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Complex Analysis Are My Answers Correct?

Mathematics Asked by Nerdy Goat on November 19, 2020

Good Evening,

I am posting this on behalf of a friend, she has a complex analysis maths exam tomorrow morning and wanted some clarification on her answers however this is not my field, so I am hoping someone with the correct knowledge can confirm or assist with the correct solution.

She has provided me with some pictures of her working out solutions which I have attached below, along with the questions 4 and 5 that need answering, English is her second language so I will convey to her anything not understood, however in a maths sense she can interpret it very well.

Many thanks in advance, any input is greatly appreciated.

Questions 4 and 5
questions 4 and 5

working out of question number 5
working out of question number 5

working out of question number 4
working out of question number 4

One Answer

Number 5 is correct.

In question 4, you wrote that $$f(overline{z})=u(z)-iv(z)$$ which isn't necessarily true. What's true is that $$f(overline{z})=f(x,-y)=u(x,-y)+iv(x-y)$$ You can however use the definition of $frac{d}{dz} f(z)$ to show that $$frac{d}{dz} overline{f(overline{z})}=overline{f'(overline{z})}$$

Correct answer by Mandelbrot on November 19, 2020

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