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

$\begin{array}{l} \int_{0}^{\infty} {[coth (\frac{1}{2} x)]}^{2 λ} e^{- β cosh x} Y_{2, μ} (α sinh x) d x \\ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? - \frac{\sec [(μ + λ) π]}{α} W_{λ, μ} (\sqrt{α^{2} + β^{2}} + β) W_{- λ, μ} (\sqrt{α^{2} + β^{2}} - β) \\ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? - \frac{tan [(μ + λ) π] Γ (\frac{1}{2} - λ + μ)}{α Γ (2 μ + 1)} W_{λ, μ} (\sqrt{α^{2} + β^{2}} + β) M_{- λ, μ} (\sqrt{α^{2} + β^{2}} - β) \end{array}$

si1279_e BU 86(5b)a, ET II 363(34)

$[Re β > | Re α |, ? ? ? Re λ < \frac{1}{2} - | Re μ |]$ si1280_e

3.¹²

$\begin{array}{l} \int_{0}^{\infty} e^{- \frac{1}{2} (a_{1} + a_{2}) t cosh x} {[coth (\frac{1}{2} x)]}^{2 υ} K_{2 μ} (t \sqrt{a_{1} a_{2}} sinh x) d x \\ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = \frac{Γ (\frac{1}{2} + μ - v) Γ (\frac{1}{2} - μ - v)}{2 t \sqrt{a_{1} a_{2}}} W_{v, μ} (a_{1} t) W_{v, μ} (a_{2} t) \end{array}$

si1281_e BU 85(4a)

$[Re v < Re \frac{1 \pm 2 μ}{2}$

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