Purely Real-valued Analytic Functions?

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Homework Help Overview

The discussion revolves around the question of whether a function that is purely real-valued can be considered analytic, specifically in the context of the Cauchy-Riemann conditions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the Cauchy-Riemann conditions for purely real-valued functions, questioning whether infinite differentiability guarantees analyticity. There is a focus on the relationship between the real part and the imaginary part of the function.

Discussion Status

Participants have engaged in examining the conditions under which a purely real-valued function could be analytic, with some suggesting that if the imaginary part is zero, it leads to the conclusion that the function must be constant. This line of reasoning has prompted further exploration of the implications of the Cauchy-Riemann conditions.

Contextual Notes

There is an ongoing discussion about the nature of real-valued functions and their differentiability, particularly in relation to the Cauchy-Riemann conditions, which may not be fully resolved.

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Homework Statement


Can a function which is purely real-valued be analytic? Describe the behavior of such functions?


Homework Equations


The Cauchy-Riemann conditions
ux=vy, vx=-uy

The Attempt at a Solution



I can't think of any pure real-valued equations off the top of my head which satisfy the CR conditions. However, could any function like sin(x), cos(x), e^x, which is infinitely differentiable be considered analytic? Doesn't infinite differentiability imply that the CR conditions are already satisfied?
 
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jtleafs33 said:

Homework Statement


Can a function which is purely real-valued be analytic? Describe the behavior of such functions?


Homework Equations


The Cauchy-Riemann conditions
ux=vy, vx=-uy

The Attempt at a Solution



I can't think of any pure real-valued equations off the top of my head which satisfy the CR conditions. However, could any function like sin(x), cos(x), e^x, which is infinitely differentiable be considered analytic? Doesn't infinite differentiability imply that the CR conditions are already satisfied?

No. If your function is real valued then v=0. What does that tell you about u?
 
If v=0, then vx=vy=0. So then, u must be constant, so that it's derivative will also be zero, and thus satisfy the CR conditions?
 
jtleafs33 said:
If v=0, then vx=vy=0. So then, u must be constant, so that it's derivative will also be zero, and thus satisfy the CR conditions?

Yes, that's it. The only purely real analytic functions are constant.
 

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