Whittaker functions
The functions $ M _ {\lambda , \mu } ( z) $ and $ W _ {\lambda , \mu } ( z) $ which are solutions of the Whittaker equation
$$ \tag{* } w ^ {\prime\prime} + \left (
\frac{ {1 / 4 } - \mu ^ {2} }{z ^ {2} }
+
{ \frac \lambda {z}
} -
{ \frac{1}{4}
}
\right ) w = 0. $$
The function $ W _ {\lambda , \mu } $ satisfies the equation
$$ W _ {\lambda , \mu } ( z) = \
\frac{\Gamma (- 2 \mu ) }{\Gamma \left ( { \frac{1}{2}
} - \lambda - \mu \right ) }
M _ {\lambda , \mu } ( z) +
\frac{\Gamma ( 2 \mu ) }{\Gamma \left ( { \frac{1}{2}
} - \lambda + \mu \right ) }
M _ {\lambda , - \mu } ( z). $$
The pairs of functions $ M _ {\lambda , \mu } ( z) , M _ {\lambda , - \mu } ( z) $ and $ W _ {\lambda , \mu } ( z) , W _ {- \lambda , \mu } ( z) $ are linearly independent solutions of the equation (*). The point $ z = 0 $ is a branching point for $ M _ {\lambda , \mu } ( z) $, and $ z = \infty $ is an essential singularity.
Relation with other functions:
with the degenerate hypergeometric function:
$$ M _ {\lambda , \mu } ( z) = \ z ^ {\mu + 1/2 } e ^ {-z/2} \Phi \left ( \mu - \lambda + \frac{1}{2}
; \
2 \mu + 1; z \right ) , $$
with the modified Bessel functions and the Macdonald function:
$$ M _ {0, \mu } ( z) = \ 2 ^ {2 \mu } \Gamma ( \mu + 1) \sqrt z I _ \mu \left ( { \frac{z}{2}
} \right ) ,
$$
$$ W _ {0, \mu } ( z) = \sqrt { \frac{z} \pi
} K _ \mu \left ( {
\frac{z}{2}
} \right ) ;
$$
with the probability integral:
$$ W _ {- {1 / 4 } , {1 / 4 } } ( z) = \ 2 z ^ {1/4} e ^ {z/2}
\mathop{\rm Erfc} ( \sqrt z );
$$
with the Laguerre polynomials:
$$ W _ {n + \mu + 1/2, \mu } ( z) = \ n! (- 1) ^ {n} z ^ {\mu + 1/2 } e ^ {-z/2} L _ {n} ^ {2 \mu } ( z). $$
References
| [1] | H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) |
| [2] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) |
Comments
The Whittaker function $ W _ {\lambda , \mu } $ can be expressed in terms of the $ \Psi $-function introduced in confluent hypergeometric function:
$$ W _ {\lambda , \mu } ( z) = e ^ {- z/2 } z ^ {\mu + 1/2 } \Psi ( \mu - \lambda + 1/2; 2 \mu + 1; z). $$
Thus, the special cases discussed in confluent hypergeometric function can be rewritten as special cases for the Whittaker functions. See also the references given there.
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