Suppose that p(z) and q(z) are polynomials with a complex coefficient with the property that deg q(z) is greater than or equal to deg p(z)+2. If C is a positively oriented simple closed contour containing all of the roots of q(z) on its interior, then prove that

Integral C of p(z)/q(z) dz = 0

Solution Preview

Solution:

Let p(z)=a(0)x^m+a(1)x^(m-1)+...+a(m)
q(z)=b(0)x^n+b(1)x^(n-1)+...+b(n)
where n>=m+2

We define the complex function f(z)=p(z)/q(z) and we have to compute Int on (C)-closed of f(z)dz.
If all the roots of q(z) are inside (C), we can apply the residues theorem:
Int on (C)-closed of f(z)dz = 2(pi)i.sum of Res ...

Solution Summary

This solution clearly demonstrates how to complete a full proof regarding integrals and simple closed contours. All of the necessary steps are included, along with the mathematical operations which are required.

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