WW Whittaker W

WW.1 Introduction

Let $x$ be a complex variable of $\mathbb{C} \setminus \{0,\infty\}$ and let $\mu,\nu$ denote a set of parameters (independent of $x$ ).The function Whittaker W (noted $\operatorname{WW} _{\mu , \nu}$ ) is defined by the following second order differential equation

$\begin{equation*} \begin{split} -x^{2} - 4 \mu x - 1 + 4 \nu^{2}y (x) + 4 x^{2} \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}$

WW.1.1

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

$\begin{equation*} \begin{split} \Biggl[x^{\Bigl(\nu + \frac{1}{2}\Bigr)}\Biggr] \operatorname{WW} _{\mu , \nu} (x)& =\frac{\pi}{\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \Bigl(\frac{1}{2} - \nu - \mu\Bigr)}, \\ \Biggl[x^{\Bigl(-\nu + \frac{1}{2}\Bigr)}\Biggr] \operatorname{WW} _{\mu , \nu} (x)& =\frac{\pi}{\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)}. \end{split} \end{equation*}$

WW.1.2

The formulae of this document are valid for $2 \nu \not\in \mathbb{Z} .$

Related function: Whittaker M

WW.2 Series and asymptotic expansions

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WW.2.1 Asymptotic expansion at $\infty$

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WW.2.1.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{WW} _{\mu , \nu} (x)\approx \frac{\operatorname{e} ^{\bigl(\frac{-1}{2 x}\bigr)} y _{0} (x)}{x^{\mu}}, \end{split} \end{equation*}$

where

$\begin{equation*} \begin{split} y _{0} (x)& =1 + \biggl(\nu^{2} + \mu - \mu^{2} - \frac{1}{4}\biggr) x - \\ & \quad{}\quad{}\frac{-\bigl(4 \nu^{2} + 12 \mu - 4 \mu^{2} - 9\bigr) \bigl(4 \nu^{2} + 4 \mu - 4 \mu^{2} - 1\bigr) x^{2}}{32} + \\ & \quad{}\quad{}\bigl(4 \nu^{2} + 20 \mu - 4 \mu^{2} - 25\bigr) \bigl(4 \nu^{2} + 12 \mu - 4 \mu^{2} - 9\bigr) \\ & \quad{}\quad{}\bigl(4 \nu^{2} + 4 \mu - 4 \mu^{2} - 1\bigr) x^{3}\Big/384 + 2 \ldots \end{split} \end{equation*}$

WW.2.1.2 General form

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WW.2.1.2.1 Auxiliary function $y _{0} (x)$

The coefficients $u (n)$

of $y _{0} (x)$ satisfy the following recurrence

$\begin{equation*} \begin{split} & 4 u (n) n + \\ & u (n - 1) \bigl(-4\nu^{2} - 4 \mu + 4 \mu^{2} - 3 + 4 n - 8 (n - 1) \mu + 4 (n - 1)^{2}\bigr)=0 \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (0)& =1 \end{split} \end{equation*}$

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (n)& =\frac{(-2)^{n} \Gamma \Bigl(n + \frac{1}{2} - \nu - \mu\Bigr) 2^{n} \Gamma \Bigl(n + \frac{1}{2} + \nu - \mu\Bigr)}{4^{n} \Gamma (n + 1) \Gamma \Bigl(\frac{1}{2} - \nu - \mu\Bigr) \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)}. \end{split} \end{equation*}$

