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I have a couple questions on complex variables:

1.)If you have a complex function defined as follows:

[tex]f(z)=u(x,y)+iv(x,y)[/tex]

with x,y real, what do you get if you take the complex conjugate of the variable z?

[tex]f(z^*)=?[/tex]

I was thinking that it wouldn't change since the complex variable has been replaced with two real variables, but that doesn't seem right.

If I take the conjugate of the entire function, is this what I should get:

[tex]f^*(z)=u(x,y)-iv(x,y)[/tex]

2.) If you are trying to prove that a function is not analytic at a specific point, is it sufficient to show that the Cauchy-Riemann conditions do not hold? I'm trying to show that the derivative of a function at zero is dependant on the direction that you approach zero. I've shown that the C-R conditions are not met, but I'm not sure how to show explicitly for that point that they are not met.

EDIT: NEW QUESTION

I figured I would just edit this topic instead of starting a new one. I'm trying to prove that if a function [tex]f(z)[/tex] is analytic then the function [tex]f^{*}(z^{*})[/tex] is also analytic.

I'm not sure how to get this started. It makes sense that taking the conjugate wouldn't affect the differentiability, but I don't know how to prove that. Any hints on how to get this problem started?

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# Homework Help: Complex conjugates

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