# Complex integral

1. Jun 3, 2008

### Belgium 12

Hi,

problrm with complex integral.Consider the integral

$$\int_{0}^{\infty}\frac{(x^a-1)}{x^3-1}\,dx$$

use the branch 0<phi<3pi/2 and an idented contour at z=0 and z=1.(circular contour in the

upper half plane)

a)show that the integral can be written in terms of the integral:

$$\pi\frac{(e^{2ia\pi/3}-1)}{(e^{2i\pi/3}-1)sin(2\pi/3)}+\int_{0}^{\infty}\frac{(x^ae^{ia\pi}-1)}{x^3+1}$$

b)evaluate the second integral in part a)and find the value of the orginal integral

$$\frac{pi.sin(a\pi/3)}{3.sin(\pi/3)sin[\pi/3(a+1)]}$$

For the last integral I used a contour z=x(0<x<R)the sector circle in the upper half plane

0<phi<2pi/3)and the line z=xe^2pi/3(0<x<R)

but I can't find the integral question b)

Can somebody give me a help.Thanks

Last edited: Jun 3, 2008