NAG Library Function Document

nag_jacobian_elliptic (s21cbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_jacobian_elliptic (s21cbc) evaluates the Jacobian elliptic functions

sn z

cn z

and

dn z

for a complex argument

z

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_jacobian_elliptic (Complex z, double ak2, Complex sn, Complex cn, Complex dn, NagError fail)

3 Description

nag_jacobian_elliptic (s21cbc) evaluates the Jacobian elliptic functions

sn (z ∣ k)

cn (z ∣ k)

and

dn (z ∣ k)

given by

\begin{array}{lcl} sn (z ∣ k) & = & \sin ϕ \\ cn (z ∣ k) & = & \cos ϕ \\ dn (z ∣ k) & = & \sqrt{1 - k^{2} \sin^{2} ϕ}, \end{array}

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{array}{lcl} sn (z ∣ k) & = & k_{1} sn (k z ∣ k_{1}) \\ cn (z ∣ k) & = & dn (k z ∣ k_{1}) \\ dn (z ∣ k) & = & cn (k z ∣ k_{1}), \end{array}

where

k_{1} = 1 / k

Special values include

\begin{array}{lcl} sn (z ∣ 0) & = & \sin z \\ cn (z ∣ 0) & = & \cos z \\ dn (z ∣ 0) & = & 1 \\ sn (z ∣ 1) & = & \tanh z \\ cn (z ∣ 1) & = & sech z \\ dn (z ∣ 1) & = & sech z . \end{array}

These functions are often simply written as

sn z

cn z

and

dn z

, thereby avoiding explicit reference to the argument

k

. They can also be expressed in terms of Jacobian theta functions (see nag_jacobian_theta (s21ccc)).

Another nine elliptic functions may be computed via the formulae

\begin{array}{lcl} cd z & = & cn z / dn z \\ sd z & = & sn z / dn z \\ nd z & = & 1 / dn z \\ dc z & = & dn z / cn z \\ nc z & = & 1 / cn z \\ sc z & = & sn z / cn z \\ ns z & = & 1 / sn z \\ ds z & = & dn z / sn z \\ cs z & = & cn z / sn z \end{array}

(see Abramowitz and Stegun (1972)).

The values of

sn z

cn z

and

dn z

are obtained by calls to nag_real_jacobian_elliptic (s21cac). Further details can be found in Section 9.

4 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

5 Arguments

1: z – ComplexInput

On entry: the argument

z

of the functions.

Constraints:

$abs (z . re) \leq = \sqrt{λ}$ ;
$abs (z . im) \leq \sqrt{λ}$ , where $λ = 1 / nag_real_safe_small_number$ .

2: ak2 – doubleInput

On entry: the value of

k^{2}

Constraint:

0.0 \leq ak2 \leq 1.0

3: sn – Complex *Output

4: cn – Complex *Output

5: dn – Complex *Output

On exit: the values of the functions

sn z

cn z

and

dn z

, respectively.

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_COMPLEX: On entry, $|z . im|$ is too large: $|z . im| = ⟨value⟩$ . It must be less than $⟨value⟩$ .
On entry, $|z . re|$ is too large: $|z . re| = ⟨value⟩$ . It must be less than $⟨value⟩$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $ak2 = ⟨value⟩$ .
Constraint: $ak2 \leq 1.0$ .
On entry, $ak2 = ⟨value⟩$ .
Constraint: $ak2 \geq 0.0$ .

7 Accuracy

In principle the function is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the standard elementary functions such as SIN and COS.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The values of

sn z

cn z

and

dn z

are computed via the formulae

\begin{array}{lcccl} sn z & = & \frac{sn (u, k) dn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} & + & i \frac{cn (u, k) dn (u, k) sn (v, k^{'}) cn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} \\ cn z & = & \frac{cn (u, k) cn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} & - & i \frac{sn (u, k) dn (u, k) sn (v, k^{'}) dn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} \\ dn z & = & \frac{dn (u, k) cn (v, k^{'}) dn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})} & - & i \frac{k^{2} sn (u, k) cn (u, k) sn (v, k^{'})}{1 - {dn}^{2} (u, k) {sn}^{2} (v, k^{'})}, \end{array}

where

z = u + i v

and

k^{'} = \sqrt{1 - k^{2}}

(the complementary modulus).

10 Example

This example evaluates

sn z

cn z

and

dn z

z = - 2.0 + 3.0 i

when

k = 0.5

, and prints the results.

NAG Library Function Documentnag_jacobian_elliptic (s21cbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_jacobian_elliptic (s21cbc)