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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
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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### Problems for Section 3.5 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) \(\frac{e^{z^2} - 1}{z^2}\) - (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
Transcribed Image Text:### Problems for Section 3.5 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) \(\frac{e^{z^2} - 1}{z^2}\) - (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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