Formula:3422

Digital Library of Mathematical Functions ID 8.21.E3

${\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{\pm\mathrm{i}t}\mathrm{d}t=e^{\pm\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right),}}$

Constraint(s)

${\displaystyle{\displaystyle 0<\Re a<1}}$

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
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle a}}$: parameter
 

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