Complex integrals

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3)

(2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2

(3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i

(4) Evaluate the integral between 2pi and 0 of
e^[email protected] . e ^e^[email protected] [email protected]
where @=theta

(5) Show that the integral over |z|=2 of ( z/z-1)^n = 2pi ni

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...(c), is the (-1)st coefficient in the Laurent expansion of the function at c,
f(z) = ... + a(z)/(z-c) + ...
For the function in this problem, f(z)=1/(z^2-1) = 1/((z-1)(z+1)), the poles within the contour are c=1 and c=-1, with the corresponding residues a=1/(1+1)=1/2 and a=1/(-1-1)=-1/2, respectively. So the integral is
int(dz 1/(z^2-1)) = 2 Pi i (1/2 - 1/2) = 0
This integral may be done simply by doing the anti-derivative with respect to z and pluggin in the limits, as you would for a real ...