Formula:4671

Digital Library of Mathematical Functions ID 11.5.E7

${\displaystyle{\displaystyle I_{-\nu}\left(x\right)-\mathbf{L}_{\nu}\left(x\right)=\frac{2(\tfrac{1}{2}x)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\infty}(1+t^{2})^{\nu-\frac{1}{2}}\sin\left(xt\right)\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle\Re\nu<\tfrac{1}{2}}}$
${\displaystyle{\displaystyle x>0}}$
${\displaystyle{\displaystyle\Re\nu<\tfrac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Struve function
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\nu}}$: real or complex order
 

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