 #1
Teymur
 16
 3
 Homework Statement:

Use the Residue Theorem to show:
$$\int \:\frac{z^{\frac{1}{2}}}{1+\sqrt{2}z+z^2}dz=2^{\frac{2}{3}}\pi isin\left(\frac{8\pi }{3}\right)$$
for a keyhole contour where ##z=re^{i\theta }## and ##\pi <\theta <\pi##
 Relevant Equations:
 the standard residue theorem
I'm really struggling with this one. A newbie to using the residue theorem. I'm trying to solve this by factorising the denominator to find values for z0 and I have:
##z=\frac{\sqrt{2}+i\sqrt{2}}{2}## and ##z=\frac{\sqrt{2}i\sqrt{2}}{2}##
I also know that sin(3π/8)= ##\frac{\sqrt{2+\sqrt{2}}}{2}##
##z=\frac{\sqrt{2}+i\sqrt{2}}{2}## and ##z=\frac{\sqrt{2}i\sqrt{2}}{2}##
I also know that sin(3π/8)= ##\frac{\sqrt{2+\sqrt{2}}}{2}##
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