Complex integration Definition and 77 Threads

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...

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...

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...

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...

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}...

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...

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...

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!

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...

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!

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...

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...

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

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...

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.

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"...

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...

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: =...

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.)...

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...

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...

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...

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...

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...

\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?

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...

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...