Formula:3656

Digital Library of Mathematical Functions ID 9.12.E30

${\displaystyle{\displaystyle\int_{0}^{z}\mathrm{Gi}\left(t\right)\mathrm{d}t\sim\frac{1}{\pi}\ln z+\frac{2\gamma+\ln 3}{3\pi}-\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{(3k-1)!}{k!(3z^{3})^{k}},}}$

Constraint(s)

${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{3}\pi-\delta}}$

Symbols List

• ${\displaystyle{\displaystyle\gamma}}$: Euler’s constant
• ${\displaystyle{\displaystyle\mathrm{Gi}\left(\NVar{z}\right)}}$: Scorer function (inhomogeneous Airy function)
• ${\displaystyle{\displaystyle\sim}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\delta}}$: small positive constant
 

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