Contour integral Definition and 122 Threads

Homework Statement let t be the triangle with vertices at the points -3, 2i, and 3, oriented counterclockwise. compute \int \frac {z+1}{z^2 + 1} dz Homework Equations f(z) = \frac {1}{2 \pi i} * \int \frac {f(z)}{z - z_o} dz The Attempt at a Solution the integrand fails to be analytic at...

Hi there! I am trying to prove the following 2 identities using complex analysis methods and contour integration and I'm really stuck on defining the integration paths. \int_{0}^{1}\frac{\log(x+1)}{x^2+1}d x=\frac{\pi\log2}{8} \int_{0}^{\infty}\frac{x^3}{e^x-1}d x=\frac{\pi^2}{15}...

Homework Statement http://img243.imageshack.us/img243/4339/69855059.jpg I can't seem to get far. It makes use of the Exponentional Taylor Series: Homework Equations http://img31.imageshack.us/img31/6163/37267605.jpg The Attempt at a Solution taylor series expansions for cos...

Is there a way to perform a contour integral around zero of something like f(z)/z e^(1/z), where f is holomorphic at 0? If you expand you get something like: \frac{1}{z} \left( f(0) + z f'(0) + \frac{1}{2!} z^2 f''(0) + ... \right) \left( 1 + \frac{1}{z} + \frac{1}{2!} \frac{1}{z^2} + ...

so suppose i wanted to calculate the antiderivatives of e^x\sin{x} and just for the hell of it also e^x\cos{x}. well i could perform integration by parts twice recognize that the original integral when it reappears, subtract from one side to the other blah blah blah. or i could pervert a...

Find an asymptotic approximation as p goes to infinity: f_{\lambda}(p)=\oint_{C}exp(-ipsinz+i\lambda z)dz where C is a square contour and p, lambda are real. Taking C to be of side length pi and centered at the origin, I applied the method of steepest descent at the point z=-pi/2...

Homework Statement Use contour integration to obtain the result int(sin(x)^2/x^2, x=-Inf..Inf) = Pi Homework Equations The Attempt at a Solution I defined a contour that encircles the pole at z=0. It looks like a bridge over the pole. The outer integral is easily shown to be zero...

We have an integral over q from -\infty to +\infty as a contour integral in the complex q plane. If the integrand vanishes fast enough as the absolute value of q goes to infinity, we can rotate this contour counterclockwise by 90 degrees, so that it runs from -i\infty to +i\infty. In making...

Can anyone recommend a good introduction to contour integrals for someone not taking complex analysis? We are doing these integrals in a physics class and I'm terribly confused. I know that I have to choose contours that "go around" my poles, but I don't understand how to do this (I can't seem...

Using Cauchy's integral theorem how could we compute \oint _{C}dz D^{r} \delta (z) z^{-m} since delta (z) is not strictly an analytic function and we have a pole of order 'm' here C is a closed contour in complex plane

I have two related questions. First of all, we have the identity: \int_{-\infty}^{\infty} e^{ikx} dk = 2 \pi \delta(x) I'm wondering if it's possible to get this by contour integration. It's not hard to show that the function is zero for x non-zero, but the behavior at x=0 is bugging...

Homework Statement Find the contour integral of e(iqz)/z^4 The Attempt at a Solution Is it infinite as the pole at z=0 is too high?

I'm trying to find \int_{-\infty}^{\infty} \frac{exp(ax)}{cosh(x)} dx where 0<a<1 and x is taken to be real. I'm doing this by contour integration using a contour with corners +- R, +- R + i(pi), and I'm getting an imaginary answer which is \frac{2i\pi}{sin (a \pi)}. I'm thinking this is...

Contour integral How would you deal with this? \int \frac{\rho \sin{\theta} d \rho d \theta}{\cos{\theta}} \frac{K^2}{K^2 + \rho^2} e^{i \rho \cos{\theta} f(\mathbf{x})} if the cos(theta) were'nt on the bottom I'd have no problem; I'd simply substitute for cos(theta) and the sin(theta)...

\int \frac{\rho^4 \sin^3{\theta} d \rho d \theta e^{i \rho r \cos{\theta}}}{(2 \pi)^2 [K^2 + \rho^2]} I am confused about where the singularities are in this function. Will they simply be at \rho = iK and -ik or does the \rho^4 factor make a difference? Also the sin^3(\theta) e^(i \rho cos...

I was w=kind of confused as of how to go about solving this integeal using complex methods. it is the Integral from 0 to infinity of{dx((x^2)(Sin[xr])}/[((x^2)+(m^2))x*r] where m and r are real variables. I tried to choose a half "donut" in the upper part of the plane with radii or p and R...

For my homework I am told: "Evaluate $z^(1/2)dz around the indicated not necessarily circular closed contour C = C1+C2. (C1 is above the x axis, C2 below, both passing counter-clockwise and through the points (3,0) and (-3,0)). Use the branch r>0, -pi/2 < theta < 3*pi/2 for C1, and the branch...

How might one evaluate an integral equation like the following: I = lim k-> 0+ {ClosedContourIntegral around y [1/(z^2 + k^2)]}, where the contour y is a simple, closed, and positively oriented curve that encloses the simple pole at z = i*K? Is it possible to evaluate integrals of this...

is this the right section to post contour integral questions?

Hi, I've typed up my work. Please see the attached pdf. Basically, I am trying to sovle this problem. \int_0^\infty \frac{x^\alpha}{x^2+b^2} \mathrm{d}x for 0 <\alpha < 1. I follow the procedure given in Boas pg 608 (2nd edition)...and everything seems to work. However, when I...

Hi, I'm having a bit of trouble with this question. Use the property |integral over c of f(z)dz|<=ML to show |integral over c of 1/(z^2-i) dz|<=3pi/4 where c is the circle |z|=3 traversed once counterclockwise thanks in advance for any tips.

I'm trying to solve this contour integral shown on the attached file, I know usually that they involve curved lines. I know that this is trivial but I need some help with the problem. Please take a look.