Answered: log r xp- (1+x²)(4+x²) 0,

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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I asked this question previously, but I don't think I got a correct answer. The tutor used only the pole at i, but doesn't the pole at 2i need to be considered also? I got a final answer of 0 and am wondering if you will get the same thing. Thanks!

Expert Solution

Solution:

Let $f (z) = \frac{\log z}{(1 + z^{2}) (4 + z^{2})}$

Use the contour shown as

Then, by the residue theorem, $\int_{C_{1} + C_{R}} f (z) d z = 2 πi \sum_{}^{} Residues of f inside the contour$ .

Examine each of the pieces in the above equation.

Then, we have

$\lim_{R \to \infty} \int_{C_{R}} f (z) d z = 0$ as $|f (z)| < \frac{M}{{|z|}^{4}} w h e r e M > 0$

Next left is,

$\begin{array}{rcl} \lim_{R \to \infty} \int_{C_{1}} f (z) d z & = & 0 = \lim_{R \to \infty} \int_{- R}^{R} f (z) d z \\ = & \int_{- \infty}^{\infty} f (x) d x \\ = & \int_{0}^{\infty} f (x) d x as f is defined on (0, \infty) \\ = & I \end{array}$

Then, obtain as follows: $\begin{array}{rcl} \int_{0}^{\infty} f (x) d x & = & 2 π i \sum_{}^{} Residues of f inside the contour \end{array}$

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