Complex integration Definition and 77 Threads

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.

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  1. D

    Integral of g(z) around |z-i|=2: Cauchy Integral Formula

    Homework Statement Find the value of the integral of g(z) around the circle |z-i|=2 in the positive sense when g(z)=\frac{1}{z^2+4}. Answer: pi\2 Homework Equations Cauchy Integral Formula f(z_0)=\frac{1}{2\Pi i}\int \frac{f(z) dz}{z-z_0} The Attempt at a Solution I tried...
  2. C

    Complex integration: which path should i choose?

    Hi there, this is a problem I'm having since last year, and should be "very easy", and i guess it is. 3 long years ago i took the complex analysis course, and I've used so little of it until now, that i almost forgot everything of it. The problem is this, I have the following integration...
  3. U

    Complex Integration: Questions About Non-Zero Derivatives, Line Integrals & More

    Homework Statement I was told to post this kind of question on the homework help section by one of the mentors,even though I'm not sure it is appropiate. Anyway,I'm doing complex integration now,so I need to get some important concepts cleared. I'll post my doubts in points...
  4. M

    Complex integration, branch cuts

    Hi guys, I need to show that: \int_{0}^{\infty } \frac{x^{a}}{(x+1)^2} \dx = \frac{\pi a}{\sin(\pi a)} , where -1<a<1. The problem is, that although a hint is given,the path of integrating it, I have difficulty what they really mean with "cut line, branch points, multivalued functions" etc...
  5. M

    Thermal green function and complex integration

    green function and complex integration Homework Statement By reading this paper http://arxiv.org/pdf/hep-ph/0610391v4 I cannot proof the following relation on page 9 equation (23) , by a suitable choice of a contour in the complex omega plane: \int\frac{d^3 p}{(2\pi)^2}...
  6. G01

    Complex Integration / Residue Problems

    Homework Statement Hi everyone. I'm currently taking a graduate math physics course and complex integrals are beating the crap out of me. Some of my questions may be relatively basic. Forgive me, I'm trying to teach myself and am regretting not taking a course on complex analysis as an...
  7. O

    Possible Values of Complex Integration for Contours in D and from 0 to 1

    Homework Statement 1)a Let D = C\{-i,i} and let γ be a closed contour in D. Find all the possible values of : (∫(1/(1+z²))dz around γ) b) If σ is a contour from 0 to 1, determine all possible values of: (∫(1/(1+z²))dz ( around σ) Homework Equations The Attempt at a...
  8. L

    Drawing complex integration contours

    Hi! I'm writing some lecture notes and I need to draw some complex integration contours. Is there any software to make this task easier? Thanks for your attention!
  9. K

    Need help in complex integration

    Can somebody help me with this. Homework Statement \int_{-\infty}^{\infty} \, \frac{\sin{x}}{x} \, dx Could u pls advice me with the procedure to follow not only the answer? The Attempt at a Solution 1. Use complex numbers as there is a pole of order=0 at x=0...
  10. K

    Complex Integration Homework: Evaluate Intergral Gamma (y)dz

    Homework Statement Evaluate the following intergral: Homework Equations Intergral from gamma of (y)dz, where gamma is the union of the line segments joining 0 to i and then i to i+2 The Attempt at a Solution I have no idea how to do this!
  11. N

    A physical meaning to complex integration

    When it comes to integration of some function f(t) where t is real, I would just treat everything as a constant and integrate it. Even with complex functions f(t) = u(t) + iv(t) why can't I just treat i as a constant and just integrate? Here is an example: \int_{0}^{\pi/2}e^{t+it}dt...
  12. L

    Complex Integration: Evaluating Line Integrals on a Circle

    Evaluate the Line Integral (assume counterclockwise orientation) \oint _{|z| = 2 } z^n \bar{z}^m dz for all m, n \in Z I have no freaken clue about how to even attempt this...
  13. L

    Solving Complex Integrals: Can I Treat i as Any Other Constant?

    How do I solve an integral of the type \int f(v) e^{iavx} dv ? Can I just treat i as any other constant?
  14. S

    Complex Integration by Parts Help. Please

    Hi Can anyone help me with this integration. I will use I to symbol the integer sign Limits between 1 and 0 Ie^-x.Sinxdx I understand I have to integrate by parts and get the following answer, ignoring the limits for the time being. Ie^-x.Sinx = e^-x.Cosx I -Cosx.e^-x dx then i...
  15. G

    Maple Solving Complex Integration with Maple and Mathematica

    I want to solve this complex integration: -\frac{1}{\sqrt{2\pi}} \int(\frac{1}{\sqrt{s+w^2}} \exp [2d\sqrt{s+w^2}+iwL]dw I have try it to solve with Maple and with Mathematica, but they cannot solve it.
  16. C

