(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The function f(z) is analytic. Show that the derivative of f(z) with respect to z* does not exist unless f(z) is a constant.

Hint: Use chain rule and take x = (z+z*)/2, y = (z-z*)/2.

2. Relevant equations

[tex]\frac{d f}{d z*} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial z*} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial z*}[/tex].

3. The attempt at a solution

Well, I used this relation, considering that the analyticity of f guarantees this. I'm not sure of this procedure, but it was the only way i figured out to use the hint of the author. Then, the result was this:

[tex]\frac{d f}{d z*} = \frac{1}{2} \left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial x} \right)[/tex].

Next, I used another relation which I seriously doubt of:

[tex]\frac{\partial f}{\partial x} = \frac{d f}{d z} \frac{\partial z}{\partial x} = 1 [/tex].

Analogously, I deduced that

[tex]\frac{\partial f}{\partial y} = \frac{d f}{d z} \frac{\partial z}{\partial y} = i [/tex].

With these results, the previous equation becomes:

[tex]\frac{d f}{d z*} = \frac{1}{2} \left( \frac{d f}{d z} - \frac{d f}{d z} \right) = 0 [/tex].

This result sounds like an absurd to me, and this could be the answer by "reductio ad absurdum" but my hypothesis doesn't seem correct (or rigorous). Someone has any idea?

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# Homework Help: Derivative of f(z) with respect to z* does not exist

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