What is Contour integral: Definition and 122 Discussions
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a method of complex analysis.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.Contour integration methods include:
direct integration of a complex-valued function along a curve in the complex plane (a contour);
application of the Cauchy integral formula; and
application of the residue theorem.One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.
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...
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...
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.