Question

1. Determine all the isolated singularities of each of the following functions and compute the residue at each singularity.\\ (a) $\frac{e^{3z}}{z - 2}$ (b) $\frac{z + 1}{z^2 - 3z + 2}$ (c) $\frac{\cos z}{z^2}$ (d) $(\frac{z - 1}{z + 1})^3$\\ (e) $\frac{e^z}{z(z + 1)^3}$ (f) $\sin(\frac{1}{3z})$ (g) $\tan z$ (h) $\frac{z - 1}{\sin z}$ \\ (i) $z^2 / (1 - \sqrt{z})$, where $\sqrt{z}$ denotes the principal branch.

          1. Determine all the isolated singularities of each of the following functions and compute the residue at each singularity.\\
(a) $\frac{e^{3z}}{z - 2}$ (b) $\frac{z + 1}{z^2 - 3z + 2}$ (c) $\frac{\cos z}{z^2}$ (d) $(\frac{z - 1}{z + 1})^3$\\
(e) $\frac{e^z}{z(z + 1)^3}$ (f) $\sin(\frac{1}{3z})$ (g) $\tan z$ (h) $\frac{z - 1}{\sin z}$ \\
(i) $z^2 / (1 - \sqrt{z})$, where $\sqrt{z}$ denotes the principal branch.

1. Determine all the isolated singularities of each of the following functions and compute the residue at each singularity.

(a) (e^3z)/(z - 2) (b) (z + 1)/(z^2 - 3z + 2) (c) (cos z)/(z^2) (d) ((z - 1)/(z + 1))^3

(e) (e^z)/(z(z + 1)^3) (f) sin((1)/(3z)) (g) tan z (h) (z - 1)/(sin z)

(i) z^2 / (1 - √(z)), where √(z) denotes the principal branch.

Added by Jose Antonio K.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/12/2023

Step 1

The residue at $$z=2$$ is given by: $$Res_{z=2} \frac{e^{3z}}{z-2} = \lim_{z \to 2} (z-2) \frac{e^{3z}}{z-2} = e^6$$ Show more…

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1. Determine all the isolated singularities of each of the following functions and compute the residue at each singularity. (a) $$ \frac{e^{3z}}{z-2} $$ (b) $$ \frac{z+1}{z^2-3z+2} $$ (c) $$ \frac{cos z}{z^2} $$ (d) $$ \left( \frac{z-1}{z+1} \right)^3 $$ (e) $$ \frac{e^z}{z(z+1)^3} $$ (f) $$ sin \left( \frac{1}{3z} \right) $$ (g) $$ tan z $$ (h) $$ \frac{z-1}{sin z} $$ (i) $$ z^2/(1-\sqrt{z}), $$ where $$\sqrt{z}$$ denotes the principal branch.