrom Laboratories)
A Brief History of Elliptic Integral Addition Theorems

[4] $K$ can be analytically extended to the complex plane.

[FOOTNOTECarlson201019.8-1] Carlson 2010, 19.8.

[2] Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 296

[3] Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 298

[5] Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. Bibcode:1949PNAS...35..371C. doi:10.1073/PNAS.35.7.371. PMC 1063041. PMID 16588908. S2CID 45071481.

[6] Chowla, S.; Selberg, A. (1967). "On Epstein's Zeta-Function". Journal für die Reine und Angewandte Mathematik. 227: 86–110.

[7] "Legendre elliptic integrals (Entry 175b7a)".

[8] N.Bagis,L.Glasser.(2015)"Evaluations of a Continued fraction of Ramanujan". Rend.Sem.Mat.Univ.Padova, Vol.133 pp 1-10

[9] "Complete elliptic integral of the second kind: Series representations (Formula 08.01.06.0002)".

[10] Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 26, 161

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[2]

[3]

[note 1]

[4]

[5]

[6]

[7]

[8]

[9]

Summary

Argument notationEdit

Incomplete elliptic integral of the first kindEdit

Incomplete elliptic integral of the second kindEdit

Incomplete elliptic integral of the third kindEdit

Complete elliptic integral of the first kindEdit

Relation to the gamma functionEdit

Relation to Jacobi theta functionEdit

Asymptotic expressionsEdit

Differential equationEdit

Continued fractionEdit

Complete elliptic integral of the second kindEdit

ComputationEdit

Derivative and differential equationEdit

Complete elliptic integral of the third kindEdit

Partial derivativesEdit

Jacobi zeta functionEdit

Functional relationsEdit

See alsoEdit

ReferencesEdit

NotesEdit

ReferencesEdit

SourcesEdit

External linksEdit