WW Whittaker W

WW.1 Introduction

Let x be a complex variable of ℂ\{0, ∞} and let μ, ν denote a set of parameters (independent of x).The function Whittaker W (noted WW_{μ,ν}) is defined by the following second order differential equation

\begin{aligned}& −x^{2} − 4μx − 1 + 4ν^{2}y(x) + 4x^{2} \frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (WW.1.1) \end{aligned}

Although 0 is a singularity of WW.1.1, the initial conditions can be given by

[x^{(ν+ \frac{1}{2})}] WW_{μ,ν}(x) = \frac{π}{sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)},

[x^{(−ν+ \frac{1}{2})}] WW_{μ,ν}(x) = \frac{π}{sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)}.

(WW.1.2)

The formulae of this document are valid for 2ν ∉ ℤ. Related function: Whittaker M

WW.2 Series and asymptotic expansions

WW.2.1 Asymptotic expansion at ∞. WW.2.1.1 First terms.

WW_{μ,ν}(x) ≈ \frac{e^{(\frac{−1}{2x})} y_{0}(x)}{x^{μ}},

where

y_{0}(x) = 1 + (ν^{2} + μ − μ^{2} − \frac{1}{4})x−

\frac{−(4ν^{2} + 12μ − 4μ^{2} − 9)(4ν^{2} + 4μ − 4μ^{2} − 1)x^{2}}{32} +

(4ν^{2} + 20μ − 4μ^{2} − 25)(4ν^{2} + 12μ − 4μ^{2} − 9)

(4ν^{2} + 4μ − 4μ^{2} − 1)x^{3}/384 + 2 …

WW.2.1.2 General form. WW.2.1.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(x) satisfy the

following recurrence

\begin{matrix}4u(n)n+ \\ u(n − 1)(−4ν^{2} − 4μ + 4μ^{2} − 3 + 4n − 8(n − 1)μ + 4(n − 1)^{2}) = 0 \end{matrix}

whose initial conditions are given by

u(0) = 1

This recurrence has the closed form solution

u(n) = \frac{(−2)^{n}Γ(n + \frac{1}{2} − ν − μ)2^{n}Γ(n + \frac{1}{2} + ν − μ)}{4^{n}Γ(n + 1)Γ(\frac{1}{2} − ν − μ)Γ(\frac{1}{2} + ν − μ)}.

WW.2.2 Asymptotic expansion at 0.

WW.2 SERIES AND ASYMPTOTIC EXPANSIONS 3

WW.2.2.1 First terms.

WW_{μ,ν}(x) ≈ x^{(ν+ \frac{1}{2})}(\frac{π}{sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)}−

\frac{μxπ}{(2ν + 1) sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)}+

\frac{(2ν + 1 + 4μ^{2})x^{2}π}{16(2ν + 1)(ν + 1) sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)}−

(μ(6ν + 5 + 4μ^{2})x^{3}π)/(48(2ν + 1)(ν + 1)(2ν + 3)

sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ))+

((12ν^{2} + 24ν + 9 + 48μ^{2}ν + 56μ^{2} + 16μ^{4})x^{4}π)/(1536 (2ν + 1)(ν + 1)(2ν + 3)(ν + 2) sin(π(2ν + 1))Γ(2ν + 1)

Γ(\frac{1}{2} − ν − μ))−

(μ(60ν^{2} + 160ν + 89 + 80μ^{2}ν + 120μ^{2} + 16μ^{4})x^{5}π)/(7680 (2ν + 1)(ν + 1)(2ν + 3)(ν + 2)(2ν + 5) sin(π(2ν + 1))

Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)) + ((120ν^{3} + 540ν^{2} + 690ν + 225+

720μ^{2}ν^{2} + 2400μ^{2}ν + 1756μ^{2} + 480μ^{4}ν + 880μ^{4} + 64μ^{6}) x^{6}π)/(368640(2ν + 1)(ν + 1)(2ν + 3)(ν + 2)(2ν + 5)(ν + 3)

sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)) − (μ(840ν^{3} + 4620ν^{2}+

