Entire Function with Negative Imaginary Values: Proving Constantness

In summary, the conversation discusses how to prove that an entire function f:C->C is constant when its imaginary part, Imf(z), is less than or equal to 0 for all z in C. It is suggested to use the Cauchy-Riemann equations and Liouville's Theorem, and to consider the function 1/(f(z)-i) to show that f is bounded. This leads to the conclusion that f must be constant.
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Homework Statement


Let f:C->C be an entire function such that Imf(z) <= 0 for all z in C. Prove that f is constant.


Homework Equations


Cauchy-Riemann equations??


The Attempt at a Solution


I don't know why I haven't been able to get anywhere with this problem. I feel like I have to use the fact that Imf(z) is harmonic or satisfies the Cauchy-Riemann equations, or something like that. And then somehow show that f is bounded. From there I just apply Liouville's Theorem. But I just need a slight push in the right direction. I mean, if Imf(z) <= 0 for all z, what does that say about its derivatives? This is really frustrating.
 
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  • #2
Hint: Consider the function 1/(f(z)-i). Is it bounded?
 

FAQ: Entire Function with Negative Imaginary Values: Proving Constantness

What are bounded entire functions?

Bounded entire functions are functions that are defined and holomorphic over the entire complex plane and are limited in their growth. This means that the values of the function do not go to infinity as the input approaches infinity.

Why are bounded entire functions important?

Bounded entire functions are important because they have many useful properties and applications in mathematics and physics. They are used to solve differential equations, to approximate other functions, and to represent complex systems.

What is the relationship between bounded entire functions and the Cauchy-Riemann equations?

Bounded entire functions satisfy the Cauchy-Riemann equations, which are a set of conditions that must be met for a function to be holomorphic. This means that the function has a complex derivative at every point in its domain, which is a defining characteristic of bounded entire functions.

How can one determine if a function is bounded entire?

A function can be determined to be bounded entire by examining its properties and using theorems such as Liouville's Theorem and Cauchy's Integral Theorem. If a function is holomorphic and its values do not go to infinity, then it is bounded entire.

What are some examples of bounded entire functions?

Some examples of bounded entire functions include polynomials, exponential functions, trigonometric functions, and rational functions. These functions have finite values and are holomorphic over the entire complex plane, satisfying the definition of bounded entire functions.

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