Chapter 14: Q26P (page 687)

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{2 πiz}}{1 - z^{3}}$ at $z = e^{\frac{2 πi}{3}}$

Short Answer

Expert verified

The residue of the function at $z = e^{\frac{2 π i}{3}}$ is $R (z_{0}) = - e^{- \sqrt{3} π} (\frac{1}{6} - i \frac{\sqrt{3}}{6})$

Step by step solution

Determine the formula:

Residue of a function at simple poles is given by:

$R [f (z)] = \underset{z \to z_{0}}{l i m} (z - z_{0}) f (z)$

Determine the residue of simple pole

Consider the function is written as:

$\begin{array}{rcl} f (z) & = & \frac{q (z)}{p (z)} \\ = & \frac{e^{2 π i z}}{1 - z^{3}} \end{array}$ …….. (1)

At $z_{0} = e^{2 π i / 3}$

The function has simple pole at $z_{0} = e^{\frac{2 π i}{3}}$ and the residue of simple pole is given by:

$\begin{array}{rcl} R (z_{0}) & = & \lim_{z \to z_{0}} (z - z_{0}) f (z) \\ = & \frac{q (z_{0})}{p' (z_{0})} \end{array}$ ……. (2)

Determine the residue of the function:

From equation (2), solve as:

$\begin{array}{rcl} R (z_{0}) & = & {[\frac{e^{2 π i z}}{3 z^{2}}]}_{z = e^{\frac{2 π i}{3}}} \\ = & \frac{\exp [2 π i (- \frac{1}{2} + i \frac{\sqrt{3}}{2})]}{3 {[\exp (\frac{2 π i}{3})]}^{2}} \\ = & - \frac{e^{- \sqrt{3} π} e^{- π i}}{3 e^{\frac{4 π i}{3}}} \\ = & - \frac{e^{- \sqrt{3} π} e^{- \frac{7 π i}{3}}}{3} \end{array}$

Solve further to obtain,

$\begin{array}{rcl} R (z_{0}) & = & - \frac{e^{- \sqrt{3} π}}{3} [\cos (\frac{7 π i}{3}) - i \sin (\frac{7 π i}{3})] \\ = & - e^{- \sqrt{3} π} (\frac{1}{6} - i \frac{\sqrt{3}}{6}) \end{array}$

Therefore, the residue of a function at $z = e^{\frac{2 π i}{3}}$ is $R (z_{0}) = - e^{- \sqrt{3} π} (\frac{1}{6} - i \frac{\sqrt{3}}{6})$ .

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Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{2 πiz}}{1 - z^{3}}$ at $z = e^{\frac{2 πi}{3}}$

Short Answer

Step by step solution

Determine the formula:

Determine the residue of simple pole

Determine the residue of the function:

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Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.e2πiz1-z3at z=e2πi3

Short Answer

Step by step solution

Determine the formula:

Determine the residue of simple pole

Determine the residue of the function:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Thermodynamics

Kinematics Physics

Fields in Physics

Fluids

Quantum Physics

Study anywhere. Anytime. Across all devices.

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.
$\frac{e^{2 πiz}}{1 - z^{3}}$ at $z = e^{\frac{2 πi}{3}}$