Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

262

Elliptic functions are double periodic with regard to the variable u : sn ( u , k ) periodic 4 K and 2 iK ′ , cn ( u , k ) periodic 4 K and 2 K + 2 iK ′ ,

(5.213)

dn ( u , k ) periodic 2 K and 4 iK ′ .

This is not hard to prove. For example, according to (5.211) it follows that sn ( v + 4 K )= sinam ( v + 4 K )= sin ( am ( v + 2 π )) , = sinam v = sinsn v

and similarly for the remaining functions. These three elliptic functions were also introduced by Jacobi, while the short denotement was introduced by Gudermann 14 . Beside the real period 4 K these functions have also and imaginary period. They are thus duble periodical. Derivatives of elliptic functions

dsin ϕ d v

d ϕ d v

d ( sn v ) d v

1 d v d ϕ

= cos ϕ

= cos ϕ

=

=

= cos ϕ q 1 − k 2 sin 2 ϕ = cn v dn v ,

(5.214)

d ( cn v ) d v d ( dn v ) d v

= − sn v dn v , = − k 2 sn v cn v .

Integrals of elliptic functions

Z sn ( u ) d u = Z cn ( u ) d u =

1 k ( dn ( u ) − k cn ( u )) ,

1 k

cos − 1 ( dn ( u )) , Z dn ( u ) d u = am ( u )= sin − 1 ( sn ( u )) .

(5.215)

Expansion of elliptic functions into series

u 3 3!

u 5 5!

+ 1 + 135 k 2 + 135 k 4 + k 6 u 7 7!

+ 1 + 14 k 2 + k 4

sn ( u , k )= u − ( 1 + k 2 )

+ ···

u 2 2!

u 4 4! −

u 6 6!

(5.216)

+( 1 + 4 k 2 )

( 1 + 44 k 2 + 16 k 4 )

cn ( u , k )= 1 − dn ( u , k )= 1 − k 2

+ ···

u 2 2!

u 4 4! −

u 6 6!

+ k 2 ( 4 + k 2 )

k 2 ( 16 + 44 k 2 + k 4 )

+ ···

If the following substitutions are introduced

q = e − piK / K ′

= π u / ( 2 K ) ,

and v

14 Gudermann