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

“Approximation by tangents”

8.452¹¹ For large values of the index (where the argument is less than the index).

Suppose that x < 0 and ν < 0. Let us set ν/x = cosh α. Then, for large values of ν, the following expansions are valid:

1. 

$\begin{array}{l}Jv\left(\frac{v}{\cosh \alpha }\right)~\frac{\exp (v\tanh \alpha -v\alpha )}{\sqrt{2v\pi \tanh \alpha }}\{1+\frac{1}{v}\left(\frac{1}{8}\coth \alpha -\frac{5}{24}{\coth }^3\alpha \right)\\ \begin{array}{c}+\frac{1}{v^2}\left(\frac{9}{128}{\coth }^2\alpha -\frac{231}{576}{\coth }^4\alpha +\frac{1155}{3456}{\coth }^6\alpha \right)+\cdots \end{array}\}\end{array}$

  WA 269(3)

2. 

$\begin{array}{l}Yv\left(\frac{v}{\cosh \alpha }\right)~-\frac{\exp (v\alpha -v\tanh \alpha )}{\sqrt{\frac{\pi }{2}v\tanh \alpha }}\{1+\frac{1}{v}\left(\frac{1}{8}\coth \alpha -\frac{5}{24}{\coth }^3\alpha \right)\\ \begin{array}{c}+\frac{1}{v^2}\left(\frac{9}{128}{\coth }^2\alpha -\frac{231}{576}{\coth }^4\alpha +\frac{1155}{3456}{\coth }^6\alpha \right)+\cdots \end{array}\}\end{array}$

  WA 270(5)

8.453 For large values of the index (where the argument is greater than ...