What is Complex integral: Definition and 125 Discussions
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as
W
=
F
⋅
s
{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }
, have natural continuous analogues in terms of line integrals, in this case
W
=
∫
L
F
(
s
)
⋅
d
s
{\displaystyle \textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }
, which computes the work done on an object moving through an electric or gravitational field F along a path
I’ve attached my attempt. I’ve tried to use triangle inequality formula to attempt, but it seems I got the value which is larger than 1. Which step am I wrong? Also, it seems I cannot neglect the minus sign in front of e^(N+1/2)*2pi. How can I deal with that?
Hi,
I have some question about evaluating a cosine function from ##-\infty## to ##\infty##.
I saw for a cosine function evaluate from ##-\infty## to ##\infty## I can change the limits from 0 to ##\infty##. I have a idea why, but I can't convince myself, furthermore, is it always the case no...
I was trying to calculate an integral of form:
$$\int_{-\infty}^\infty dx \frac{e^{iax}}{x^2}$$
using contour integration, with ##a>0## above. So I would calculate a contour integral with contour being a semicircle that goes along the real axis, closing it in positive direction in the upper...
I have a complicated function to integrate from -\infty to \infty .
I = \int_{-\infty}^{\infty}\frac{(2k^2 - \Omega^2)(I_0^2(\Omega) + I_2(\Omega)^2) - \Omega^2 I_0(\Omega) I_2(\Omega)}{\sqrt{k^2 - \Omega^2}} \Omega d\Omega Where I0I0 and I2I2 are functions containing Hankel functions as...
Homework Statement
The following is a problem from "Applied Complex Variables for Scientists and Engineers"
It states:
The following integral occurs in the quantum theory of collisions:
$$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$
where p is real. Show that
$$I=\begin{cases}0 &...
Homework Statement
##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks!
Homework Equations
Cauchy-Goursat's Theorem
The Attempt at a Solution
Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...
Hey,
I tried to construct the derivation of the integral C with respect to Y:
$$ \frac{\partial C}{\partial Y} = ? $$
$$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$
with
$$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...
The problem
I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$
Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##.
The attempt
I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...
Homework Statement
I have the following integral I wish to solve (preferably analytically):
$$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$
where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...
Homework Statement
I have an integral
$$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$
where j and c are complex, which I'd like to write in terms of ## \text{erf}##
I'd like to know what would happen to the integral limits as I make the change of variables ##t = jp - c##.
1) As ##p## tends...
Hello! If I have a real integral between ##-\infty## and ##+\infty## and the function to be integrated is holomorphic in the whole complex plane except for a finite number of points on the real line does it matter how I make the path around the poles on the real line? I.e. if I integrate on the...
Hello! I found a proof in my physics books and at a step it says that: ##\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx \sim_{t \to \infty} e^{-imt}##. Any advice on how to prove this?
I am trying to do the following integral:
$$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)}.$$
Wolfram alpha - http://www.wolframalpha.com/input/?i=integrate+(cos(x))^(1/2)+dx+from+x=pi+to+3pi gives me
$$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)} = 4f E(2) = 2.39628f + 2.39628if,$$
where E is the...
I have this problem with a complex integral and I'm having a lot of difficulty solving it:
Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$
Where a > 0, k...
Homework Statement
I'm having some trouble evaluating the integral
$$\int^\infty_{-\infty} \frac{\sqrt{2a}}{\sqrt{\pi}}e^{-2ax^2}e^{-ikx}dx$$
Where a and k are positive constants
Homework Equations
I've been given the following integral results which may be of help
$$\int^\infty_{-\infty}...
Hey everyone,
I am trying to evaluate the following integral: \int z(z+1)cosh(1/z) dz with a C of |z| = 1. Can someone please guide me with how to start? I have tried to parametrise the integral in terms of t so that z(t) = e^it however the algebra doesn't seem to work...
Hi,
I have a difficult time trying to perform the following integral,
$$ j({T}, \Omega)=\int_0^{ T} d\tau \frac{\tau^2\exp(-i\Omega\tau)}{(\tau-i\epsilon)^2(\tau+i\epsilon)^2} $$
The problem is that the poles ##\pm i\epsilon## when taking the limit ##\epsilon\rightarrow 0## are located at...
