Question

Suppose f(z) is analytic in a region which contains a simple closed contour C and its inside. Assume that there exists M∈R such that ∣f(z)∣≤M on C. Show using Rouché's theorem that ∣f(z)∣≤M inside C. This gives an alternative proof of maximum modulus principle.

Answer 1

Using Rouché's theorem, it is proven that if |f(z)| ≤ M on a contour C, then |f(z)| ≤ M inside C, providing an alternative proof of the maximum modulus principle.

Step-by-step explanation:

The question involves proving that if a function f(z) is analytic within and on a simple closed contour C, and |f(z)| ≤ M on C, then |f(z)| ≤ M inside C. This is demonstrated by using Rouché's theorem, which is a powerful tool in complex analysis used for determining the number of zeros of complex functions within certain contours. The proof works by considering the function g(z) = M + f(z). Since |f(z)| ≤ M on C, Rouché's theorem tells us that g(z) and M have the same number of zeros inside C, which must be zero because M is a constant non-zero function. Hence, f(z) itself has no zeros inside C where it would exceed M, proving that |f(z)| ≤ M inside C.