Complex integral Definition and 125 Threads

How can I find the primitive of $\int_{\gamma}ze^{z^2}dz$ from $i$ to $2-i$?

$\gamma(t)=1+it+t^2, \ 0\leq t\leq 1$ $\displaystyle\int_0^1 (1+it+t^2)(i+2t)dt=\int_0^1(2t^3+t)dt+i\int_0^1(1+3t^2)dt = 1 + 2i$ I was told that was wrong. What is wrong with it?

$\gamma$ is the unit circle oriented counterclockwise. $\displaystyle\int_{\gamma}\dfrac{e^z}{z}dz$ $\gamma(t) = e^{it}$ for $0\leq t\leq 2\pi$ $\gamma'(t) = ie^{it}$ Using $\int_{\gamma} f(\gamma(t))\gamma'(t)dt$, I obtain $\displaystyle i\int_0^{2\pi}e^{e^{it}}dt$ Not quite sure how to...

$$ \int_{\gamma}ze^{z^2}dz $$ $\gamma(t) = 2t + i -2ti$, for $0\leq t\leq 1$. $ \int_{\gamma} f(\gamma(t))\gamma'(t)dt $ But $ \int_{\gamma}ze^{z^2}dz \Rightarrow \frac{1}{2}\int e^wdw $ So then I would be solving $$ \frac{1}{2}\int\exp(4t-1+4ti-8t^2i)(4+4i-16ti)dw $$ Correct? And how...

Hi all! I have to perform this complex integration over three curves, the first one is \( C=\{ z \in \mathbb{C} : |z|=2 \} \) and the function to integrate is $$ f(z)=\frac{z^2}{e^{2z}+1}$$ If I do the usual change of variables \(z=2e^{i\theta} \) and integrate from \( \theta = 0 \rightarrow...

I read in some text or book that the integral \int_{-\infty}^{\infty} \cos(x^2) + \sin(x^2) \, \mathrm{d}x = \sqrt{2\pi} I was wondering how this is possible. I read on this site that one such possible way was to start by integrating e^{-i x^2} = \cos(x^2) - i \cos(x^2) My...

Homework Statement The problem is given in the following image. http://img46.imageshack.us/img46/2972/lskfjsf.png Homework Equations ∫h(r)*dr = ∫h[r(u)]*r'(u)du The Attempt at a Solution I was able to figure out plugging in (t^2+3) at every x and sin(1/2πt) at every y. I then set...

The problem is as follows: lim_{n\rightarrow\infty} \int^{1}_{0}√(1+n^{2}x^{2n}) My issue is that I'm unsure as to where to start. We just went over DE's in my calculus class, so I assume that they are relevant, but we never attempted integrals that weren't explicitly defined. Any help...

Homework Statement \int^{√x}_{1}\frac{t^{3}+t-1}{t^{2}(t^{2}+1)} dt Homework Equations The Attempt at a Solution So I first start by expanding the bottom part of the fraction to t^{4}+t^{2}, and letting u equal to that. Then du=4t^{3}+2t dt. I move the common multiple of 2 over...

Homework Statement Let C be a loop around \pi/2. Find the value of \frac{1}{2\pi i} \int_C \frac{\sin(z)}{(z-\pi/2)^3} dz.Homework Equations Thm: If f is analytic in its simply connected domain D, and C is a simply closed positively oriented loop that lies in D, and if z lies in the inside of...

Homework Statement Assuming a counterclockwise orientation for the unit circle, calculate ∫ \frac{z+i}{z^3+2z^2} dz |z|=1 Homework Equations f'(a)=\frac{n!}{2i\pi}=∫\frac{f(z)}{{z-a}^(n+1)} ? The Attempt at a Solution I don't understand these types of questions. What does the |z| have to...

Hey, Got stuck studying complex stuff again, I am trying to find out how i can get rid of the isin2bx in my result, here is the question [PLAIN]http://img832.imageshack.us/img832/3199/unledkcv.jpg The integral of e^(-z^2) = 0 as C is a closed curve and e^(-z^2) is analytic So first...

Hey, I've been trying to do this integral without cauchy's theorem (with the theorem i get 6ipi in like 2 steps). I am stuck at this point, I have found afew ways to do the integral I am stuck on but they all involve multiple variable changes and I was wondering if there is a simple way to do...

Homework Statement I need to establish the estimate and inequality |\int_{C} e^{iz^{2}}dz| \leq\frac{\pi(1-e^{-R^{2}}}{4R} < \frac {\pi}{4R} where C={z(t)=Re^{it},t\in[0,\fraq{\pi}{4}] Homework Equations The Attempt at a Solution I thought perhaps I could use the ML equality but...

I've come across an complex integral that I just can't seem to figure out. Basically I need to integrate f(z) = 1/(z^6-1) around the circle |z+1|=1. At first glance the radius of the circle must be zero in order to satisfy |z+1|=1 and therefore the function, f(z), is analytic in D so...

