Complex integrals

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(1)
f = u + i v is analytic if and only if it satisfies the Cauchy-Riemann equations,
du/dx = dv/dy
dv/dx = -du/dy
where z = x + i y and d is a partial derivative.
since in the case specified f is real valued, v=0 and hence, by the first C-R equation, du/dx=0. So f(3)=u(3)=u(1)=f(1)=1.
(2)
We use the theorem of residues:
For integrating a function counter-clockwise over a closed contour,
int(dz f(z)) = 2 Pi i Sum(a(c))
where the sum is over the singular points, c, within the contour, and the residue, ...