Proving Non-Existence of f'(z) for f(z) = e^x * e^-iy | Cauchy-Riemann Question

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Homework Help Overview

The discussion revolves around the function f(z) = e^x * e^-iy and the task of demonstrating that its derivative does not exist. The context is rooted in complex analysis, specifically relating to the Cauchy-Riemann equations.

Discussion Character

  • Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants are exploring how to express the function in the form f(z) = u + iv, with some questioning the definitions of u(x,y) and v(x,y). There is a discussion about the nature of the components of the function and their representation.

Discussion Status

Some participants have provided guidance on how to express the function correctly, while others are still seeking clarification on the definitions of the real and imaginary parts. Multiple interpretations of the function's components are being explored.

Contextual Notes

There is a noted confusion regarding the representation of the function, particularly in distinguishing between real and imaginary parts, as well as the correct mathematical operations involved.

tylerc1991
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Homework Statement


Show that f'(z) DNE

f(z) = e^x * e^-iy

Homework Equations



so I have to show that u_x =/= v_y or v_x =/= -u_y

The Attempt at a Solution



my question is this: what is u(x,y) and v(x,y)? is it e^x and e^-iy respectively? Thank you for your help!
 
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You have to write f=u+iv where u and v are real functions. e^(-iy) isn't a real function. And there's a '*' between them, not a '+'.
 
That is the basis of my question, how would I write that function in f(z) = u + iv? e^(x-iy)?
 
tylerc1991 said:
That is the basis of my question, how would I write that function in f(z) = u + iv? e^(x-iy)?

Use deMoivre. e^(it)=cos(t)+i*sin(t).
 
ahh I see, thank you!
 

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