naginterfaces.library.specfun.jacellip_complex¶

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

jacellip_complex evaluates the Jacobian elliptic functions $s n (z)$ , $c n (z)$ and $d n (z)$ for a complex argument $z$ .

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

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s21cbf.html

Parameters

zcomplex: The argument $z$ of the functions.
ak2float: The value of $k^{2}$ .

Returns

sncomplex: The value of the function $s n (z)$ .
cncomplex: The value of the function $c n (z)$ .
dncomplex: The value of the function $d n (z)$ .

Raises

NagValueError

(errno $1$ )

On entry, $| I m (z) |$ is too large: $| I m (z) | = ⟨ v a l u e ⟩$ . It must be less than $⟨ v a l u e ⟩$ .

(errno $1$ )

On entry, $| R e (z) |$ is too large: $| R e (z) | = ⟨ v a l u e ⟩$ . It must be less than $⟨ v a l u e ⟩$ .

(errno $1$ )

On entry, $a k 2 = ⟨ v a l u e ⟩$ .

Constraint: $a k 2 \leq 1.0$ .

(errno $1$ )

On entry, $a k 2 = ⟨ v a l u e ⟩$ .

Constraint: $a k 2 \geq 0.0$ .

Notes

jacellip_complex evaluates the Jacobian elliptic functions $s n (z | k)$ , $c n (z | k)$ and $d n (z | k)$ given by

\begin{matrix} \begin{matrix} s n (z | k) & = & sin (ϕ) c n (z | k) & = & cos (ϕ) d n (z | k) & = & \sqrt{1 - k^{2} {sin}^{2} (ϕ)}, \end{matrix} \end{matrix}

where $z$ is a complex argument, $k$ is a real argument (the modulus) with $k^{2} \leq 1$ and $ϕ$ (the amplitude of $z$ ) is defined by the integral

z = \int_{0}^{ϕ} \frac{d θ}{\sqrt{1 - k^{2} {sin}^{2} (θ)}} .

The above definitions can be extended for values of $k^{2} > 1$ (see Salzer (1962)) by means of the formulae

\begin{matrix} \begin{matrix} s n (z | k) & = & k_{1} s n (k z | k_{1}) c n (z | k) & = & d n (k z | k_{1}) d n (z | k) & = & c n (k z | k_{1}), \end{matrix} \end{matrix}

where $k_{1} = 1 / k$ .

Special values include

\begin{matrix} \begin{matrix} s n (z | 0) & = & sin (z) c n (z | 0) & = & cos (z) d n (z | 0) & = & 1 s n (z | 1) & = & tanh (z) c n (z | 1) & = & s e c h (z) d n (z | 1) & = & s e c h (z) . \end{matrix} \end{matrix}

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

Another nine elliptic functions may be computed via the formulae

\begin{matrix} \begin{matrix} c d (z) & = & c n (z) / d n (z) s d (z) & = & s n (z) / d n (z) n d (z) & = & 1 / d n (z) d c (z) & = & d n (z) / c n (z) n c (z) & = & 1 / c n (z) s c (z) & = & s n (z) / c n (z) n s (z) & = & 1 / s n (z) d s (z) & = & d n (z) / s n (z) c s (z) & = & c n (z) / s n (z) \end{matrix} \end{matrix}

(see Abramowitz and Stegun (1972)).

The values of $s n (z)$ , $c n (z)$ and $d n (z)$ 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

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