# Spherical functions

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solid spherical harmonics, associated Legendre functions of the first and second kinds

Two linearly independent solutions $P _ \nu ^ \mu ( z)$ and $Q _ \nu ^ \mu ( z)$ of the differential equation

$$( 1- z) ^ {2} \frac{d ^ {2} y }{dz ^ {2} } - 2z \frac{dy}{dz} + \left [ \nu ( \nu + 1) - \frac{\mu ^ {2} }{1- z ^ {2} } \right ] = 0,$$

where $\mu$ and $\nu$ are complex constants, which arises in the solution of a class of partial differential equations by the method of separation of variables (cf. Separation of variables, method of). The points $z = \pm 1 , \infty$ are branching points of the solutions, in general. The spherical functions are particular cases of the hypergeometric functions (cf. Hypergeometric function):

$$P _ \nu ^ \mu ( z) = \frac{1}{\Gamma ( 1- \mu ) } \left ( z+ \frac{1}{z-} 1 \right ) ^ {\mu /2 } {} _ {2} F _ {1} \left ( - \nu , \nu + 1; 1- \mu ; 1- \frac{z}{2} \right )$$

$$\left ( \mathop{\rm arg} z+ \frac{1}{z-} 1 = 0 \ \textrm{ if } \mathop{\rm Im} z = 0, z > 1 \right ) ,$$

$$Q _ \nu ^ \mu ( z) = \frac{e ^ {\mu \pi i } \sqrt \pi \Gamma ( \mu + \nu + 1) }{2 ^ {\nu + 1 } \Gamma ( \nu + 3/2) } \frac{( z ^ {2} - 1) ^ {\mu /2 } }{z ^ {\mu + \nu + 1 } } \times$$

$$\times {} _ {2} F _ {1} \left ( \frac{\mu + \nu + 1 }{2} , \frac{\mu + \nu + 2 }{2} ; \nu + \frac{3}{2} ; \frac{1}{z ^ {2} } \right )$$

$$\textrm{ ( } \mathop{\rm arg} z = 0 \ \textrm{ if } \mathop{\rm Im} z = 0, z > 0;$$

$$\mathop{\rm arg} ( z ^ {2} - 1) = 0 \ \textrm{ if } \mathop{\rm Im} z = 0, z > 1 \textrm{ ) }.$$

The spherical functions $P _ \nu ^ \mu ( z)$ and $Q _ \nu ^ \mu ( z)$ are defined and single-valued in the domains $| 1- z |< 2$ and $| z | > 1$, respectively, of the complex plane cut by the real axis from $- \infty$ to $+ 1$.

If $\mathop{\rm Im} z = 0$, $z = x$, $- 1 < x < 1$, then the following functions are usually taken as solutions:

$$P _ \nu ^ \mu ( z) = \frac{1}{2} [ e ^ {\mu \pi i/2 } P _ \nu ^ \mu ( x+ i0) + e ^ {- \mu \pi i/2 } P _ \nu ^ \mu ( x- i0) ] =$$

$$= \ \frac{1}{\Gamma ( 1- \mu ) } \left ( 1+ \frac{x}{1-} x \right ) ^ {\mu /2 } {} _ {2} F _ {1} \left ( - \nu , \nu + 1 ; 1- \mu ; 1- \frac{x}{2} \right ) ,$$

$$Q _ \nu ^ \mu ( z) = \frac{1}{2} e ^ {\mu \pi i } [ e ^ {- \mu \pi i/2 } Q _ \nu ^ \mu ( x+ i0) + e ^ {\mu \pi i/2 } Q _ \nu ^ \mu ( x- i0) ] =$$

$$= \ \frac \pi {2 \sin \mu \pi } \left [ \cos \mu \pi P _ \nu ^ \mu ( x) - \frac{\Gamma ( \nu + \mu + 1) }{\Gamma ( \nu - \mu + 1) } P _ \nu ^ {- \mu } ( x) \right ] ,$$

where $f( x+ i0)$ $( f( x- i0))$ are the values of the function $f( z)$ on the upper (lower) boundary of the cut.

When $\mu = 0$, $\nu = n = 0, 1 \dots$ $P _ {n} ( z) \equiv P _ {n} ^ {0} ( z)$ are the Legendre polynomials. For zonal spherical functions see Spherical harmonics.

How to Cite This Entry:
Spherical functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_functions&oldid=48775
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article