7518ν + 3429 + 1680μ^{2}ν^{2} + 6720μ^{2}ν + 6076μ^{2} + 672μ^{4}ν+ 1456μ^{4} + 64μ^{6})x^{7}π)/(2580480(2ν + 1)(ν + 1)(2ν + 3)(ν + 2)

(2ν + 5)(ν + 3)(2ν + 7) sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)) + ((

1680ν^{4} + 13440ν^{3} + 36120ν^{2} + 36960ν + 11025 + 13440μ^{2}ν^{3}+ 87360μ^{2}ν^{2} + 172256μ^{2}ν + 99760μ^{2} + 13440μ^{4}ν^{2}+ 62720μ^{4}ν + 67424μ^{4} + 3584μ^{6}ν + 8960μ^{6} + 256μ^{8})x^{8}π)/( 165150720(2ν + 1)(ν + 1)(2ν + 3)(ν + 2)(2ν + 5)(ν + 3)

(2ν + 7)(ν + 4) sin(π(2ν + 1))Γ(2ν + 1)Γ(\frac{1}{2} − ν − μ)) …) + (

\frac{π}{sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)}+

\frac{μxπ}{(2ν − 1) sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)}−

\frac{(2ν − 1 − 4μ^{2})x^{2}π}{16(2ν − 1)(ν − 1) sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)}−

(μ(6ν − 5 − 4μ^{2})x^{3}π)/(48(2ν − 1)(ν − 1)(2ν − 3)

sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ))+

((12ν^{2} − 24ν + 9 − 48μ^{2}ν + 56μ^{2} + 16μ^{4})x^{4}π)/(1536 (2ν − 1)(ν − 1)(2ν − 3)(ν − 2) sin(π(2ν + 1))Γ(1 − 2ν)

Γ(\frac{1}{2} + ν − μ))+

(μ(60ν^{2} − 160ν + 89 − 80μ^{2}ν + 120μ^{2} + 16μ^{4})x^{5}π)/(7680 (2ν − 1)(ν − 1)(2ν − 3)(ν − 2)(2ν − 5) sin(π(2ν + 1))

Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)) − ((120ν^{3} − 540ν^{2} + 690ν − 225−

720μ^{2}ν^{2} + 2400μ^{2}ν − 1756μ^{2} + 480μ^{4}ν − 880μ^{4} − 64μ^{6}) x^{6}π)/(368640(2ν − 1)(ν − 1)(2ν − 3)(ν − 2)(2ν − 5)(ν − 3)

sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)) − (μ(840ν^{3} − 4620ν^{2}+

7518ν − 3429 − 1680μ^{2}ν^{2} + 6720μ^{2}ν − 6076μ^{2} + 672μ^{4}ν− 1456μ^{4} − 64μ^{6})x^{7}π)/(2580480(2ν − 1)(ν − 1)(2ν − 3)(ν − 2)

(2ν − 5)(ν − 3)(2ν − 7) sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)) + ((

1680ν^{4} − 13440ν^{3} + 36120ν^{2} − 36960ν + 11025 − 13440μ^{2}ν^{3}+ 87360μ^{2}ν^{2} − 172256μ^{2}ν + 99760μ^{2} + 13440μ^{4}ν^{2}− 62720μ^{4}ν + 67424μ^{4} − 3584μ^{6}ν + 8960μ^{6} + 256μ^{8})x^{8}π)/( 165150720(2ν − 1)(ν − 1)(2ν − 3)(ν − 2)(2ν − 5)(ν − 3)

(2ν − 7)(ν − 4) sin(π(2ν + 1))Γ(1 − 2ν)Γ(\frac{1}{2} + ν − μ)) …)/x^{(ν− \frac{1}{2})}.

(WW.2.2.1.1)

4 WW WHITTAKER W

WW.2.2.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).