I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof.
He is evaluating an integral along a contour which makes up the imaginary axis. He has the following:
$$ -i\int_{-iR}^{iR}...
Homework Statement
The following is a problem from "Applied Complex Variables for Scientists and Engineers"
It states:
The following integral occurs in the quantum theory of collisions:
$$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$
where p is real. Show that
$$I=\begin{cases}0 &...
Homework Statement
Show that
$$\int_C e^zdz = 0$$
Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i.
Homework Equations
$$z = x + iy$$
The Attempt at a Solution
I know that if a function is analytic/holomorphic on a domain and the contour lies...
I'm trying to make sense of the derivation of the Klein-Gordon propagator in Peskin and Schroeder using contour integration. It seems the main step in the argument is that ## e^{-i p^0(x^0-y^0)} ## tends to zero (in the ##r\rightarrow\infty## limit) along a semicircular contour below (resp...
As part of the work I'm doing, I'm evaluating a contour integral:
$$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$
along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...
I am trying to numerically integrate the following complicated expression:
$$\frac{-2\exp{\frac{-4m(u^2+v^2+vw+w^2+u(v+w))}{\hbar^2\beta}-\frac{\hbar\beta(16\epsilon^2-8m\epsilon(-uv+uw+vw+w^2-4(u+w)\xi...
Homework Statement
∫-11 dx/(√(1-x2)(a+bx)) a>b>0
Homework Equations
f(z0)=(1/2πi)∫f(z)dz/(z-z0)
The Attempt at a Solution
I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed...
As I understand it, in order to compute a contour integral one can deform the contour of integration, such that it doesn't pass through any poles of the integrand, and the result is identical to that found using the original contour of integration considered. However, I have seen applications...
Hello.
I have a difficulty to understand the branch cut introduced to solve this integral.
\int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]}
here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
Homework Statement
Find the solution of the following integral
Homework Equations
The Attempt at a Solution
I applied the above relations getting that
Then I was able to factor the function inside the integral getting that
From here I should be able to get a solution by simply finding the...
I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##.
Using their notation, consider a contour ##\mathcal{C}## with...
I'm having a tough time with this integral:
$$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$
where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck.
What contour should I...
Homework Statement
State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals.
\begin{eqnarray*}
(i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\
(ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...
I am trying to teach myself complex analysis . There seems to be multiple ways of achieving the same thing and I am unsure on which approach to take, I am also struggling to visualise the problem...Would someone show me step by step how to solve for example...
When one uses a contour integral to evaluate an integral on the real line, for example \int_{-\infty}^{\infty}\frac{dz}{(1+x)^{3}} is it correct to say that one analytically continues the integrand onto the complex plane and integrate it over a closed contour ##C## (over a semi-circle of radius...
Homework Statement
I'm using the text-book Student Friendly Quantum Field Theory [Robert D. Klauber] and I don't know how to get the function (k0) (3-134) present in the figure below.
http://[url=http://postimg.org/image/5j62c27nz/][PLAIN]http://s11.postimg.org/5j62c27nz/help.jpg [Broken]...
Hi everyone,
in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like:
I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)}
where a,b,c are...
I want to solve this contour integral
$$J(\omega)= \frac{1}{2\pi}\frac{\gamma_i\lambda^2}{(\lambda^2+(\omega_i-\Delta-\omega)^2)} $$
$$N(\omega)=\frac{1}{e^{\frac{-\omega t}{T}}-1}$$
$$\int_0^\infty J(\omega)N(\omega)$$
there are three poles I don't know how I get rid of pole on zero (pole...