# Residues and Closed Contours : Solve the Integral

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Calculate the following integral...

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Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C.

Let . By residual Theorem, we have

...............................(1)

Note:

(1) are the only poles of .

(2)

(2) is very useful when we compute the following residues.

.....................................(2)

.....................................(3)

Now we evaluate . We know that

Note: We use an inequality

So,

When R goes to infinity, we know that

....................(4)

By (1), (2) , (3) and (4), we take limit as R goes to infinity, we can obtain

ie.,

i.e.,

i.e.,

Comparing the real part of both sides, we get

Note:

So, we get

So, by the fact of (since is an even function)

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An integral of a closed contour is solved using residues are examined.

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Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C.

Let . By residual Theorem, we have

...............................(1)

Note:

(1) are ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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