# What is Complex integration: Definition and 77 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.

View More On Wikipedia.org
1. ### Complex Integration Along Given Path

From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...

40. ### MHB Evaluating Complex Integration I_c |z^2|

How can I evaluate I_c |z^2|,where I is the integral and c is the square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) traversed anti-clockwise...?
41. ### MHB Complex Integration: Solving $\int_0^1\frac{2t+i}{t^2+it^2+1}dt$

$\displaystyle\int_0^1\frac{2t+i}{t^2+it^2+1}dt = \int_0^1\frac{2t^3+3t+i-it^2}{t^4+3t^2+1}dt =\int_0^1\frac{2t^3+3t}{t^4+3t^2+1}dt+i\int_0^1 \frac{1-t^2}{t^4+3t^2+1}dt$ I tried multiplying through by the conjugate but that didn't seem fruitful and left me with the above expression. Is there a...
42. ### Integrate $\Sigma \frac{1}{n!}\int z^{n}e^{1/z}dz$

Homework Statement Integrate $\Sigma \frac{1}{n!}\int z^{n}e^{1/z}dz$ Homework Equations The Attempt at a Solution Wrote out the first couple of terms, with $\frac{1}{z}=w$, making the integral $\Sigma \frac{1}{n!} (-w^{2-n}e^{w}+(2-n)(w^{1-n}e^{w}+(1-n)(w^{1-n}e^w)-(1-n)^2(w^(-n)e^w)...)$...
43. ### Complex integration problem using residues .

Homework Statement I =\int \frac{cosx}{x^{2}-2x+2}dx the integral runs from -inf to inf evaluate the integral using the calculus of residues. Homework Equations shown in my attempt The Attempt at a Solution Re \oint\frac{e^{iz}}{z^{2}-2z+2} with singularities at...
44. ### Complex integration via parametrization

Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction using parametrization. Show all work...
45. ### Complex Integrals: Sketching Paths & Computing Integrals

Homework Statement Sketch the C1 paths a: [0; 1] -> C, t -> t + it2 and b: [0; 1 + i]. Then compute the following integrals. ∫Re(z)dz over a ∫Re(z)dz over b Homework Equations The Attempt at a Solution Sketching a seems ok, y-axis is Imaginary, x-axis is Real, and the...
46. ### Complex integration over a square contour (part b)

Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction using parametrization. Show all work...
47. ### Complex integration over a square contour

Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction. You must use two different methods to solve...
48. ### Complex integration of real-valued trig function

Homework Statement Integrate: \int \frac{1}{(3+2cos(θ))} dθ evaluated from zero to pi. Homework Equations I can't think of any. All of the integration formulas in the text rely on the existence of a singularity somewhere in the complex plane. This thing is analytic everywhere...
49. ### Complex Integration: Find g(2)=8πi, g(z) when |z|>3

Homework Statement Let C be the circle |z|=3, described in the positive sense. Show that if g(z)= \int_C \frac{2s^2-s-2}{s-z} ds such that |z| does not equal 3, then g(2)=8 \pi i . What is the value of g(z) when when |z|>3? Homework Equations Cauchy Integral Formula Deformation of...
50. ### Complex Analysis Complex Integration Question

Its question 1(g) in the picture. My work is shown there as well. This has to do with independence of path of a contour. Reason I am suspicious is that first there is a different answer online and second it says "principal branch" which I have not understood. Does that mean a straight line for...