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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

Just_A_User
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Nerdy Goat
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    Curious sort of exam, where the students are allowed to ask online for help... – David C. Ullrich Aug 08 '20 at 21:17
  • @DavidC.Ullrich Is it not common practice to ask for help before exams? the reason of asking is to help one understand the solving of questions like these, there is no guarantee any of these questions will show up in the exam, thus there is no form of cheating. – Nerdy Goat Aug 08 '20 at 22:38
  • Typeset mathematical terms using MathJax rather than posting image. Here's the tutorial. – SarGe Aug 09 '20 at 07:02

1 Answers1

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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 $$\frac{d}{dz} \overline{f(\overline{z})}=\overline{f'(\overline{z})}$$

Mandelbrot
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  • Another way is to pick $w$ in $G$ and consider the Taylor expansion of $f$ at $w$ with coefficients $a_n$; then show that the conjugate function has an expansion at $\bar w$ with coefficients $\bar a_n$ so is analytic – Conrad Aug 08 '20 at 21:27
  • As can be seen here https://math.stackexchange.com/questions/551458/complex-conjugates-of-holomorphic-functions – Mandelbrot Aug 08 '20 at 21:29