Residues and Closed Contours : Solve the Integral
Not what you're looking for?
Calculate the following integral...
Please see attached for full question.
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)
Purchase this Solution
Solution Summary
An integral of a closed contour is solved using residues are examined.
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
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.
Recent Feedback
- "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."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"
Purchase this Solution
Free BrainMass Quizzes
Probability Quiz
Some questions on probability
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.