Let C be the contour|z|= 1 traversed once counter-clockwise and a ∈R. (a) Compute the value of the integral ∫$\frac{e^{(iaz)}}{z^2}$dz. (b) Use your answer to part (a) to compute the values of the integrals ∫(from 0 to 2π) $e^{(−asinθ)}$cos(a cosθ−θ)dθ, and ∫(from 0 to2π)e$^{(−asinθ)}$sin(acosθ−θ)dθ.
My solution: for part (a) I applied Cauchy integral theorem and got =2πi$f'(0)$= -2πa. and for part(b) by putting z= e$^{iθ}$= cosθ+i sinθ., dz=i e$^{iθ}$ dθ. i get $\frac {e^{aicosθ-asinθ} (ie^{iθ})dθ}{e^{2iθ}}$; $\frac {(e^{aicosθ}.e^{asinθ})i dθ}{e^{iθ}}$;$\frac {(cos(acosθ+isin(acosθ)e^{-asinθ})idθ}{e^{iθ}}$. This is where im stuck at.. i cant figure out how to get just $θ$ instead of $isin(acosθ)$. Am i missing any formula or shortcut or trick i can use there?