Formula:4817

Digital Library of Mathematical Functions ID 12.5.E6

${\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2\pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle-\tfrac{1}{2}\pi<\operatorname{ph}t<\tfrac{1}{2}\pi}}$
${\displaystyle{\displaystyle-\tfrac{1}{2}\pi<\operatorname{ph}t<\tfrac{1}{2}\pi}}$
${\displaystyle{\displaystyle c>0}}$

Symbols List

• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
 

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