Hi, could you please help me solving this integral: \oint \frac{e^{-(a+b)+az+\frac{b}{z}}}{z(z-1)}dz over the unit circle, where a, b are two positive constants (it is not a homework) thanks a lot in advance

Homework Statement Evaluate the integral along the path given: integral(along a(t) of (b^2-1)/(b^2+1) db ) where a(t)=2*e^(it) , 0 <= t <= 2*pi Homework Equations none The Attempt at a Solution I am thinking of using the Residue Theorem. I think there are poles at -i...

I'm struggling to work out how to integrate the following \int_0^t(\gamma^{1/\kappa}-i\zeta{w}(1-t/s)_+^{H-1/2})^{\kappa}ds here (.)_+ denotes the positive part if I did not have the ^(H-1/2) I can do it, alas it does have it! and so it stumps me on how to evaluate this integral. any...

Homework Statement "Evaluate the integral of f(z)=\frac{z^5}{1-z^3} around the circle |z|=2 in the positive sense. Homework Equations Cauchy's residue theorem? The Attempt at a Solution Truthfully, I don't know where to begin. I've done others of these using Cauchy's theorem...

I have attached a pdf of my problem and attempted solution. I seem to be a factor of f'(z) out from the required solution, can anyone see where I've gone wrong?

In our Complex Methods lecture today, our lecturer went through the example of evaluating the integral \int_{-1}^1 \frac{(1-x^2)^{1/2}}{x^2+1}dx and then proceeded to do the whole contour calculation using the complex function \frac{(z^2-1)^{1/2}}{z^2+1}. I'm concerned that the answer will be a...

Homework Statement \oint_{L} \frac{ \mbox{d} z}{ z(z+3) } and L:|z|=4 The Attempt at a Solution what is assumption, is it oriented positive or negative? and Cauchy formula, can it be done like this? \frac{ 1 }{ 3 } \left( \oint_{L} \frac{ \mbox{d} z}{ z } - \oint_{L} \frac{ \mbox{d} z}{ z+3...

Hi everybody while trying to find the pdf of user in CDMA, I got stuck up with an integral, which is given below: \int_0^1\frac{1}{x (\ln x)^{\frac{n-1}{n}}\sqrt{y^2-x^2}}dx where y is a constant and n is an integer. Please help me to solve this integral.

Homework Statement I need to solve: \int_{-\infty}^{\infty}xe^{(a-x)^2}dx Homework Equations The Attempt at a Solution My first intuition would be to rewrite this as: \oint_cze^{(a-z)^2}dz and then use Cauchy's Residue theorem to calculate the integral. There is one singularity at x_o=0...

\int_{|z|=3}^{nothing}\frac{dz}{z^3(z^{10}-2)}\\ f=\frac{1}{z^3(z^{10}-2)}\\ f(\frac{1}{z})=\frac{1}{(\frac{1}{z})^3((\frac{1}{z})^{10}-2)}\frac{z^{13}}{1-2z^{10}}=\\ res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=\frac{1}{z^2}\sum_{n=0}^{\infty}(2z^{10})^n\\ res(f,\infty)=...

\int_{|z|=1}^{nothing } \frac{1}{z}e^{\frac{1}{z}} in this integral there is no upper bound its around |z|=1 there are no poles here only singular significant what to do here when calclating the residium ??

I'm studying the non-linearity effect of power amplifier on multicarrier signal. While modeling the behavior of power amplifier I came across the following integral; I'm not able to figure out how to solve it...

Homework Statement I'm studying the non-linear effect of power amplifier on multicarrier siganl. I have come across an complex integral which is given below, but not able to figure out how to solve it analytically.Homework Equations...

Homework Statement We know sin(z) has zeros at integral multiples of pi. Let f(z) = z2/sin2(z) How do I find the integral of f(z) dz around C1 (C1 is the circle |z| = 1 orientated anti-clockwise) and how do I find the integral of f(z) dz around C2 (C2 is the circle |z - pi| = 1 orientated...

Hi... I have an integral over a contour. The contour is a semicircle with vanishing radius around the origin and situated in the upper half plane. The integrand is \frac{(lnx)^2}{1+x^2}. The integral is supposed be zero. I don't see how. Taking the modulus and letting the radius...

hi, this is really frustrating because I've been working on this one integral for the last 2 1/2 horus and i can't figure out what I did wrong... it's the Integral sin^2(x)/(3-2cos(x)) dx from x=0 to x=2pi I tried the substitution z=e^(itheta) plugging in sin and cos as functions of z...

Homework Statement Let U be open in C, f : U -> C continuous. Prove that \lim_{R\rightarrow 0} \int_0^{2\pi} f(Re^{it}) dt = 2\pi f(0) Homework Equations \lim_{R\rightarrow 0} f(Re^{it}) dt = f(0) Also \int_0^{2\pi} \lim_{R\rightarrow 0} f(Re^{it}) dt =...

