Chapter 14: Q28P (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.
$\frac{z + 2}{(z^{2} + 9) (z^{2} + 1)}$ at $z = 3 i$

Short Answer

Expert verified

The residue of the function at $z = 3 i$ is $R (3 i) = - \frac{1}{16} + \frac{1}{24} i$

Step by step solution

Determine the formula

Consider the formula for 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 the simple pole

Consider the expression for the given function as:

$f (z) = \frac{z + 2}{(z^{2} + 9) (z^{2} + 1)}$

At $z = 3 i$

The function has simple pole at $z = 3 i$ and the residue of simple pole is given by:

$R (3 i) = \lim_{z \to 3 i} (z - z_{0}) f (z)$ ……. (1)

Determine the residue of the function:

Substitute the values in equation (1) and solve as:

$\begin{array}{rcl} R (3 i) & = & \lim_{z \to z_{0}} (z - 3 i) \frac{z + 2}{(z^{2} + 9) (z^{2} + 1)} \\ = & \lim_{z \to z_{0}} (z - 3 i) \frac{z + 2}{(z + 3 i) (z - 3 i) (z^{2} + 1)} \\ = & \frac{(3 i) + 2}{(3 i + 3 i) ({(3 i)}^{2} + 1)} \\ = & \frac{2 + 3 i}{6 i (- 8)} \end{array}$

Solve further as:

$\begin{array}{rcl} R (3 i) & = & \frac{2 + 3 i}{- 48 i} \\ = & - \frac{1}{64} + \frac{1}{24} i \end{array}$

Therefore, the residue of a function at $\begin{array}{rcl} z & = & 3 i \end{array}$ is $R (3 i) = - \frac{1}{16} + \frac{1}{24}$ .

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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.
$\frac{z + 2}{(z^{2} + 9) (z^{2} + 1)}$ at $z = 3 i$

Short Answer

Step by step solution

Determine the formula

Determine the residue of the simple pole

Determine the residue of the function:

One App. One Place for Learning.

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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.z+2z2+9z2+1atz=3i

Short Answer

Step by step solution

Determine the formula

Determine the residue of the 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

Further Mechanics and Thermal Physics

Space Physics

Fundamentals of Physics

Astrophysics

Electrostatics

Solid State 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.
$\frac{z + 2}{(z^{2} + 9) (z^{2} + 1)}$ at $z = 3 i$