Complex Analysis: Maximum Modulus Principle

  • Thread starter tylerc1991
  • Start date
  • #1
166
0

Homework Statement



Let f(z) = u(x,y) + iv(x,y) be a continuous, non-constant function that is analytic on some closed bounded region R. Prove that the component function u(x,y) reaches a minimum value on the boundary of R.

The Attempt at a Solution



By the minimum modulus principle, f(z) reaches its minimum value on the boundary of R. => u(x,y) reaches its minimum value on the boundary of R or v(x,y) reaches its minimum value on the boundary of R (or both). Assume that v(x,y) reaches its minimum value on the boundary of R (and that u(x,y) does not). Then u(x,y) is constant throughout R (since f(z) is bounded on R and hence u(x,y) is bounded on R. I have shown in another problem that if this case arises that u(x,y) is constant). Consider the function exp(z). If u(x,y) is constant then |exp(z)| is constant and hence a contradiction is reached.
 

Answers and Replies

Related Threads on Complex Analysis: Maximum Modulus Principle

Replies
4
Views
1K
Replies
5
Views
6K
Replies
5
Views
1K
Replies
2
Views
761
  • Last Post
Replies
0
Views
5K
Replies
1
Views
7K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
0
Views
1K
Top