naginterfaces.library.specfun.jacellip_​complex

naginterfaces.library.specfun.jacellip_complex(z, ak2)[source]

jacellip_complex evaluates the Jacobian elliptic functions , and for a complex argument .

For full information please refer to the NAG Library document for s21cb

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s21cbf.html

Parameters
zcomplex

The argument of the functions.

ak2float

The value of .

Returns
sncomplex

The value of the function .

cncomplex

The value of the function .

dncomplex

The value of the function .

Raises
NagValueError
(errno )

On entry, is too large: . It must be less than .

(errno )

On entry, is too large: . It must be less than .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Notes

jacellip_complex evaluates the Jacobian elliptic functions , and given by

where is a complex argument, is a real argument (the modulus) with and (the amplitude of ) is defined by the integral

The above definitions can be extended for values of (see Salzer (1962)) by means of the formulae

where .

Special values include

These functions are often simply written as , and , thereby avoiding explicit reference to the argument . They can also be expressed in terms of Jacobian theta functions (see jactheta_real()).

Another nine elliptic functions may be computed via the formulae

(see Abramowitz and Stegun (1972)).

The values of , and are obtained by calls to jacellip_real(). Further details can be found in Further Comments.

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Salzer, H E, 1962, Quick calculation of Jacobian elliptic functions, Comm. ACM (5), 399