So I'm currently taking a 3rd year University maths course which includes Residue theorem. I've been able to work out most of the questions but I've been getting stuck with the expansion of (sin(z))^2. Since by Euler's formula sin(z) = (exp(iz) - exp(-iz)) / 2i, I would expect that (sin(z))^2 = ((exp(iz) - exp(-iz)) / 2i) ^ 2 = (exp(2iz) - exp(-2iz) - 2) / (-4) using normal expansion rules. In order to get the correct result the actual expansion needs to be (2 * exp(2iz) - 2) / (-4). What am I missing here?
Any help is appreciated!
Suppose the following:
- U is a bounded domain such that \partial U = \Gamma is the boundary. In particular, closure of U=U \cup \Gamma.
- f: C \ closure(U) --> C is holomorphic and extends continuously to \Gamma.
- A= lim f(z) as z --> infinity is finite
Now, I want to show that for any u in C \ closure(U),
f(u) = A - (1/2πi) * \int_\Gamma (f(z)dz)/(z-u)
I definitely recognize that Cauchy integral formula as well as the residue theorem could be useful here, but I am unsure how to apply it to C \ closure(U).
I was thinking I could consider the function g(z) = f(1/z) instead and then extend it to zero such that g(0) = A. Then apply Cauchy integral formula to this, but I am stuck...
Thank you so much in advance!
So guys.. i have this doubt and i don't know from where i have to begin..
Hello,
I am trying to compute a real integral using the residue theorem.
It is: Integral from -1 to 1 of sqrt(1-x^2) / (1+x^2) dx
I have noticed the function is symmetric about the origin, that could be useful.
I can't seem to find a nice curve to use the residue theorem on, so I suspect a (real) change of variables is needed first!
Any hints? Thank you!
This might be a dumb question but is there a field within C.S. in which the Residue Theorem from Complex Analysis is used in? I know some other areas from Complex Analysis are used within C.S. but I'm having trouble finding something pertaining to the Residue Theorem.
Thank you
Hello,
I'm about to embark on my senior project in college. My topic is using the Residue Theorem in order to compute multiple integrals. I'm having lots of trouble as to how my topic can go beyond what is covered in a Complex Analysis course. I've been trying to apply the Theorem within some fields of Physics but I can't see how it applies. How can computing multiple integrals with the Residue Theorem go beyond what is covered in an undergrad Complex Analysis course. Any ideas on any applications to expand my knowledge about the Theorem?
Thank you
Please, how can I compute the following integral using the residue theorem? It appears to me it has no singularity...
https://imgur.com/cONwH0Q
Thanks in advance for your help.
I'm trying this problem: https://i.imgur.com/cOV50pU.jpg
I've converted the integral to a contour integral on the complex plane, but I believe I'm running into a problem finding the residues, I just don't think my work is correct. Did I convert the function incorrectly, did I calculate the poles wrong, or maybe both? Or am I on the right track, and I should continue and take L'Hopital's rule on the limit.
I know we start with an improper integral defined on the Real Axis.
All the sources I have read then make a large semicircular contour on the top of the Imaginary Plane, and say that the arc goes to zero, and then evaluate the complex integral using the residue theorem on the top two quadrants.
In essence, to my understanding, you take the real integral; replace x with z and then integrate using the residue theorem over the top two quadrants.
I'm not sure why the arc goes to zero, if this method works in all cases, if we just consider the residues where imaginary part > 0.
Any help/summary/sources is much appreciated. :)
https://i.imgur.com/ekXOq0j.jpg
I'm solving this integral and turning it into a contour integral over the unit circle. The poles are z = -3/7 and z = i.
This may be a dumb question, but since z = i intersects the circle, do I need to find that residue to calculate the integral?
If you have a simple closed curve i the complex plane, i.e. a loop, and a singularity within the loop, i.e. holes, then you can integrate over that loop by taking the residues within those singularities, adding them up and then multiplying by 2πi, which is actually pretty simple. Getting the residues is pretty easy ypu just have to take a limit depending on the order of the singularity.
However, since Captain Holt has never taken a loophole, he probably despises this method. He would probably say something like, "I say this without any doubt or falsehood, but I would rather integrate numerically" if he's confronted with this and Kevin would probably react with shock
Hello.
I am attempting to solve the following contour integral (along a circle around the origin with radius 1) using the Cauchy Residue Theorem:
https://imgur.com/a/k1o9r
However, I am quite unsure of how to calculate the residue at z_0=0 because of the high power involved. Any advice for how to get started?
Any help is highly appreciated!
