In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an elliptic integral as any function f which can be expressed in the form

$f(x)\; =\; int\_\{c\}^\{x\}\; R(t,P(t))\; dt\; ,!$

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

In general, elliptic integrals 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. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the forms given below, the elliptic integrals may also be expressed in Legendre form and 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, and this one in particular: we have F (sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.

Notation

Elliptic integrals are often expressed as functions for a variety of different arguments. These different arguments are completely equivalent (they give the same elliptic integral), but they can be confusing due to their different appearance. Most texts adhere to a canonical naming scheme. Before defining the integrals, we review the naming conventions for the arguments:

• $o!varepsilon,,!$ the modular angle;
• $k=sin\; o!varepsilon,,!$ the elliptic modulus;
• $m=k^2=[sin(o!varepsilon)]^2,,!$ the parameter;

Note that the above three conventions are completely determined by one another. Specifying one is the same as specifying another. The elliptic integrals also depend on another argument, which can be specified in a number of different ways:

• $phi,!$ the amplitude
• x where $x=sin\; phi=\; textrm\{sn\}\; ;\; u,!$
• u, where x = sn u and sn is one of the Jacobian elliptic functions

Specifying any one of these arguments determines the others. Thus, they may be used interchangeably in the notation. Note that u also depends on m. Some additional relationships involving u include

$cos\; phi\; =\; textrm\{cn\};\; u,!$

and

$sqrt\{1-msin^2\; phi\}\; =\; textrm\{dn\};\; u.,!$

The latter is sometimes called the delta amplitude and written as $Delta(phi)=textrm\{dn\};\; u,!$. 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.

Incomplete elliptic integral of the first kind

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

$F(phisetminus\; o!varepsilon\; )\; =\; F(phi|m)\; =$

int_0^phifrac{dtheta}{sqrt{1-(sinthetasin o!varepsilon)^2}}.,!

Equivalently, using notation in Jacobi's form, one sets $x=sin\; phi\; ~,~\; t=sin\; theta;!$; then

$F(phisetminus\; o!varepsilon\; )\; =\; F(x;k)\; =$

int_{0}^{x} frac{dt}{sqrt{(1-t^2)(1-k^2 t^2)} },! where it is understood that when there is a vertical bar used, the argument following the vertical bar is the parameter (as defined above); and, when a backslash is used, it is followed by the modular angle. In this sense, $F(sinphi;sin\; o!varepsilon)\; =\; F(phi|sin\; (o!varepsilon)^2)\; =\; F(phisetminus\; o!varepsilon\; )~\; ,!$, with the notations directly borrowed from the reference book of standards, Abramowitz and Stegun. The use of the delimiters ; | is traditional in elliptic integrals.

However, there remain different conventions for the notation of elliptic integrals! The differences can be very confusing, especially to a novice . The functions that evaluate the elliptic integrals do not have standard and unique names and meanings (like sqrt, sin and erf have). Even the literatures on the subject use differentiated notations. Gradstein, Ryzhik , Eq.(8.111)] and the Wikipedia article "Legendre form" use $F(phi,k)\; ,!$. The notation is equivalent to our $F(phi|k^2)~\; ,!$; also $E(phi,k)=E(phi|k^2)~\; ,!$ below.

Accordingly, if one translates the code from the Mathematica language into the language used by Maple, one should replace the argument of the EllipticK function with its square root. Correspondingly, in the translation from Maple to Mathematica, the argument should be replaced by its square. EllipticK(x) in Maple is almost equivalent to EllipticK[x^2] in Mathematica; one may expect to get the same result in both systems, at least while 0Note that

$F(x;k)\; =\; u\; ,!$

with u as defined above: thus, the Jacobian elliptic functions are inverses to the elliptic integrals.

Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind E is

$E(phisetminus\; o!varepsilon)\; =\; E(phi|m)\; =$

int_0^phi!E'(theta) dtheta = int_0^phisqrt{1-(sinthetasin o!varepsilon)^2} dtheta.,!

Equivalently, using an alternate notation (substituting $t=sintheta,!$),

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

Additional relations include

$E(phi|m)\; =\; int\_0^u\; textrm\{dn\}^2\; w\; ;dw\; =$

u-mint_0^u textrm{sn}^2 w ;dw = (1-m)u+mint_0^u textrm{cn}^2 w ;dw.,!

Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind $Pi,!$ is

$Pi(n;\; phi|m)\; =\; int\_0^phi\; frac\{1\}\{1-nsin^2\; theta\}$

frac {dtheta}{sqrt{1-(sinthetasin o!varepsilon)^2}},,!

or

$Pi(n;\; phi|m)\; =\; int\_\{0\}^\{x\}\; frac\{1\}\{1-nt^2\}$

frac{dt}{sqrt{(1-k^2 t^2)(1-t^2) }},,!

or

$Pi(n;\; phi|m)\; =\; int\_0^u\; frac\{dw\}\{1-n\; textrm\{sn\}^2\; (w|m)\}.\; ;\; ,!$

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value $Pi(1;pi/2|m),!$ is infinite, for any $m,!$.

