Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality)

Let be an entire function such that

Show that is a constant.

I don't know if is analytic at , therefore I cannot use the Maximum Modulus Principle.On the other hand, I cannot prove that the maximum value exists, therefore I cannot use the Cauchy's Inequality.

Thank you.

Re: Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality

Quote:

Originally Posted by

**fareastmovement** Let

be an entire function such that

Show that

is a constant.

I don't know if

is analytic at

, therefore I cannot use the Maximum Modulus Principle.On the other hand, I cannot prove that the maximum value exists, therefore I cannot use the Cauchy's Inequality.

Thank you.

If f is an entire function, then it can be written as a Taylor series . If only the first term of the Taylor series is nonzero (i.e. you have a constant), then you will have .

If the first two terms are nonzero, then you will have

.

If any other terms are nonzero, then you will have

So the only way for is if is constant.

Re: Problem about Complex Analysis (Maximum Modulus Principle and Cauchy's Inequality

Thank you very much(Happy)

Didnt think it was that easy