Elliptic integral

2020 Mathematics Subject Classification: Primary: 33E05 [MSN][ZBL]

An integral of an algebraic function of the first kind, that is, an integral of the form

$$ \tag{1 } \int\limits _ { z _ {0} } ^ { {z _ 1 } } R ( z , w ) d z , $$

where $ R ( z , w ) $ is a rational function of the variables $ z $ and $ w $. These variables are connected by an equation

$$ \tag{2 } w ^ {2} = f ( z) \equiv a _ {0} z ^ {4} + a _ {1} z ^ {3} + a _ {2} z ^ {2} + a _ {3} z + a _ {4} , $$

in which $ f ( z) $ is a polynomial of degree 3 or 4 without multiple roots. Here it is usually understood that the integral (1) cannot be expressed in terms of only one elementary function. When such an expression is possible, then (1) is said to be a pseudo-elliptic integral.

The name elliptic integral stems from the fact that they appeared first in the rectification of the arc of an ellipse and other second-order curves in work by Jacob and Johann Bernoulli, G.C. Fagnano dei Toschi, and L. Euler, who at the end of the 17th century and the beginning of the 18th century laid the foundations of the theory of elliptic integrals and elliptic functions (cf. Elliptic function), which arise in the inversion of elliptic integrals (cf. Inversion of an elliptic integral).

To the equations (2) corresponds a two-sheeted compact Riemann surface $ F $ of genus $ g = 1 $, homeomorphic to a torus, on which $ z $ and $ w $, and hence also $ R ( z , w ) $, regarded as functions of a point of $ F $, are single-valued. The integral (1) is given as the integral $ \int _ {L} \omega $ of the Abelian differential $ \omega = R ( z , w ) d z $ on $ F $, taken along some rectifiable path $ L $. The specification of the beginning $ z _ {0} $ and the end $ z _ {1} $ of this path $ L $ does not determine completely the value of the elliptic integral (1), generally speaking; in other words, (1) is a many-valued function of $ z _ {0} $ and $ z _ {1} $.

Any elliptic integral can be expressed as a sum of elementary functions and linear combinations of canonical elliptic integrals of the first, second and third kinds. The latter can be written, for example, in the following form:

$$ I _ {1} = \int\limits \frac{dz}{w} ,\ I _ {2} = \int\limits z \frac{dz}{w} ,\ I _ {3} = \int\limits \frac{dz}{( z - c ) w } , $$

where $ c $ is the parameter of the elliptic integral of the third kind.

The differential $ dz / w $ corresponding to $ I _ {1} $ is finite everywhere on the Riemann surface $ F $, the differentials of the second kind and third kinds have a pole-type singularity with residue zero or a simple pole, respectively. Regarded as functions of the upper limit of integration with a fixed lower limit, these three elliptic integrals are many-valued on $ F $. If one cuts $ F $ along two cycles of a homology basis, then on the resulting simply-connected domain $ F ^ { * } $ the integrals $ I _ {1} $ and $ I _ {2} $ are single valued, while $ I _ {3} $ still has a logarithmic singularity that arises on going around the simple pole. On passing through a cut each integral changes by an integer multiple of the corresponding period or modulus of periodicity, while $ I _ {3} $ has in addition a third logarithmic period $ 2 \pi i $ corresponding to a circuit around the singular point. Thus, the computation of an integral of type (1) reduces to that of an integral along the path $ L ^ {*} $ on $ F ^ { * } $ joining the points $ z _ {0} $ and $ z _ {1} $, and the addition of the corresponding linear combination of periods.