An old exam question is: Evaluate $ \oint \frac{e^{iz}}{z^3}dz $ where the contour is a square of sides a, centered at 0. This has a simple pole of order 3 at z = 0
Perhaps using residues, $ Res(f,0) = \frac{1}{2!}\lim_{{z}\to{0}}\d{^2{}}{{z}^2}z^2 \frac{e^{iz}}{z^3} =...
An old exam has: Evaluate $ \oint\frac{dz}{z(2z+1)} $, where the contour is a unit circle
This look good for the residue theorem, it has 2 simple poles at 0, $-\frac{1}{2}$
$ Res(f, 0)= \lim_{{z}\to{0}}z\frac{1}{z(2z+1)}=1$
$ Res(f, -\frac{1}{2})=...
This relates to z-transform causality, but I'll try to phrase it as a complex analysis question. Suppose I have a function ##X(z)## whose poles are all inside the unit circle, and which has the property
\lim_{|z|\to\infty} \frac{X(z)}{z} = 0
Is that sufficient to guarantee that
\frac{1}{2\pi...
Homework Statement
Describe all the singularities of the function ##g(z)=\frac{z}{1-\cos{z}}## inside ##C## and calculate the integral
## \int_C \frac{z}{1-\cos{z}}dz, ##
where ##C=\{z:|z|=1\}## and positively oriented.
Homework Equations
[/B]
Residue theorem: Let C be a simple closed...
How can I solve the integral below?
## \int_{-\infty}^{\infty} \sqrt{k^2+m^2} e^{izk} dk ##
I thought about contour integration but, as you can see, it doesn't satisfy Jordan's lemma. Also no substitution comes to my mind!
Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{b}^{a} f(t)dt \ \bigg| \leq \int_{b}^{a} \bigg|f(t)\bigg| dt,$$ where the first integral is a complex integral, and the second integral is a definite real integral...
Hello,
I am trying to understand how to get the residue as given by wolfram :
http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2
The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...
I'm solving two different definite integrals of functions
\frac{sin(z)}{z} and \frac{cos(z)}{e^z+e^{-z}}
with complex analysis and the residue theorem, and in the solutions they replace both
sin(z) and cos(z) with e^{iz}
why is this possible?
integrate e^z/(1-cosz) dz over circle of radius, say 2
i can't seem to recall how it is done.
singularity at z=0
2*pi*i * res (at z=0) would be the solution
any shortcut to find this residue?
Homework Statement .
Let ##\gamma_r:[0,\pi] \to \mathbb C## be given by ##\gamma_r(t)=re^{it}##. Prove that ##\lim_{r \to \infty} \int_{\gamma_r} \dfrac{e^{iz}}{z}dz=0##.
The attempt at a solution.
The only thing I could do was:
## \int_{\gamma_r} \dfrac{e^{iz}}{z}dz=\int_0^{\pi}...
\int(z^2+1)^2dz
Evaluate this over the cycloidx=a(\theta-sin\theta) and y=a(1-cos\theta) for \theta =0 to \theta = 2\pi
Am I on the right track, or do I need to approach this a different way?
for z^2 we have (x+iy)^2, so x^2-y^2 + i2xy for the real part...
[b]1. Homework Statement
Evaluate the following real integral using complex integrals:
\int_0^\infty \frac{cos(2x)}{x^2+4}dx
Homework Equations
Cauchy's Residue Theorem for simple pole at a: Res(f;a)=\displaystyle\lim_{z\rightarrow a} (z-a)f(z)The Attempt at a Solution
Since the function...
Homework Statement
I have an integral of the form:
∫0∞exp(ax+ibx)/x dx
What is the general method for solving an integral of this kind.
Homework Equations
Maybe residual calculus?
The Attempt at a Solution
hello everybody
I'd like to understand what mean the result of a complex integral. For example, integrate f(z) = z² from 0 to 2+i results 2/3 + 11/3 i. But, what is this? What 2/3 + 11/3 i represents geometrically? Is it possivel view this result?
Thx!
I need help to solve this problem from Complex variables, Arthur A. Hauser, Ch. 5. pag. 122. Problem 5.42
show that ∫ csc(z)dz/z = 0
where C is the unit circle around the origin.
