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
working out of question number 5
working out of question number 4
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 $$\frac{d}{dz} \overline{f(\overline{z})}=\overline{f'(\overline{z})}$$