This is an exercise in a course on complex analysis I am taking:
Determine the function $f$ using complex contour integration: $$\lim_{R\to\infty}\frac{1}{2\pi i}\int_{c-iR}^{c+iR}\frac{\exp(tz)}{(z-i)^{\frac{1}{2}}(z+i)^{\frac{1}{2}}} dz$$ Where $c>0$ and the branch cut for $z^\frac{1}{2}$ is to be chosen on $\{z;\Re z=0, \Im z \leq0\}$. Make a distinction between: $$t>0, \quad t=0, \quad t<0$$ I think I showed that for $t<0$, $f(t)=0$ by using Jordan's Lemma. For $t=0$ I think the answer must be $f(0)=\frac{1}{2}$. For $t>0$ however, I have no idea what contour I have to define, nor how I have to calculate the residues in $i$ and $-i$.