Solve it without using The Cauchy Integral Formula...
Homework Statement
We need to calculate this complex integral as line integral:
Homework Equations
The Attempt at a Solution
This is correct, I guess:
But not sure about this part:
Are dx, dy, x, y chages correct or there is other method to use?
Hi
I have been using a textbook which shows that ∫cos z/z^2 around the circle |z|=1 is zero by doing a Laurent expansion and finding the residue is zero.
I was under the impression that only analytic functions have a integral of zero around a closed surface. ( the Cauchy-Goursat Theorem )...
Though it is not homework I posted this here, hopefully it'll get more action. Thanks
given \int_{\Lambda} \frac{e^{i w}}{w^2 + 1} \mathrm{d}w where ##w \in \mathbb{C}## with ##Im(w) \geqslant 0## and ##|w| = \xi##
want to evaluate the behavior of the Integral as ##\xi \rightarrow \infty##...
Homework Statement
A semi-infinite bar (0 < x < 1) with unit thermal conductivity is fully insulated
at x = 0, and is constantly heated at x = 1 over such a narrow interval that the
heating may be represented by a delta function:
\frac{\partial U}{\partial t}=\frac{\partial^2 U}{\partial...
Hi,
i'm studing the divergent/convergent behavior of some feynman diagrams that emerge from the study of luttinger liquid. One of this diagrams has a loop inside it and loop-integrals has the following form:
\int_{-\Lambda}^{+\Lambda}dQ\int...
greetings , we have the following integral :
I(x)=\lim_{T\rightarrow \infty}\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\sin(n\pi s)\frac{x^{s}}{s}ds
n is an integer . and \gamma >1
if x>1 we can close the contour to the left . namely, consider the contour :
C_{a}=C_{1}\cup C_{2}\cup...
Homework Statement
Use Cauchy's Integral Theorem to evaluate the following integral
##\int_0^{\infty} \frac{x^2+1}{(x^2+9)^2} dx##
Homework Equations
Res ##f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}##
The Attempt at a Solution
I determine the roots of...
Homework Statement
z is a complex variable.
What is antiderivative of \frac{e^{-iz}}{z^2+(\mu r)^2}?
Homework Equations
The Attempt at a Solution
To caluculate the Fourier transform encounterd in reading quantum phsycis i have to calcualte this integral. I have little...
1. let C be the circle |z| = 2 traveled once in the positive sense. Computer the following integrals...
a.∫c zez/(2z-3) dz
Homework Equations
I am confused as to a step in my solution, but i believe a relevant equation is if i am integrating over a circle and the function is analytic...
I have been working in complex functions and this is a new animal I came across.
Let γ be a piecewise smooth curve from -1 to 1, and let
A=∫γa(x2-y2) + 2bxy dz
B=∫γ2axy - b(x2-y2) dz
Prove A + Bi = (2/3)(a-bi)
In the past anything like this defined γ and I would find a parametric...
Is my solution to the following problem correct?
Evaluate $$ \int_\gamma \frac{z^3}{z-3} dz$$ where $\gamma$ is the circle of radius 4 centered at the origin.
Solution
Form the cauchy integral formula we have that:
$$ f(3)=\frac{1}{2\pi i} \int_\gamma \frac{z^3}{z-3}dz$$ and so $$...
Homework Statement
Consider the integral of the function (1) around a large circle of radius R>>b which avoids the singularities of (e^{z}+1)^{-1}. Use this result to determine the sum (2) and (3).
Homework Equations
(1) - f(z) = \frac{1}{(z^2-b^2)(e^z+1)}
(2) -...
Is there a problem with the following evaluation?\displaystyle \int e^{-ix^{2}} \ dx = \frac{1}{\sqrt{i}} \int e^{-u^{2}} \ du = \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf}(u) + C = \left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf} (\sqrt{i}x) + C So...
$\displaystyle \int_{0}^{\infty} e^{-ix^{2}} \ dx = \lim_{R \to \infty} \int_{0}^{R} e^{-ix^{2}} \ dx $
I think I'm perfectly justified in treating $i$ like a constant since we're integrating with respect to a real variable.$ \displaystyle = \frac{1}{\sqrt{i}} \ \lim_{R \to \infty}...