Complex analysis/entire function question

In summary, Complex analysis is a branch of mathematics that deals with functions of complex variables, while an entire function is a complex-valued function defined and analytic on the entire complex plane. Key concepts in complex analysis include analyticity, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem, which are used to study the behavior and properties of complex functions. It has various applications in science, particularly in physics and engineering, and involves techniques such as power series, Laurent series, and residues to solve problems and determine the behavior of complex functions.
  • #1
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Homework Statement



Suppose f is an entire function, satisfying

f(z + a) = f(z) = f(z + b), for all z [itex]\in[/itex] C; where a; b are nonzero, distinct complex numbers.

Prove that f is constant.

Homework Equations



Loville's theorem: if f is bounded & entire, then f is constant.

The Attempt at a Solution



where would I begin to prove this function is bounded? any hint would be appreciated!
 
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  • #2
Yes, show that the function is bounded. This isn't particularly hard to do because it is periodic!
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex variables. It involves the manipulation and analysis of functions that have complex numbers as their inputs and outputs.

2. What is an entire function?

An entire function is a complex-valued function that is defined and analytic on the entire complex plane. In other words, it does not have any singularities or poles. Examples of entire functions include polynomials and trigonometric functions.

3. What are the key concepts in complex analysis?

Some key concepts in complex analysis include analyticity, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem. These concepts are used to study the behavior of complex functions and their properties.

4. What is the importance of complex analysis in science?

Complex analysis has various applications in science, particularly in physics and engineering. It is used to solve problems involving electric circuits, fluid dynamics, and quantum mechanics. It also plays a crucial role in the development of mathematical models and theories in these fields.

5. What are some common techniques used in complex analysis?

Some common techniques used in complex analysis include power series, Laurent series, and residues. These techniques are used to find the behavior of complex functions, calculate integrals, and determine the presence of singularities.

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