ELLIPTIC INTEGRAL

(Redirected from Complete elliptic integral of the second kind)
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 (a cubic or quartic) 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.

Contents
Notation
Incomplete elliptic integral of the first kind
Incomplete elliptic integral of the second kind
Incomplete elliptic integral of the third kind
Complete elliptic integral of the first kind
Special values
The derivative of the complete elliptic integral of the first kind
Complete elliptic integral of the second kind
Special values
The derivative of the complete elliptic integral of the second kind
Complete elliptic integral of the third kind
The partial derivatives of the complete elliptic integral of the third kind
See also
References

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! arepsilon,,! the 'modular angle';

★ k=sin o! arepsilon,,! the 'elliptic modulus';

★ m=k^2=sin(o! arepsilon)^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= extrm{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 = extrm{cn}; u,!
and
:sqrt{1-msin^2 phi} = extrm{dn}; u.,!
The latter is sometimes called the 'delta amplitude' and written as Delta(phi)= extrm{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! arepsilon ) = F(phi|m) =
int_0^phi rac{d heta}{sqrt{1-(sin hetasin o! arepsilon)^2}}.,!
Equivalently, using notation in Jacobi's form, one sets
x=sin phi ~,~ t=sin heta;!; then
: F(phisetminus o! arepsilon ) = F(x;k) =
int_{0}^{x} rac{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! arepsilon) = F(phi|sin (o! arepsilon)^2) = F(phisetminus o! arepsilon )~ ,!, 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 (as discussed ). The functions that evaluate the elliptic integrals do not have standard and unic names and meanings (like sqrt, sin and erf have). Even the literatures on the subject use differentiated notations. Gradstein, Ryzhik [1], 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 0 Note 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! arepsilon) = E(phi|m) =
int_0^phi!E'( heta) d heta = int_0^phisqrt{1-(sin hetasin o! arepsilon)^2} d heta.,!
Equivalently, using an alternate notation (substituting t=sin heta,!),
: E(x;k) = int_{0}^{x} rac{sqrt{1-k^2 t^2} }{sqrt{1-t^2}} dt. ,!
Additional relations include
:E(phi|m) = int_0^u extrm{dn}^2 w ;dw =
u-mint_0^u extrm{sn}^2 w ;dw =
(1-m)u+mint_0^u extrm{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 rac{1}{1-nsin^2 heta}
rac {d heta}{sqrt{1-(sin hetasin o! arepsilon)^2}},,!
or
: Pi(n; phi|m) = int_{0}^{x} rac{1}{1-nt^2}
rac{dt}{sqrt{(1-k^2 t^2)(1-t^2) }},,!
or
: Pi(n; phi|m) = int_0^u rac{dw}{1-n extrm{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


The 'complete elliptic integral of the first kind' ''K'' may be defined as
:K(k) = int_0^{ rac{pi}{2}} rac{d heta}{sqrt{1-k^2 sin^2 heta}}
or
:K(k) = int_{0}^{1} rac{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( rac{pi}{2},|,k^2)!
The special case can be expressed as a power series
:K(k) = rac{pi}{2} sum_{n=0}^{infty} left[ rac{(2n)!}{2^{2 n} n!^2}
ight]^2 k^{2n}!
which is equivalent to
:K(k) = rac{pi}{2}left{1 + left( rac{1}{2}
ight)^2 k^{2} + left( rac{1 cdot 3}{2 cdot 4}
ight)^2 k^{4} + cdots + left[ rac{left(2n - 1
ight)!!}{left(2n
ight)!!}
ight]^2 k^{2n} + cdots
ight}.!
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) = rac{pi}{2} F left( rac{1}{2}, rac{1}{2}; 1; k^2
ight).,!
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) = rac 1 2 pi!
:K(1) = infty!
:Kleft( racsqrt{2} 2
ight) = rac{Gammaleft( rac 1 4
ight)^2}{4 sqrt{pi}}!
:Kleft( rac{sqrt{6} - sqrt{2}}{4}
ight) = rac{2^{- rac 7 3} 3^{ rac 1 4} Gammaleft( rac 1 3
ight)^3}{pi}!
:Kleft( rac{sqrt{6} + sqrt{2}}{4}
ight) = rac{2^{- rac 7 3} 3^{ rac 3 4} Gammaleft( rac 1 3
ight)^3}{pi}!
The derivative of the complete elliptic integral of the first kind

: rac{mathrm{d}K(k)}{mathrm{d}k} = rac{E(k)}{k(1-k^2)}- rac{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^{ rac{pi}{2}}sqrt {1-k^2 sin^2 heta} d heta!
or
: E(k) = int_{0}^{1} rac{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( rac{pi}{2},|,k^2)!
that can be expressed as a power series
:E(k) = rac{pi}{2} sum_{n=0}^{infty} left[ rac{(2n)!}{2^{2 n} n!^2}
ight]^2 rac{k^{2n}}{1-2 n}!
which is
:E(k) = rac{pi}{2}left{1 - left( rac{1}{2}
ight)^2 rac{k^2}{1} - left( rac{1 cdot 3}{2 cdot 4}
ight)^2 rac{k^4}{3} - cdots - left[ rac{left(2n - 1
ight)!!}{left(2n
ight)!!}
ight]^2 rac{k^{2n}}{2 n-1} - cdots
ight}.!
In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
:E(k) = rac{pi}{2} F left( rac{1}{2}, - rac{1}{2}; 1; k^2
ight).,!
Special values

:E(0) = rac pi 2!
:E(1) = 1!
:Eleft( racsqrt{2} 2
ight) = pi^{ rac 3 2} Gammaleft( rac 1 4
ight)^{-2}+ rac{Gammaleft( rac 1 4
ight)^2}{8 sqrt pi}!
:Eleft( rac{sqrt{6} - sqrt{2}}{4}
ight) = 2^{ rac 1 3} 3^{- rac 3 4} pi^2 Gammaleft( rac 1 3
ight)^{-3} + 2^{- rac {10} 3} 3^{- rac {1} 4} rac{sqrt3 + 1}{pi} Gammaleft( rac 1 3
ight)^3!
:Eleft( rac{sqrt{6} + sqrt{2}}{4}
ight) = 2^{ rac 1 3} 3^{- rac 1 4} pi^2 Gammaleft( rac 1 3
ight)^{-3} + 2^{- rac {10} 3} 3^{ rac 1 4} rac{sqrt3 - 1}{pi} Gammaleft( rac 1 3
ight)^3!
The derivative of the complete elliptic integral of the second kind

: rac{mathrm{d}E(k)}{mathrm{d}k}= rac{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^{ rac{pi}{2}} rac{ d heta}{(1+nsin^2 heta)sqrt {1-k^2 sin^2 heta}}
The partial derivatives of the complete elliptic integral of the third kind

: rac{partialPi(n,k)}{partial n}=
rac{1}{2(k^2-n)(n-1)}left(E(k)+ rac{(k^2-n)K(k)}{n}+ rac{(n^2-k^2)Pi(n,k)}{n}
ight)
: rac{partialPi(n,k)}{partial k}=
rac{k}{n-k^2}left( rac{E(k)}{k^2-1}+Pi(n,k)
ight)

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)

★ George Greenhill The applications of elliptic functions (New York, Macmillan, 1892)

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