    Parametrization in Complex Integration

    I have a complex analysis final exam on Wednesday, and I am studying the section on complex integration. I am having trouble seeing how to parametrize an equation. "\Gamma is the line segment from -4 to i" In the homework solutions our TA said, "Parametrize \Gamma by z = -4 +t(i+4), 0<t<1"...
  17. S

    Complex Integration Homework: Compute Integrals w/ Principle Value of z^i

    Homework Statement Compute the following integrals using the principle value of z^{i} a. \int z^{i} dz where \gamma_{1}(t)=e^{it} and \frac{-\pi}{2}\leq t \leq \frac{\pi}{2} b. \int z^{i} dz where \gamma_{1}(t)=e^{it} and \frac{\pi}{2}\leq t \leq \frac{3\pi}{2} Homework...
  18. F

    Validating Complex Integration Using Polar Form and the Unit Circle

    I've been asked to find the value of: \int_{-1}^{1}z^{\frac{1}{2}}dz Here is how I did it: I want to change to polar form. To do that I'll use the fact that: z=e^{i\theta} For a unit circle radius 1. dz=ie^{i\theta}d\theta I replace z and dz: =...
  19. B

    Integrating Complex Variables - Types & Solutions

    I'm a little confused about integration with complex variables. Are there two types of integrals?: 1. Integrands with complex numbers but the variable of integration is real. 2. Intregands with complex numbers and the variable of integration is also complex. But can't (2.) be made into (1.)...
  20. J

    How can the arc length of a cycloid be calculated using parametric equations?

    Integrate \int_C \sec^2 z dz \ \mbox{any path from } \ \frac{\pi}{4} \mbox{ to } \frac{\pi\iota}{4} \\ \sec^2 z = \frac{1}{\cos^2 z} \ \mbox{ which is equal to } \ \frac{1}{2(1+\cos 2z)} \\ Therefore \frac{2}{1+\cos 2z} = \frac{2}{1 + \cos2x\cosh2y -\iota \sin2x\sinh2y}\\ How do you split...
  21. J

    Is Cosh(4z) Integrable Along Any Path Between -πi/8 and πi/8?

    Integrate cosh(4z) w.r.t., z, for any path from \frac{-\pi i}{8} \inbox{to } \frac{\pi i}{8} . If the function is analytic, i.e., obeys Cauchy - Riemann equations we can integrate as in standard calculus. \frac{{\partial \cosh(4(x+iy)}}{{\partial x}} = a\sinh(4x + 4iy) \\ \frac{{\partial...
  22. malawi_glenn

    How to Solve a Complex Integration Problem with a Unit Circle?

    Homework Statement evaluate \int_{c} | z - 1 | |dz| where c is the positive oriented unit circle.Homework Equations The Attempt at a Solution | z - 1 | = \left[ ( z-1)( \overline{z} - 1 ) \right] ^{1/2} = \left[ |z|^{2} - z - \overline{z} +1 \right] ^{1/2} c : z(t) = e^{it} ; 0...
  23. Reshma

    How Do I Apply Cauchy's Theorems to Complex Integrals with Boundary Conditions?

    I am trying to teach myself some complex analysis. I am using Complex Numbers by Churchill & Brown as my reference. I have reached the integration section and I am encountering certain difficulties. For e.g. I have this problem: \oint \frac{dz}{z^2 - z -2}, |z|\leq 3 I can split up the...
  24. S

    Calculating Complex Integration: f(z)=e^z/(1+e^(4z))

    I have the function f(z)=\frac{e^z}{1+e^{4z}}. and the loop \gamma_{r_1,r_2}=I_{r_1,r_2}+II_{r_2}+III_{r_1,r_2}+IV_{r_1},\quad r_1,r_2>0, which bounds the domain A_{r_1,r_2}=\{z\in\mathbb{C}\mid -r_1<\Re(z)< r_2\wedge0<\Im(z)<\pi\}. Now I have to show that...
  25. Y

    Complex Integral: Existence of Formula for 𝑒^(-𝑎𝑥2+𝑏𝑥)

    \int_0^\infty e^{-ax^2+bx} dx, a and b may be complex. Does exist any formula for this integral? Or for \int_{-\infty}^\infty e^{-ax^2+bx} dx?
  26. W

    Find dx in a Complex Integration Problem: Anti-Derivative of Cos(2x)

    how would i find the anti-derv. of (cos(2x)). I am really confused with sub. ok i'll use u-du u = 2x du = 2 dx dx = 1/2 du k so... cos(u)*dx since the dx is there, you plug in the dx that i found right? so... sin(2x)1/2 ok so that's a simple integration problem, can someone...
  27. E

    Solving a Complex Integration Problem: Finding the Work to Empty a Tank

    I apologize that I don't know how to make the math equations. Alright it's going to be kind of complicated trying to describe this in words, but I'll do my best. There is a tank shaped like a right cylinder on it's side. The length of the tank (or height of the cylinder) is 6m, and the radius...
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