naginterfaces.library.specfun.mathieu_​ang_​periodic_​real

naginterfaces.library.specfun.mathieu_ang_periodic_real(ordval, q, parity, mode, x=None)

mathieu_ang_periodic_real calculates real-valued periodic angular Mathieu functions ($c{e}_m(x,q)$ or $s{e}_m(x,q)$) and/or their first derivatives, where $c{e}_m(x,q)$ and $s{e}_m(x,q)$ are solutions to the Mathieu differential equation $\frac{d^{2}y}{d{x}^2}+[a-2q\cos \left(2x\right)]y=0$.

For full information please refer to the NAG Library document for s22ca

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s22caf.html

Parameters

ordvalint

$m$, the order number of the Mathieu function to be computed.

qfloat

$q$, the Mathieu function parameter.

parityint

Specifies whether to compute even or odd Mathieu function.

$parity=0$

Compute even Mathieu function, $c{e}_m(x,q)$.

$parity=1$

Compute odd Mathieu function, $s{e}_m(x,q)$.

modeint

Specifies whether the Mathieu function or its derivative is required.

$mode=0$

Compute Mathieu function values.

$mode=1$

Compute derivative values of Mathieu function.

$mode=2$

Compute both Mathieu function and derivative values.

$mode=3$

Compute neither Mathieu functions nor derivative values, returns only $a$ (the characteristic value).

xNone or float, array-like, shape $(:)$, optional

Note: the required length for this argument is determined as follows: if $mode\ne 3$: $n$; otherwise: $0$.

The values of $x$ at which to compute Mathieu function or derivative values.

Returns

fNone or float, ndarray, shape $(:)$

If $mode=0$ or $2$, the Mathieu function values $c{e}_m(x,q)$ or $s{e}_m(x,q)$. If $mode=1$ or $3$, $f$ is not used.

f_derivNone or float, ndarray, shape $(:)$

If $mode=1$ or $2$, the Mathieu function derivative values $c{e}_m^{\prime }(x,q)$ or $s{e}_m^{\prime }(x,q)$. If $mode=0$ or $3$, $f\_deriv$ is not used.

afloat

$a$, the characteristic value for the Mathieu function.

Raises

NagValueError

(errno $1$)

On entry, $ordval=⟨value⟩$ and $parity=⟨value⟩$.

Constraint: if $parity=0$, $ordval\ge 0$ or if $parity=1$, $ordval\ge 1$.

(errno $2$)

On entry, $parity=⟨value⟩$.

Constraint: $parity=0$ or $1$.

(errno $3$)

On entry, $mode=⟨value⟩$.

Constraint: $mode=0$, $1$, $2$ or $3$.

(errno $4$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Notes

mathieu_ang_periodic_real calculates an approximation to $c{e}_m(x,q)$ and/or ${ce}_m^{\prime }(x,q)$, or $s{e}_m(x,q)$ and/or ${se}_m^{\prime }(x,q)$, where $c{e}_m(x,q)$ and $s{e}_m(x,q)$ are respectively the even and odd parity real-valued periodic angular Mathieu functions, for an array of values of $x$, and for integer order value $m$, where $m\ge 0$ for even parity, and $m\ge 1$ for odd parity. The function also returns values of $a$ for these periodic Mathieu functions, this is known as the characteristic value or eigenvalue.

The solutions are computed by approximating Mathieu functions as Fourier series, where expansion coefficients are obtained by solving the eigenvalue problem generated from the relevant recurrence relation, see Module 28 in NIST Digital Library of Mathematical Functions.

References

NIST Digital Library of Mathematical Functions