Mathematical Physics Vol 1

5.8 Elliptic integrals

259

3. Legendre normal elliptic integral of the first order is an odd function

F ( − ϕ , k )= − F ( ϕ , k )

(5.196)

Proof.

− ϕ Z 0

d ϕ q 1 − k 2 sin 2 ϕ

by substitution − ϕ = θ becomes

ϕ Z 0

d θ p 1 − k 2 sin 2 θ

= − F ( ϕ , k )

−

From (5.195), and according to (5.196), we obtain ϕ = − π / 2 F π 2 , k = F − π 2 , k + F ( π , k )= − F π 2

, k + F ( π , k )

and from that

F ( π , k )= 2 F

, k .

π 2

Definite integral (usually denoted by K )

π / 2 Z 0

1 Z 0

K = F

, k =

π 2

d ϕ q 1 − k 2 sin 2 ϕ

d u p ( 1 − u 2 )( 1 − k 2 u 2 )

(5.197)

=

is called a complete elliptic integral of the first kind .

5.8.2 Elliptic functions

Using the property (5.195) we obtain the following relation

F ( ϕ + n π , k )= F ( ϕ , k )+ 2 nK ,

(5.198)

where n is a positive or negative integer. The variable ϕ in equation (5.289), taken as a function of λ t obtained by an inversion of the integral, namely ϕ = ϕ ( λ , t ) (5.199)

represents the amplitude of the normal elliptic integral of the first order

ϕ = am λ t

(5.200)

and it is called the Jacobi elliptic function . It has a dimension of angle.