For example, If I have (z-2)^2 * cos[(z^2 -4z)/(z-2)^2 ], the function has an essential singularity at Z=2, but how do I apply the residue theorem to calculate the integral?
I was thinking I could do the Laurent series of the function in (z-2) and check for the coefficient of (z-2)^-1 and that would be it. But is that really the best way?
Solved The problem is "Integrate z^(3)·cos(3/z) dz on the circle centered at zero with the radius 3."
The correct answer is 27(pi)i/4. I rewrote cos(3/z) as a Laurent series to 1 - 9/z^2 + 27/(8·z^4) - 81/(8·z^6) + ... so after multiplying with z^3 and (z-0)^1 (since we only have one pole) i get z^4 - 9·z^2 + 27/8 - 81/(8·z^2) + ... and as we take the limit of this to 0 i suppose the book gets it to 27/8 but i disagree, am i missing something?
edit: I saw my fault now, i didnt realize that the residue is the first term in the series that has a z to the power of a negative number
How do I compute an integral with limits of -inf/+inf using Residue Theorem?
I am told to do this for integral of
[; cos(3x)dx/(1+x^2)^2 ;]
I do not ask for exact solution here but for what pieces of theory needs to be used in order to compute it.
Can someone solve this an tell me what they get?
I keep getting 1/2, while my professor is saying it is -1/2
find the residue of: https://i.imgur.com/rnsw4Wt.jpg
I'm having a hard time wrapping my head around this topic, this is the second time I've seen it in a class and I don't understand it very well, it seems overly complicated. I was wondering if somebody could explain it as simply as possibly for me, I'd rather not just memorize the steps. For example, one of the integrals I'm supposed to evaluate is (x^2 )/(x^4 + x^2 +1) from 0 to infinity.
Thank you.
I am a physics grad student (high energy), I've come across a problem while doing a certain loop integral that I don't understand. I've removed as much of the physics as I can so that this is just a math problem, hopefully someone can spot my error:
I need to solve this integral using standard contour integration and the residue theorem:
[; \int_ \infty^ \infty dx\int_ \infty^ \infty dy \frac{1}{(A+x y+i \epsilon ) (A+x (y-b)+i \epsilon )} ;]
where [; A, b, \epsilon ;] are real constants and [; i ;] is the imaginary unit.
Suppose I do the x integral by completing a contour in one of the half-planes. I can do this because the denominator has degree 2, so the contribution from the arc goes to zero. The roots are:
[; x = -\frac{A+i \epsilon }{y}, \frac{A+i \epsilon }{b-y} ;]
For [; y<0 ;] and [; y>b ;], both roots are in the same half-plane, so I can close the contour in the other half-plane and get zero by the residue theorem. When [; 0<y<b ;], the roots are in different half-planes, I choose to pick up the residue from the either root listed above, and get
[; \int_ 0^ b dy \frac{2 i \pi }{b (A+i \epsilon )} = \frac{2 i \pi }{(A+i \epsilon )} ;]
This looks pretty good so far, but now the problem: I should be able to do the y integral by contours instead. In this case the roots are
[; y = \frac{-A-i \epsilon }{x}, \frac{-A+b x-i \epsilon }{x} ;]
And now for any non-zero x, the roots are in the same half-plane, so I should get zero for the contour integral, and then zero as my final answer.
Now, I'm pretty sure that the second line of reasoning is wrong, and that it has something to do with the fact that the roots are both singular at x=0, while in the first case only one root is singular at a time. The final result doesn't depend on b in the first case (and in particular would be the same no matter how small b is), which suggests to me that in the second case there is some contribution from x=0 that I'm missing.
I would like to understand exactly what has gone wrong in the second case, because I've written scripts to perform integrals of this kind automatically, and I need to modify them to recognize this issue.
Also, I know that there are tricks for solving a large class of loop integrals using Feynman parameters, but the one's I'm working with are very unusual, such that I'm forced to use alternative methods. What I'm really interested in is the math question, i.e. "why can't I solve this integral this way?"
Thanks for your help.
... keep reading on reddit ➡I am trying to find the least nonnegative residue of 3^2016 modulo 22. I am trying to approach this using fermat's little theorem,
So I have 3^21 congruent 1 (mod 22)
So I can rewrite 3^2016 as (3^21)^96 * 3^0, and that becomes 1^96 * 1 = 1 (mod 22), so the answer I get is 1, but it is really 3^2016 congruent 3 (mod 22). Is there something I am doing wrong? or is Fermat's little theorem not the correct way to approach this?
