Formula:4221

Digital Library of Mathematical Functions ID 10.32.E4

${\displaystyle{\displaystyle I_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{z\cos\theta}\cos\left(\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-z\cosh t-\nu t}\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}$

Symbols List

• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\cosh\NVar{z}}}$: hyperbolic cosine function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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