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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^-i@ . e ^e^i@ d@
where @=theta

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

Solution Preview

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.
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, ...

Solution Summary

This shows how to evaluate given complex integrals.