Formula:4159

Digital Library of Mathematical Functions ID 10.23.E8

${\displaystyle{\displaystyle\frac{\mathscr{C}_{\nu}\left(w\right)}{w^{\nu}}=2^{\nu}\Gamma\left(\nu\right)\*\sum_{k=0}^{\infty}(\nu+k)\frac{\mathscr{C}_{\nu+k}\left(u\right)}{u^{\nu}}\frac{J_{\nu+k}\left(v\right)}{v^{\nu}}C^{(\nu)}_{k}\left(\cos\alpha\right),}}$

Constraint(s)

${\displaystyle{\displaystyle|ve^{\pm i\alpha}|<|u|}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)}}$: cylinder function
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)}}$: ultraspherical (or Gegenbauer) polynomial
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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