naginterfaces.library.specfun.legendre_​p

naginterfaces.library.specfun.legendre_p(mode, x, m, nl)

legendre_p returns a sequence of values for either the unnormalized or normalized Legendre functions of the first kind $P_n^m\left(x\right)$ or $\overline{P_n^m}\left(x\right)$ for real $x$ of a given order $m$ and degree $n=0,1,\dots ,N$.

For full information please refer to the NAG Library document for s22aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s22aaf.html

Parameters

modeint

Indicates whether the sequence of function values is to be returned unnormalized or normalized.

$mode=1$

The sequence of function values is returned unnormalized.

$mode=2$

The sequence of function values is returned normalized.

xfloat

The argument $x$ of the function.

mint

The order $m$ of the function.

nlint

The degree $N$ of the last function required in the sequence.

Returns

pfloat, ndarray, shape $(nl+1)$

The required sequence of function values as follows:

if $mode=1$, $p\left[n\right]$ contains $P_n^m\left(x\right)$, for $\text{n}=0,1,\dots ,N$;

if $mode=2$, $p\left[n\right]$ contains $\overline{P_n^m}\left(x\right)$, for $\text{n}=0,1,\dots ,N$.

Raises

NagValueError

(errno $1$)

On entry, $nl=⟨value⟩$ and $\left|m\right|=⟨value⟩$.

Constraint: $nl+\left|m\right|\le 55$.

(errno $1$)

On entry, $\left|m\right|=⟨value⟩$.

Constraint: $\left|m\right|\le 27$.

(errno $1$)

On entry, $nl=⟨value⟩$.

Constraint: when $\left|m\right|\ne 0$, $nl\ge 0$.

(errno $1$)

On entry, $nl=⟨value⟩$.

Constraint: $nl\le 100$ when $m=0$.

(errno $1$)

On entry, $nl=⟨value⟩$.

Constraint: $nl\ge 0$.

(errno $1$)

On entry, $\left|x\right|=⟨value⟩$.

Constraint: $\left|x\right|\le 1.0$.

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $mode\le 2$.

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $mode\ge 1$.

Notes

legendre_p evaluates a sequence of values for either the unnormalized or normalized Legendre ($m=0$) or associated Legendre ($m\ne 0$) functions of the first kind $P_n^m\left(x\right)$ or $\overline{P_n^m}\left(x\right)$, where $x$ is real with $-1\le x\le 1$, of order $m$ and degree $n=0,1,\dots ,N$ defined by

$$\begin{array}{c}\begin{array}{cccc}{P}_n^m\left(x\right)& =& {(1-x^2)}^{m/2}\frac{d^m}{d{x}^m}{P}_n\left(x\right)& \text{if}m\ge 0\text{,}\\ & & & \\ P_n^m\left(x\right)& =& \frac{(n+m)!}{(n-m)!}{P}_n^{-m}\left(x\right)& \text{if}m<0\text{and}\\ & & & \\ \overline{P_n^m}\left(x\right)& =& \sqrt{\frac{(2n+1)}{2}\frac{(n-m)!}{(n+m)!}}{P}_n^m\left(x\right)& \end{array}\end{array}$$

respectively; $P_n\left(x\right)$ is the (unassociated) Legendre polynomial of degree $n$ given by

$$P_n\left(x\right)\equiv P_n^0\left(x\right)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}{(x^2-1)}^n$$

(the Rodrigues formula). Note that some authors (e.g., Abramowitz and Stegun (1972)) include an additional factor of ${(-1)}^m$ (the Condon–Shortley Phase) in the definitions of $P_n^m\left(x\right)$ and $\overline{P_n^m}\left(x\right)$. They use the notation $P_{mn}\left(x\right)\equiv {(-1)}^m{P}_n^m\left(x\right)$ in order to distinguish between the two cases.

legendre_p is based on a standard recurrence relation described in Section 8.5.3 of Abramowitz and Stegun (1972). Constraints are placed on the values of $m$ and $n$ in order to avoid the possibility of machine overflow. It also sets the appropriate elements of the array $p$ (see Parameters) to zero whenever the required function is not defined for certain values of $m$ and $n$ (e.g., $m=-5$ and $n=3$).

References

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