- #1

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- 11

## Homework Statement

evaluate sinx/x^4 over the unit circle

## Homework Equations

Cauchys Residue theorem

##sinz=1/(2i)(z+1/z)##

## The Attempt at a Solution

So we have a branch point at z=0 but its of order 4 so I can't see any direct way of using Cauchys residue theorem. Ive tried changing the sin expression to as above but simply end up with poles of order 3 and 5, which again doesn't help me.

So I tried defining a new contour essentially around the unit circle, but also enclosing the branch point by travelling back along the real axis at some small value of y. By Cauchy's residue theorem the integral along this combined contour is zero since no poles are enclosed. Redefining ##z=e^(i*theta)## for the integral along the outer circle, ##z=x+ie_0## along the path from ##x to e_0##, ##z=e_0e^(i*theta)## around the small inner circle, and ##z=x+ie_0## for the line running just above the real axis from ##e_0 to x##. But I'm not sure how to proceed from here, nor how to cary out any of the integrals.

Many thanks :)