Complete elliptic integral of the first kind

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

$K(k)\; =\; int\_0^\{frac\{pi\}\{2\}\}\; frac\{dtheta\}\{sqrt\{1-k^2\; sin^2theta\}\}$

or

$K(k)\; =\; int\_\{0\}^\{1\}\; frac\{dt\}\{sqrt\{(1-t^2)(1-k^2\; t^2)\}\}.!$

It is a special case of the incomplete elliptic integral of the first kind:

$K(k)\; =\; F(1;,k)\; =\; F(frac\{pi\}\{2\},|,k^2)!$

The special case can be expressed as a power series

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

which is equivalent to

$K(k)\; =\; frac\{pi\}\{2\}left\{1\; +\; left(frac\{1\}\{2\}right)^2\; k^\{2\}\; +\; left(frac\{1\; cdot\; 3\}\{2\; cdot\; 4\}right)^2\; k^\{4\}\; +\; cdots\; +\; left[frac\{left(2n\; -\; 1right)!!\}\{left(2nright)!!\}right]^2\; k^\{2n\}\; +\; cdots\; right\}.!$

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

$K(k)\; =\; frac\{pi\}\{2\}\; ,\_2F\_1\; left(frac\{1\}\{2\},\; frac\{1\}\{2\};\; 1;\; k^2right).,!$

The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed in terms of the arithmetic-geometric mean.

Special values

$K(0)\; =\; frac\; 1\; 2\; pi!$

$K(1)\; =\; infty!$

$Kleft(fracsqrt\{2\}\; 2right)\; =\; frac\{Gammaleft(frac\; 1\; 4right)^2\}\{4\; sqrt\{pi\}\}!$

$Kleft(frac\{sqrt\{6\}\; -\; sqrt\{2\}\}\{4\}right)\; =\; frac\{2^\{-frac\; 7\; 3\}\; 3^\{frac\; 1\; 4\}\; Gammaleft(frac\; 1\; 3right)^3\}\{pi\}!$

$Kleft(frac\{sqrt\{6\}\; +\; sqrt\{2\}\}\{4\}right)\; =\; frac\{2^\{-frac\; 7\; 3\}\; 3^\{frac\; 3\; 4\}\; Gammaleft(frac\; 1\; 3right)^3\}\{pi\}!$

The derivative of the complete elliptic integral of the first kind

$frac\{mathrm\{d\}K(k)\}\{mathrm\{d\}k\}\; =\; frac\{E(k)\}\{k(1-k^2)\}-frac\{K(k)\}\{k\}$

Complete elliptic integral of the second kind

The complete elliptic integral of the second kind E may be defined as

$E(k)\; =\; int\_0^\{frac\{pi\}\{2\}\}sqrt\; \{1-k^2\; sin^2theta\}\; dtheta!$

or

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

It is a special case of the incomplete elliptic integral of the second kind:

$E(k)\; =\; E(1;,k)\; =\; E(frac\{pi\}\{2\},|,k^2)!$

that can be expressed as a power series

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

which is

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

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

$E(k)\; =\; frac\{pi\}\{2\}\; ,\_2F\_1\; left(frac\{1\}\{2\},\; -frac\{1\}\{2\};\; 1;\; k^2right).,!$

Special values

$E(0)\; =\; frac\; pi\; 2!$

$E(1)\; =\; 1!$

$Eleft(fracsqrt\{2\}\; 2right)\; =\; pi^\{frac\; 3\; 2\}\; Gammaleft(frac\; 1\; 4right)^\{-2\}+frac\{Gammaleft(frac\; 1\; 4right)^2\}\{8\; sqrt\; pi\}!$

$Eleft(frac\{sqrt\{6\}\; -\; sqrt\{2\}\}\{4\}right)\; =\; 2^\{frac\; 1\; 3\}\; 3^\{-frac\; 3\; 4\}\; pi^2\; Gammaleft(frac\; 1\; 3right)^\{-3\}\; +\; 2^\{-frac\; \{10\}\; 3\}\; 3^\{-frac\; \{1\}\; 4\}\; frac\{sqrt3\; +\; 1\}\{pi\}\; Gammaleft(frac\; 1\; 3right)^3!$

$Eleft(frac\{sqrt\{6\}\; +\; sqrt\{2\}\}\{4\}right)\; =\; 2^\{frac\; 1\; 3\}\; 3^\{-frac\; 1\; 4\}\; pi^2\; Gammaleft(frac\; 1\; 3right)^\{-3\}\; +\; 2^\{-frac\; \{10\}\; 3\}\; 3^\{frac\; 1\; 4\}\; frac\{sqrt3\; -\; 1\}\{pi\}\; Gammaleft(frac\; 1\; 3right)^3!$

The derivative of the complete elliptic integral of the second kind

$frac\{mathrm\{d\}E(k)\}\{mathrm\{d\}k\}=frac\{E(k)-K(k)\}\{k\}$

Complete elliptic integral of the third kind

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

$Pi(n,k)\; =\; int\_0^\{frac\{pi\}\{2\}\}frac\{\; dtheta\}\{(1-nsin^2theta)sqrt\; \{1-k^2\; sin^2theta\}\}$

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign in $n$, i.e.

$Pi\text{'}(n,k)\; =\; int\_0^\{frac\{pi\}\{2\}\}frac\{\; dtheta\}\{(1+nsin^2theta)sqrt\; \{1-k^2\; sin^2theta\}\}.$

The partial derivatives of the complete elliptic integral of the third kind

$frac\{partialPi(n,k)\}\{partial\; n\}=$

frac{1}{2(k^2-n)(n-1)}left(E(k)+frac{(k^2-n)K(k)}{n}+frac{(n^2-k^2)Pi(n,k)}{n}right)

$frac\{partialPi(n,k)\}\{partial\; k\}=$

frac{k}{n-k^2}left(frac{E(k)}{k^2-1}+Pi(n,k)right)

See also

• Elliptic curve
• Schwarz-Christoffel map

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. (See chapter 17).
• Harris Hancock Lectures on the theory of Elliptic functions (New York, J. Wiley & sons, 1910)
• Alfred George Greenhill The applications of elliptic functions (New York, Macmillan, 1892)
• Louis V. King On The Direct Numerical Calculation Of Elliptic Functions And Integrals (Cambridge University Press, 1924)