- #1
hmiamid
- 4
- 0
Hello PF,
I just found a curious integral. I wondered if it comes from a bigger group of integral definitions:
[tex]\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)[/tex]
Where Si(x) is the sine integral function [itex]\mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x[/itex]
I proved the equation by developing Si(x) in Taylor series and there is a nice simplification between Si Taylor coefficients and the Gamma function.
I just found a curious integral. I wondered if it comes from a bigger group of integral definitions:
[tex]\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)[/tex]
Where Si(x) is the sine integral function [itex]\mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x[/itex]
I proved the equation by developing Si(x) in Taylor series and there is a nice simplification between Si Taylor coefficients and the Gamma function.