# Proof required for integration involving analytic function.

• Apr 15th 2008, 11:00 AM
jalal0
Proof required for integration involving analytic function.
The question goes this way:

(Q) The complex variable function f(z) is analytic in the complex plane, and |f(z)| < M, where M is a constant.

Prove that

I = ∫C f(z) / [(z-a)(z-b)] dz = 0,

where a, b are two different complex number, and C is the circle |z| = R, (R > |a|, R> |b|).
Let R→∞

Using residual theorem, I have only managed to reach this far:
I = 2πi {[f(a)-f(b)]/(a-b)}

Now, how do I prove that f(a)=f(b)?

Thanks.
• Apr 15th 2008, 11:50 AM
ThePerfectHacker
Quote:

Originally Posted by jalal0
The question goes this way:

(Q) The complex variable function f(z) is analytic in the complex plane, and |f(z)| < M, where M is a constant.

Prove that

I = ∫C f(z) / [(z-a)(z-b)] dz = 0,

where a, b are two different complex number, and C is the circle |z| = R, (R > |a|, R> |b|).
Let R→∞

Using residual theorem, I have only managed to reach this far:
I = 2πi {[f(a)-f(b)]/(a-b)}

Now, how do I prove that f(a)=f(b)?

Thanks.

If $|f(z)|\leq M$ for all points in the complex plane then by Liouville's theorem the function is constant and so $f(a) = f(b)$ completing the proof.