Consequence of Cauchy's Integral Formula

  • Thread starter Nebula
  • Start date
  • #1
46
0

Homework Statement


Suppose f is analytic inside and on the unit circle |z|=1. Prove that if |f(z)|[tex]\leq[/tex]M for |z|=1, then |f(0)|[tex]\leq[/tex]M and |f'(0)|[tex]\leq[/tex]M.

Finally, what estimate can you give for |f^n(0)|?

Homework Equations


This problem is in a section dealing with Cauchy's Integral Formula and it's generalized form.
http://en.wikipedia.org/wiki/Cauchy_integral_formula

The Attempt at a Solution


Im not quite sure how to get started here. Obviously Cauchy's formulas apply to our function f considering its analiticity. I guess I don't really understand the if part of our statement. IE if |f(z)|[tex]\leq[/tex]M for |z|=1. The magnitude of the function f is bounded??? I don't know what it means "for |z|=1." I'm missing something here. A nudge in the right direction would be appreciated. Thanks!
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,263
619
The absolute value of the integral of a function is less or equal to the integral of the absolute value of the function times the length of the curve you are integrating over. The 'a' the Cauchy formula is zero. |z|=1. The function f is bounded on |z|=1 by M. That's a nudge.
 

Related Threads on Consequence of Cauchy's Integral Formula

Replies
1
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
891
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
2
Views
12K
  • Last Post
Replies
5
Views
5K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
7
Views
1K
Replies
2
Views
3K
  • Last Post
Replies
3
Views
4K
Top