WW.2.2 Asymptotic expansion at

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WW.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{WW} _{\mu , \nu} (x)\approx x^{\Bigl(\nu + \frac{1}{2}\Bigr)} \Biggl(\frac{\pi}{\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \Bigl(\frac{1}{2} - \nu - \mu\Bigr)} - \\ & \quad{}\quad{}\frac{\mu x \pi}{(2 \nu + 1) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \Bigl(\frac{1}{2} - \nu - \mu\Bigr)} + \\ & \quad{}\quad{}\frac{\bigl(2 \nu + 1 + 4 \mu^{2}\bigr) x^{2} \pi}{16 (2 \nu + 1) (\nu + 1) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \Bigl(\frac{1}{2} - \nu - \mu\Bigr)} - \\ & \quad{}\quad{}\Bigl(\mu \bigl(6 \nu + 5 + 4 \mu^{2}\bigr) x^{3} \pi\Bigr)\Bigg/\Biggl(48 (2 \nu + 1) (\nu + 1) (2 \nu + 3) \\ & \quad{}\quad{}\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \biggl(\frac{1}{2} - \nu - \mu\biggr)\Biggr) + \\ & \quad{}\quad{}\Bigl(\bigl(12 \nu^{2} + 24 \nu + 9 + 48 \mu^{2} \nu + 56 \mu^{2} + 16 \mu^{4}\bigr) x^{4} \pi\Bigr)\Bigg/\Biggl(1536 \\ & \quad{}\quad{}(2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \\ & \quad{}\quad{}\Gamma \biggl(\frac{1}{2} - \nu - \mu\biggr)\Biggr) - \\ & \quad{}\quad{}\Bigl(\mu \bigl(60 \nu^{2} + 160 \nu + 89 + 80 \mu^{2} \nu + 120 \mu^{2} + 16 \mu^{4}\bigr) x^{5} \pi\Bigr)\Bigg/\Biggl(7680 \\ & \quad{}\quad{}(2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \\ & \quad{}\quad{}\Gamma (2 \nu + 1) \Gamma \biggl(\frac{1}{2} - \nu - \mu\biggr)\Biggr) + \Bigl(\bigl(120 \nu^{3} + 540 \nu^{2} + 690 \nu + 225 + \\ & \quad{}\quad{}720 \mu^{2} \nu^{2} + 2400 \mu^{2} \nu + 1756 \mu^{2} + 480 \mu^{4} \nu + 880 \mu^{4} + 64 \mu^{6}\bigr) \\ & \quad{}\quad{}x^{6} \pi\Bigr)\Bigg/\Biggl(368640 (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5) (\nu + 3) \\ & \quad{}\quad{}\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \biggl(\frac{1}{2} - \nu - \mu\biggr)\Biggr) - \Bigl(\mu \bigl(840 \nu^{3} + 4620 \nu^{2} + \\ & \quad{}\quad{}7518 \nu + 3429 + 1680 \mu^{2} \nu^{2} + 6720 \mu^{2} \nu + 6076 \mu^{2} + 672 \mu^{4} \nu + \\ & \quad{}\quad{}1456 \mu^{4} + 64 \mu^{6}\bigr) x^{7} \pi\Bigr)\Bigg/\Biggl(2580480 (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) \\ & \quad{}\quad{} (2 \nu + 5) (\nu + 3) (2 \nu + 7) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \biggl(\frac{1}{2} - \nu - \mu\biggr)\Biggr) + \Bigl(\bigl( \\ & \quad{}\quad{}1680 \nu^{4} + 13440 \nu^{3} + 36120 \nu^{2} + 36960 \nu + 11025 + 13440 \mu^{2} \nu^{3} + \\ & \quad{}\quad{}87360 \mu^{2} \nu^{2} + 172256 \mu^{2} \nu + 99760 \mu^{2} + 13440 \mu^{4} \nu^{2} + \\ & \quad{}\quad{}62720 \mu^{4} \nu + 67424 \mu^{4} + 3584 \mu^{6} \nu + 8960 \mu^{6} + 256 \mu^{8}\bigr) x^{8} \pi\Bigr)\Bigg/\Biggl( \\ & \quad{}\quad{}165150720 (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5) (\nu + 3) \\ & \quad{}\quad{}(2 \nu + 7) (\nu + 4) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (2 \nu + 1) \Gamma \biggl(\frac{1}{2} - \nu - \mu\biggr)\Biggr)\ldots\Biggr) + \Biggl( \\ & \quad{}\quad{}\frac{\pi}{\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)} + \\ & \quad{}\quad{}\frac{\mu x \pi}{(2 \nu - 1) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)} - \\ & \quad{}\quad{}\frac{\bigl(2 \nu - 1 - 4 \mu^{2}\bigr) x^{2} \pi}{16 (2 \nu - 1) (\nu - 1) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \Bigl(\frac{1}{2} + \nu - \mu\Bigr)} - \\ & \quad{}\quad{}\Bigl(\mu \bigl(6 \nu - 5 - 4 \mu^{2}\bigr) x^{3} \pi\Bigr)\Bigg/\Biggl(48 (2 \nu - 1) (\nu - 1) (2 \nu - 3) \\ & \quad{}\quad{}\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr) + \\ & \quad{}\quad{}\Bigl(\bigl(12 \nu^{2} - 24 \nu + 9 - 48 \mu^{2} \nu + 56 \mu^{2} + 16 \mu^{4}\bigr) x^{4} \pi\Bigr)\Bigg/\Biggl(1536 \\ & \quad{}\quad{}(2 \nu - 1) (\nu - 1) (2 \nu - 3) (\nu - 2) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \\ & \quad{}\quad{}\Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr) + \\ & \quad{}\quad{}\Bigl(\mu \bigl(60 \nu^{2} - 160 \nu + 89 - 80 \mu^{2} \nu + 120 \mu^{2} + 16 \mu^{4}\bigr) x^{5} \pi\Bigr)\Bigg/\Biggl(7680 \\ & \quad{}\quad{}(2 \nu - 1) (\nu - 1) (2 \nu - 3) (\nu - 2) (2 \nu - 5) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \\ & \quad{}\quad{}\Gamma (1 - 2 \nu) \Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr) - \Bigl(\bigl(120 \nu^{3} - 540 \nu^{2} + 690 \nu - 225 - \\ & \quad{}\quad{}720 \mu^{2} \nu^{2} + 2400 \mu^{2} \nu - 1756 \mu^{2} + 480 \mu^{4} \nu - 880 \mu^{4} - 64 \mu^{6}\bigr) \\ & \quad{}\quad{}x^{6} \pi\Bigr)\Bigg/\Biggl(368640 (2 \nu - 1) (\nu - 1) (2 \nu - 3) (\nu - 2) (2 \nu - 5) (\nu - 3) \\ & \quad{}\quad{}\operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr) - \Bigl(\mu \bigl(840 \nu^{3} - 4620 \nu^{2} + \\ & \quad{}\quad{}7518 \nu - 3429 - 1680 \mu^{2} \nu^{2} + 6720 \mu^{2} \nu - 6076 \mu^{2} + 672 \mu^{4} \nu - \\ & \quad{}\quad{}1456 \mu^{4} - 64 \mu^{6}\bigr) x^{7} \pi\Bigr)\Bigg/\Biggl(2580480 (2 \nu - 1) (\nu - 1) (2 \nu - 3) (\nu - 2) \\ & \quad{}\quad{} (2 \nu - 5) (\nu - 3) (2 \nu - 7) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr) + \Bigl(\bigl( \\ & \quad{}\quad{}1680 \nu^{4} - 13440 \nu^{3} + 36120 \nu^{2} - 36960 \nu + 11025 - 13440 \mu^{2} \nu^{3} + \\ & \quad{}\quad{}87360 \mu^{2} \nu^{2} - 172256 \mu^{2} \nu + 99760 \mu^{2} + 13440 \mu^{4} \nu^{2} - \\ & \quad{}\quad{}62720 \mu^{4} \nu + 67424 \mu^{4} - 3584 \mu^{6} \nu + 8960 \mu^{6} + 256 \mu^{8}\bigr) x^{8} \pi\Bigr)\Bigg/\Biggl( \\ & \quad{}\quad{}165150720 (2 \nu - 1) (\nu - 1) (2 \nu - 3) (\nu - 2) (2 \nu - 5) (\nu - 3) \\ & \quad{}\quad{}(2 \nu - 7) (\nu - 4) \operatorname{sin} \bigl(\pi (2 \nu + 1)\bigr) \Gamma (1 - 2 \nu) \Gamma \biggl(\frac{1}{2} + \nu - \mu\biggr)\Biggr)\ldots\Biggr)\Bigg/x^{\Bigl(\nu - \frac{1}{2}\Bigr)}. \end{split} \end{equation*}$

WW.2.2.1.1

WW.2.2.2 General form

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