By subjecting the variable $ z $ to certain transformations one can bring the function $ w $ and the basic elliptic integrals to normal forms. In Weierstrass normal form the relation

$$ w ^ {2} = 4 z ^ {3} - g _ {2} z - g _ {3} $$

holds, and the integral

$$ u = - \int\limits _ { z } ^ \infty \frac{dz}{w} $$

has the periods $ 2 \omega _ {1} , 2 \omega _ {3} $. The inversion of this elliptic integral gives the Weierstrass elliptic function $ {\mathcal p} ( z) $ with periods $ 2 \omega _ {1} , 2 \omega _ {3} $ and invariants $ g _ {2} , g _ {3} $( see Weierstrass elliptic functions). The calculation of the periods $ 2 \omega _ {1} , 2 \omega _ {3} $ from given invariants proceeds by means of the modular function $ J ( \tau ) $. If in a normal integral of the second kind

$$ \int\limits \frac{z dz }{w} $$

one takes a normal integral of the first kind $ u $ as integration variable, then for a suitable choice of the integration constant the equality

$$ \int\limits \frac{z d z }{w} = - \zeta ( u) $$

holds, where $ \zeta ( u) $ is the Weierstrass $ \zeta $- function. Here the periods of the normal integral of the second kind are equal to $ - 2 \eta _ {1} = 2 \zeta ( \omega _ {1} ) $, $ - 2 \eta _ {3} = 2 \zeta ( \omega _ {3} ) $. A normal integral of the third kind in Weierstrass form has the form

$$ I ( z , w ; z _ {0} , w _ {0} ) = \frac{1}{2} \int\limits \frac{( w + w _ {0} ) dz }{( z - z _ {0} ) w } = \ \mathop{\rm log} \frac{\sigma ( u - u _ {0} ) }{\sigma ( u) \sigma ( u _ {0} ) } + u \frac{\sigma ^ \prime ( u _ {0} ) }{\sigma ( u _ {0} ) } , $$

where $ \sigma ( u) $ is the Weierstrass $ \sigma $- function, $ z _ {0} = {\mathcal p} ( u _ {0} ) $, $ w _ {0} = {\mathcal p} ^ \prime ( u _ {0} ) $, $ u _ {0} \not\equiv 0 $ $ \mathop{\rm mod} ( 2 \omega _ {1} , 2 \omega _ {3} ) $. Here the transposition rule holds:

$$ I ( z , w ; z _ {0} , w _ {0} ) - I ( z _ {0} , w _ {0} ; z , w ) = $$

$$ = \ \frac{\sigma ^ \prime ( u _ {0} ) }{\sigma ( u _ {0} ) } u - \frac{\sigma ^ \prime ( u) }{\sigma ( u) } u _ {0} + ( 2 n + 1 ) \pi i , $$

where $ n $ is an integer. The periods of a normal integral of the third kind have the form

$$ - u _ {0} \eta _ {3} + \zeta ( u _ {0} ) \omega _ {1} + 2 n _ {1} \pi i ; $$

$$ - u _ {0} \eta _ {3} + \zeta ( u _ {0} ) \omega _ {3} + 2 n _ {3} \pi i , $$

where $ n _ {1} , n _ {3} $ are integers and $ 2 \pi i $ is the logarithmic period.

In applications on often comes across the Legendre normal form. Here

$$ w ^ {2} = ( 1- z ^ {2} ) ( 1 - k ^ {2} z ^ {2} ), $$

where $ k $ is called the modulus of the elliptic integral, $ k ^ {2} $ is sometimes called the Legendre modulus, and $ k ^ \prime = \sqrt {1 - k ^ {2} } $ is called the supplementary modulus. Most frequently the normal case occurs, when $ 0 < k < 1 $ and $ z = x = \sin t $ is a real variable. An elliptic integral of the first kind in Legendre normal form has the form

$$ u = \int\limits _ { 0 } ^ { z } \frac{dx}{\sqrt {( 1- x ^ {2} ) ( 1 - k ^ {2} x ^ {2} ) } } = \int\limits _ { 0 } ^ \phi \frac{dt}{\sqrt {1 - h ^ {2} \sin ^ {2} t } } = F ( \phi , k); $$

it is also called an incomplete elliptic integral of the first kind; $ \phi = \mathop{\rm am} u $ is called its amplitude. This is an infinite-valued function of $ u $. The inversion of a normal integral of the first kind leads to the Jacobi elliptic function $ z = \mathop{\rm sn} u $( see Jacobi elliptic functions).

