Chapter 14: Q15P (page 705)

To find that the integrals by computing residue at infinity.
$\oint_{c} \frac{z^{2}}{(2 z + 1) (z^{2} + 9)}$ around $| z | = 5$ .

Short Answer

Expert verified

The answer is, $\oint_{c} f (z) d z = 2 πiRes (z = \infty) = 2 πi (\frac{- 1}{2}) = πi$

Step by step solution

Evaluate the integral by residue at t=∞

Let $I = \oint_{c} \frac{z^{2}}{(2 z + 1) (z^{2} + 9)}$ around $|z| = 5$ .

The integral by residue at t=∞

The integral by residue is expressed as,

$\oint_{c} f (z) d z = 2 πiRes (z = \infty)$

C is described as counter clock wise direction

Suppose,. $\begin{array}{rcl} f (z) & = & \frac{z^{2}}{(2 z + 1) (z^{2} + 9)} \\ = & \frac{1}{(2 z + 1) (1 + \frac{9}{z^{2}})} \\ = & \frac{1}{2 (z + \frac{1}{2}) (1 + \frac{9}{z^{2}})} \end{array}$

In regular at $\begin{array}{rcl} z & = & \infty \end{array}$ is given $\begin{array}{rcl} \lim_{z \to \infty} (- z f (z)) \end{array}$ .

$\begin{array}{rcl} \Rightarrow & \lim_{z \to \infty} \frac{1}{2 (z + \frac{1}{2}) (1 + \frac{9}{z^{2}})} = - \frac{1}{2} \end{array}$

Hence, $\begin{array}{rcl} \oint_{c} f (z) d z = 2 πiRes (z & = & \infty) = 2 πi (\frac{- 1}{2}) = πi \end{array}$

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To find that the integrals by computing residue at infinity.
$\oint_{c} \frac{z^{2}}{(2 z + 1) (z^{2} + 9)}$ around $| z | = 5$ .

Short Answer

Step by step solution

Evaluate the integral by residue at t=∞

The integral by residue at t=∞

C is described as counter clock wise direction

One App. One Place for Learning.

Most popular questions from this chapter

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To find that the integrals by computing residue at infinity.∮cz2(2z+1)(z2+9) around |z|=5 .

Short Answer

Step by step solution

Evaluate the integral by residue at t=∞

The integral by residue at t=∞

C is described as counter clock wise direction

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Thermodynamics

Medical Physics

Radiation

Conservation of Energy and Momentum

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

To find that the integrals by computing residue at infinity.
$\oint_{c} \frac{z^{2}}{(2 z + 1) (z^{2} + 9)}$ around $| z | = 5$ .