I need another easier answer for this question

Anonymous
timer Asked: Nov 16th, 2015
account_balance_wallet $5

Question Description

I need another easier answer for this question
http://www.cramster.com/answers-jun-11/physics/calculate-residuse-calculatenbspnbsp-residuse-theorem_1342637.aspx?rec=0

Tutor Answer

anusha_reddy
School: UC Berkeley

Thank you for the opportunity to help you with your question!

We will use the semicircle as the contour of integration. Just for simplicity, to justify this integration, the integral on the arc going from : π to -π, vanishes, so I only need to evaluate the integral on the x-axis, i.e. real line:

The function of choice should be : 

I need to find the roots of the following equation:Let z = (re^iθ)

write -1 as: -1 = e^(iπ +i2πn), (z^4) = = -1, becomes: (r^4)(e^(i4θ)) = (e^i(π+4nπ))

Equate magnitude to magnitude, to get : r^4 = 1, hence r = 1.

Equate argument to argument to get: , let n = 0,1,2,3 to get all the roots:

They are: 

Let the zeros be denoted: 

Rewrite f(z) as followed: f(z) = 

Only the first two zero are inside the semicircle:

The residue at , the math is messy, but it's:

The residue at 

Then 2πi times by the sum of the residue at z1 and z2 gives you the desired answer:

This is basically Cauchy's integral formula, which is a special case of the Residue theorem.


Please let me know if you need any clarification. I'm always happy to answer your questions.

flag Report DMCA
Review

Anonymous
Top quality work from this guy! I'll be back!

Similar Questions
Hot Questions
Related Tags
Study Guides

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors