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Problem 5

Discuss the singularities of $$ f(z)=\frac{z^{3}\left(z^{2}-1\right)(z-2)^{2}}{\sin ^{2}(\pi z)} e^{1 / z^{2}} $$ Classify which of these are poles, removable singularities and essential singularity.

Expert verified

All the singularities of the function are essential singularities at points \(z = n\), where \(n\) is an integer.

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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

How many roots of the equation \(z^{4}+z^{3}+1=0\) have modulus between \(3 / 4\) and \(3 / 2 ?\)

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

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

Show that the equation \(z^{3}+i z+1=0\) has neither a real root nor a purely imaginary root.

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