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#### “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.

$\begin{array}{l}Jv\left(\frac{v}{\mathrm{cosh}\alpha }\right)~\frac{\mathrm{exp}\left(v\mathrm{tanh}\alpha -v\alpha \right)}{\sqrt{2v\pi \mathrm{tanh}\alpha }}\left\{1+\frac{1}{v}\left(\frac{1}{8}\mathrm{coth}\alpha -\frac{5}{24}{\mathrm{coth}}^{3}\alpha \right)\\ \begin{array}{c}+\frac{1}{{v}^{2}}\left(\frac{9}{128}{\mathrm{coth}}^{2}\alpha -\frac{231}{576}{\mathrm{coth}}^{4}\alpha +\frac{1155}{3456}{\mathrm{coth}}^{6}\alpha \right)+\cdots \end{array}\right\}\end{array}$ WA 269(3)

2.

$\begin{array}{l}Yv\left(\frac{v}{\mathrm{cosh}\alpha }\right)~-\frac{\mathrm{exp}\left(v\alpha -v\mathrm{tanh}\alpha \right)}{\sqrt{\frac{\pi }{2}v\mathrm{tanh}\alpha }}\left\{1+\frac{1}{v}\left(\frac{1}{8}\mathrm{coth}\alpha -\frac{5}{24}{\mathrm{coth}}^{3}\alpha \right)\\ \begin{array}{c}+\frac{1}{{v}^{2}}\left(\frac{9}{128}{\mathrm{coth}}^{2}\alpha -\frac{231}{576}{\mathrm{coth}}^{4}\alpha +\frac{1155}{3456}{\mathrm{coth}}^{6}\alpha \right)+\cdots \end{array}\right\}\end{array}$ WA 270(5)

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

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