Let U be a simply connected domain with a simple closed boundary curve C oriented anticlockwise, and define for all w ∈ C \ C $$ g(w)=\oint_C \frac{e^zdz}{(z-w)^2}$$ Find a formula for g(w) which does not involve integration.
I haven't seen something like this before...We have f(z) and g(w) together. Is the procedure for Cauchy's integration formula same as ordinary ones? Or is there something special here?
The residue theorem will give the result immediately. But if you don't know the theorem, there's an informal way to see that: First we may expand $e^z=\sum_n z^n/n!$. Then the integration is just $e^{z+w}/z^2$, expand and notice that $\oint_C z^n$ vanishes when $n\not = -1$. The only term remain is just $$\oint e^w \frac{z}{1! z^2}dz = 2\pi i e^w$$ (PS: this way is informal since it does not prove that the integration and summation can be interchanged, but this can be proved because $e^z$ converges uniformly in a bounded area.)