The Legendre normal form of a normal integral of the second kind is

$$ \int\limits _ { 0 } ^ { z } \frac{\sqrt {1 - k ^ {2} x ^ {2} } }{\sqrt {1 - x ^ {2} } } dx = \int\limits _ { 0 } ^ \phi \sqrt {1 - k ^ {2} \sin \ ^ {2} t } dt = E ( \phi , k) = E ( u) ; $$

it is also called an incomplete elliptic integral of the second kind.

The integrals

$$ F \left ( \frac \pi {2} , k \right ) = K ( k) = K, $$

$$ F \left ( \frac \pi {2} , k ^ \prime \right ) = K ^ \prime ( k) = K ^ { \prime } , $$

$$ E \left ( \frac \pi {2} , k \right ) = E ( k) = E, $$

$$ E \left ( \frac \pi {2} , k ^ \prime \right ) = E ^ \prime ( k) = E ^ { \prime } , $$

are called complete elliptic integrals of the first and second kind, respectively. The Legendre integrals of the first kind have periods $ 4K $ and $ 2iK ^ { \prime } $, those of the second kind — $ 4E $ and $ 2i( K ^ { \prime } - E ^ { \prime } ) $.

The Legendre normal form of a normal integral of the third kind is

$$ \int\limits _ { 0 } ^ { z } \frac{dx}{( 1 - n ^ {2} x ^ {2} ) \sqrt {( 1 - x ^ {2} )( 1 - k ^ {2} x ^ {2} ) } } = $$

$$ = \ \int\limits _ { 0 } ^ \phi \frac{dt}{( 1 - n ^ {2} \sin ^ {2} t ) \sqrt {1 - k ^ {2} \sin ^ {2} t } } = $$

$$ = \ \Pi ( \phi ; n ^ {2} , k) = \Pi ( u; n ^ {2} ) , $$

where $ n ^ {2} $ is the parameter and, as a rule, $ - \infty < n ^ {2} < \infty $. When $ n ^ {2} < 0 $ or $ k ^ {2} < n ^ {2} < 1 $, it is called a circular integral, and when $ 0 < n ^ {2} < k ^ {2} $ or $ 1 < n ^ {2} $— a hyperbolic integral.

A normal integral of the third kind according to Jacobi is defined somewhat differently:

$$ \Pi _ {J} ( u; a) = k ^ {2} \mathop{\rm sn} a \cdot \mathop{\rm cn} a \ \cdot \mathop{\rm dn} a \int\limits _ { 0 } ^ { u } \frac{ \mathop{\rm sn} ^ {2} u du }{1 - k ^ {2} \mathop{\rm sn} ^ {2} u \cdot \mathop{\rm sn} ^ {2} a } , $$

where $ n ^ {2} = k ^ {2} \mathop{\rm sn} ^ {2} a $. The connection between Jacobi and Legendre integrals of the third kind can be expressed by the formula

$$ \Pi ( u; n ^ {2} ) = u + \frac{ \mathop{\rm sn} a }{ \mathop{\rm cn} a \cdot \mathop{\rm dn} a } \Pi _ {J} ( u; a); $$

a circular character corresponds to an imaginary $ a $ and a hyperbolic one to a real $ a $.

Side-by-side with elliptic functions, elliptic integrals have numerous and important applications in various problems of analysis, geometry and physics; in particular, in mechanics, astronomy and geodesy. There are tables of elliptic integrals and extensive guidebooks on the theory of elliptic integrals and functions, and also compendia of formulas.

For references see also Elliptic function.

References

[1]	V.M. Belyakov, R.I. Kravtsova, M.G. Rappoport, "Tables of elliptic integrals" , 1–2 , Moscow (1962–1963) (In Russian)
[2]	E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German) MR0015900 Zbl 0061.29906

References

[a1]	H. Hancock, "Theory of elliptic functions" , Dover, reprint (1958) MR0100106 Zbl 0084.07302
[a2]	M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972) MR0314236 Zbl 0543.33001