Purchase Solution

Analytic Function Proofs on Bounded Regions

Not what you're looking for?

Ask Custom Question

(a) Let f be analytic in a bounded region D and its boundary C, such that |f(z)| = 1
on C. Show that f has at least one zero inside D, unless f is a constant.
(b) Let f(z) be an analytic function in a region D except for one simple pole and
assume |f(z)| = 1 on the boundary of D. Prove that every value a with |a| > 1 is
taken by f(z) inside D once and once only.

Purchase this Solution

Solution Summary

This is two proofs regarding analytic functions.

Solution Preview

Please see the attached file.

(a) Proof:
Here we use a basic fact that if is analytic in a region , then is a constant if and only if is a constant. The "if" part can be verified by C-R equations.
Now in this problem, suppose is not a constant and has no zeros in the region . Let . Since is analytic, then is analytic. So both and can not obtain its maximum value ...

Purchase this Solution

Free BrainMass Quizzes
Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.