Table of Integrals, Series, and Products, 8th Edition

8.57 Lommel functions

8.570 Definitions of the Lommel functions s_μν(z) and S_μν(z):

1.¹²

$\begin{array}{l} s_{μ, v} (z) & = \sum_{m = 0}^{\infty} \frac{(- 1) m_{z} μ + 1 + 2 m}{[{(μ + 1)}^{2} - v^{2}] [{(μ + 3)}^{2} - v^{2}] \dots [{(μ + 2 m + 1)}^{2} - v^{2}]} \\ \begin{array}{l} \begin{array}{l} ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [μ \pm v ? is ? not ? a ? negative ? odd ? integer] \end{array} \end{array} \\ = z^{μ - 1} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} {(\frac{z}{2})}^{2 m + 2} Γ (\frac{1}{2} μ - \frac{1}{2} v + \frac{1}{2}) Γ (\frac{1}{2} μ + \frac{1}{2} v + \frac{1}{2})}{Γ (\frac{1}{2} μ - \frac{1}{2} v + m + \frac{3}{2}) Γ (\frac{1}{2} μ + \frac{1}{2} v + m + \frac{3}{2})} \\ \begin{array}{l} \begin{array}{l} ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [μ \pm v ? is ? not ? a ? negative ? odd ? integer] \end{array} \end{array} \end{array}$

si1107_e EH II 40(69), WA 377(2)

2.¹²

$\begin{array}{l} S_{μ, v} (z) = & s_{μ, v} (z) + 2^{μ - 1} Γ (\frac{1}{2} μ - \frac{1}{2} v + \frac{1}{2}) Γ (\frac{1}{2} μ + \frac{1}{2} v + \frac{1}{2}) \\ \times \frac{\cos [\frac{1}{2} (μ - v) π] J_{- v} (z) - \cos [\frac{1}{2} (μ + v) π] J_{v} (z)}{\sin v π} \\ [μ \pm v ? is ? a ? positive ? odd ? integer, v ? is ? an ? odd ? integer] \end{array}$

EH II 40(71), WA 379(2)

$\begin{array}{l} = & s_{μ, v} (z) + 2^{μ - 1} Γ (\frac{1}{2} μ - \frac{1}{2} v + \frac{1}{2}) Γ (\frac{1}{2} μ + \frac{1}{2} v + \frac{1}{2}) \\ \times {\sin [\frac{1}{2} (μ - \end{array}$

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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

8.57 Lommel functions

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