Homework Statement Prove the following identity, \frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}\ln|z|^2 = 2\pi \delta^2(z,\bar{z}) where the delta function is defined such that \int dz d\bar{z} \detla^2(z,\bar{z}) = 1 Homework Equations The Attempt at a...

1. Integrate z2/(z4-1) counterclockwise around x2 + 16y2=42. Cauchy's Integral Forumula3. Solution I found the points z=1,-1,i,-i where the function is not defined. Using partial fractions to split them up, and integral them separately. Only points z=1,-1 lies in the contour, so...

Homework Statement By applying Cauchy's theorem to a suitable contour, prove that the integral of cos(ax2) = (pi/8a)1/2 Homework Equations Cauchy's integral formula: http://en.wikipedia.org/wiki/Cauchy's_integral_formula The Attempt at a Solution I'm not sure where to go...

I feel ashamed asking this, but how do you take the integral of a complex or pure imaginary function? My sheer guess is that you take the real parts of the function and integrate them seperately, then take the imaginary part and integrate it, but I don't quite know how to do that last part...

We all know that \frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x'). i am working a problem which appears to depend on the statement \int e^{z^*(z-w)}dz^*\propto\delta(z-w) Does anyone know if this is valid? \delta(z-w) is defined in the usual way so that...

Homework Statement \int\intD\left|x\right|dA D= X2+y2<=a2 where a>0 Homework Equations \int\stackrel{\Pi/2}{0}\int\stackrel{a}{0} r cos \Theta r dr d\Theta I hope that's clear... I evaluate this to \frac{a^3}{3} sin \Theta sin \Pi/2 = 1 so I get \frac{a^3}{3} the...

Hi, I have a question about infinite limit of complex integral. Problem: Consider the function ln(1+\frac{a}{z^{n}}) for n\ge1 and a semicircle, C , defined by z=Re^{j\gamma} for \gamma\in[\frac{-\pi}{2},\frac{\pi}{2}]. Then. If C is followed clockwise, I_R = \lim_{R\rightarrow \infty}\int_C\...

Homework Statement Assume that f(z) is analytic and that f'(z) is continuous in a region that contains a closed curve \gamma. Show that \int_\gamma \overline{f(z)} f'(z) dz is purely imaginary.Homework Equations If f(z) is holomorphic on the region containing a closed curve \gamma or if...

Hi,i would be gratefull if anyone could help me with this problem. \frac{2}{\pi}\int \frac{cos(ux)}{\sqrt{x}} dx x goes from zero to infinity. thanks in advance.

Homework Statement Solve: \oint x^{99}y^{100}dx + x^{100}y^{99}dy Assuming that it satisfies the conditions for Green's theroem, and: y = \sin{t} + 2, x = \cos{t}, 0 \leq t \leq 2\pi Homework Equations Green's theorem. The Attempt at a Solution \frac{\partial P}{\partial y} =...

Ok, I am trying to integrate the following function, and not getting very far: it's s=integral between 0 and 2pi of cos^-1(arctan((2*pi/b)*a*cos(2*pi*x/b)))dx)^-1 where a and b are known variables. What I would like to know, is can this integral be evaluated directly, or must I use the trapezium...

I'm stuck trying to prove a step inside a lemma from Serre; given is 0<a<b 0<x To prove: |\int_{a}^{b}e^{-tx}e^{-tiy}dt|\leq\int_{a}^{b}e^{-tx}dt I've tried using Cauchy-Schwartz for integrals, but this step is too big (using Mathematica, it lead to a contradiction); something...

How to evaluate the following integral using residue theorem: \int_1^2 (x+1) \sqrt[6]{\frac{x-1}{2-x}}dx (The answer is \frac{31}{36}\pi ) Thanks for any help

I am going to provide my answer to a complex integral and i was just seeking a few pointers as to weather i was on the right track or was there something i completely forgot...happens quite a bit...lol \oint exp(z+(1/z)) around the path \left |z|\right=1 now i converted that to a Laurent...

Hi, problrm with complex integral.Consider the integral \int_{0}^{\infty}\frac{(x^a-1)}{x^3-1}\,dx use the branch 0<phi<3pi/2 and an idented contour at z=0 and z=1.(circular contour in the upper half plane) a)show that the integral can be written in terms of the integral...

I've been given the problem of evaluating the integral \int(exp^z)/Sinh(z) dz Over the region C which is the circle |z|=4 I can't figure out how to do this,I tried parameterizing with z(t)=4e^i\theta but the integrand just seems far too complicated. Any suggestions? (Apologies for...

Hi, I have a problem with the following complex integral. Integral from 0 to+infinty sin²(a.lnz)/(x - 1)² = pi.a coth(2.pi.a) - 1/2 a>0 I tried different contours and methods,but without result. Can you help me to find out the complex contour integration. Thanks

This integral came up while trying to find the potential of a uniformly charged rectangle. \int \log(\sqrt{a^2+x^2} + b) dx Integrator gives a pretty long expression involving inverse tangents so I'm not sure where to begin at all. I tried integrating by parts once, taking u to be the...