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- 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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