Mathematical Physics Vol 1

5.8 Elliptic integrals

261

Note that

u = Z u = Z

cn ( u , k )

d t q ( 1 − t 2 )( k ′ 2 d t q ( 1 − t 2 )( t 2 − k ′ 2 ) . + k 2 t 2 )

,

1

dn ( u , k )

1

5.8.5 Main properties of elliptic functions

Relations between elliptic functions Akin to trigonometric functions, the following relations exist for elliptic functions sn 2 v + cn 2 v = 1 ,

dn 2 v + k 2 sn 2 v = 1 , dn 2 v − k 2 cn 2 v = k 2 .

(5.206)

The relations follow directly from the definitions of these functions. Some values

sn ( 0 , k )= 0 ,

cn ( 0 , k )= 1 ,

dn ( 0 , k )= 1 , dn ( u , 0 )= 1 ,

am ( 0 , k )= 0 ,

sn ( u , 0 )= sin u , cn ( u , 0 )= cos u ,

(5.207)

sn ( u , 1 )= th u ,

cn ( u , 1 )= sech u , dn ( u , 1 )= sech u .

Symmetricity of elliptic functions

su ( − u )= − su ( u ) , cu ( − u )= cu ( u ) , du ( − u )= du ( u ) , am ( − u ) = − am ( u ) .

(5.208) (5.209)

Additivity formulas

sn ( u ) cn ( v ) dn ( u ) ± cn ( u ) sn ( v ) dn ( u ) 1 − k 2 sn 2 u sn 2 v cn ( u ) cn ( v ) ∓ sn ( u ) dn ( u ) sn ( v ) dn ( v ) 1 − k 2 sn 2 u sn 2 v , dn ( u ) dn ( v ) ∓ k 2 sn ( u ) cn ( u ) sn ( v ) cn ( v ) 1 − k 2 sn 2 u sn 2 v . ??

sn ( u ± v )= cn ( u ± v )= dn ( u ± v )=

(5.210)

Some properties and periodicity of elliptic functions The main properties of elliptic functions follow directly from equations (5.196) and (5.198): am ( − v )= − am v , am ( v + 2 nK )= am v + n π . (5.211) It can be seen that this function is odd, but that it is not "purely" periodic, but rather pseudoperiodic. From (5.200) and u = sin ϕ the following basic elliptic functions are obtained

sin ϕ = sin ( am v )= sn v , cos ϕ = cos ( am v )= cn v , q 1 − k 2 sin ( am v )= dn v .

(5.212)