question from ablowitz and folkas. ### Problems for Section 3.51. **Discuss the type of singularity** (removable, pole and order, essential, branch, cluster, natural barrier, etc.); if the type is a pole, give the strength of the pole, and give the nature (isolated or not) of all singular points associated with the following functions. Include the point at infinity. - (a) $\frac{e^{z^2} - 1}{z^2}$ - (b) $\frac{e^{2z} - 1}{z^2}$ - (c) $e^{\tan z}$ - (d) $\frac{z^3}{z^2 + z + 1}$ - (e) $\frac{z^{1/3} - 1}{z - 1}$ - (f) $\log(1 + z^{1/2})$ - (g) $f(z) = \begin{cases} z^2 & |z| \leq 1 \\ 1/z^2 & |z| > 1 \end{cases}$ - (h) $f(z) = \sum_{n=1}^{\infty}\frac{z^n}{n!}$ - (i) $\text{sech } z$ - (j) $\text{coth }\frac{1}{z}$### Notes:- **Singularity Types**: - **Removable**: Singularities that can be removed by defining the function appropriately at that point. - **Pole**: Points where a function goes to infinity. The order of the pole is the highest power in the denominator that causes the infinity. - **Essential**: Singularities where the function behaves erratically and does not have a pole or removable form. - **Branch**: Points that introduce a multi-valued nature to functions, like square roots or logarithms. - **Cluster**: Points where singularities accumulate. - **Natural Barrier**: Points beyond which the function cannot be analytically continued.- **Points at Infinity**: Analysis of the behavior of functions as $ z $ approaches infinity is crucial in complex analysis.Each function listed has its unique behaviors and singular points

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Discuss the type of singularity (removable, pole and order, essential, branch, cluster, natural barrier, etc.); if the type is a pole give the strength of the pole, and give the nature (isolated or not) of all singular points associated with the following functions. Include the point at infinity. (a) e²² z² 1 (b) e²-1 z² (c) etanz (d) z³ z²+z+1

Discuss the type of singularity (removable, pole and order, essential, branch, cluster, natural barrier, etc.); if the type is a pole give the strength of the pole, and give the nature (isolated or not) of all singular points associated with the following functions. Include the point at infinity. (a) e²² z² 1 (b) e²-1 z² (c) etanz (d) z³ z²+z+1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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