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i am stuccoed doing this basic complex integrals, i am really tired but i need to do it for my basic complex analysis course, i need to solve this integrals using basic theory of analytic functions, or residues and poles or series, you can use any method you got, i really appreciate your giant help. $$\int_\gamma e^{\dfrac{1}{z^2}}dz$$ $$\int_\gamma e^{\dfrac{1}{z}}dz$$

where $$\gamma:|z|=1 $$ $$\int_{-\pi}^\pi \frac{d\theta}{1+sin^2(\theta)}$$ Thanks

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up vote 2 down vote accepted

Hint: For the first two, expand the integrand using a Laurent series, and apply the residue theorem.

For the last integral, use the substitution $z=e^{i\theta},$ the fact that $\sin\theta=\dfrac{z-\overline z}{2i},$ and that for $|z|=1$ we have $\overline z=z^{-1}.$ Then you will get a contour integral over $\gamma=\{z:|z|=1\}$ of a function with two poles, one inside and one outside the contour, and will be able to apply the residue theorem.

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