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Problem 21

Suppose \(f(z)\) is analytic at \(z_{0}\) with $f^{\prime}\left(z_{0}\right) \neq 0 .\( Show that there exists an analytic function \)g(z)\( such that \)f(g(z))=z$ in some neighborhood of \(z_{0}\). This is known as the inverse function theorem.

Expert verified

By applying Newton's method and carefully controlling the error of each iteration, we can show that there exists an analytic function \(g\) whose composition with \(f\) equals \(z\) in some neighborhood of \(z_{0}\).

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Chapter 9

If \(0<|a|<|b|\), expand $$ f(z)=\frac{1}{z(z-a)(z-b)} $$ as a Laurent series valid in (i) \(0<|z|<|a|\) (ii) \(|a|<|z|<|b|\) (iii) \(|z|>|b|\).

Chapter 9

Evaluate the following integrals along different simple closed curves not passing through 0 and \(\pm 1\). (i) \(\int_{C} \frac{e^{z}-1}{z^{2}(z-1)} d z\) (ii) \(\int_{C} \frac{e^{z}}{z^{2}\left(1-z^{2}\right)} d z\).

Chapter 9

Determine the order of the pole at \(z=0\) for (i) \(f(z)=\frac{z}{\sin z-z+z^{3} / 3 !}\) (ii) \(f(z)=\frac{z}{\left(\sin z-z+z^{3} / 3 !\right)^{2}}\).

Chapter 9

Use "long division" method (or other method) to find the principal part in the Laurent series of \(f(z)=1 /(1-\cos z)\) about \(z=0\).

Chapter 9

Express \(\sin z \sin (1 / z)\) in a Laurent series valid for \(|z|>0\).

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