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Problem 2

Expand \(f(z)=\frac{3 z-1}{z^{2}-2 z-3}\) in Laurent series valid for (i) \(1<|z|<3\) (ii) \(|z|>3\) (iii) \(|z|<1\).

Expert verified

The Laurent series of \(f(z)\) are \(-\frac{3}{z} + \frac{7}{z^2}\) for \(|z|>3\), \(\frac{1}{z} + \frac{1}{z^2}\) for \(1<|z|<3\), and \(\sum_{n=0}^{\infty} (-z)^n + \sum_{n=0}^{\infty} (\frac{2}{3}z)^n\) for \(|z|<1\).

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Chapter 9

If \(f(z)\) is analytic at \(z_{0}\), show that \(f(z)\) has a zero of order \(k\) at \(z_{0}\) if and only if \(1 / f(z)\) has a pole of order \(k\) at \(z_{0}\).

Chapter 9

Given arbitrary distinct complex numbers \(z_{0}, z_{1}\) and \(z_{2}\), construct a function \(f(z)\) having a removable singularity at \(z=z_{0}\), a pole of order \(k\) at \(z=z_{1}\), and an essential singularity at \(z=z_{2}\).

Chapter 9

If \(f(z)\) has poles at a sequence of points \(\left\\{z_{n}\right\\}\), and \(z_{n} \rightarrow z_{0}\), show that \(f(z)\) does not have a pole at $z=z_{0} .$ Illustrate this fact by a concrete example.

Chapter 9

If \(f(z)\) is analytic and nonzero in the disk \(|z|<1\), prove that for \(0<\) \(r<1\) $$ \exp \left(\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|f\left(r e^{i \theta}\right)\right| d \theta\right)=|f(0)| . $$

Chapter 9

Expand $$ f(z)=\frac{z^{2}+9 z+11}{(z+1)(z+4)} $$ as a Laurent series about \(z=0\) valid when (i) \(|z|<1\) (ii) \(1<|z|<4\) (iii) \(|z|>4\).

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