Questions tagged [integration]

For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

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Best integration method for singular functions in Mathematica

I have a quantity, $f$, as a function of energy, $E$, that is generated in Mathematica (see figure below). The expression for f contains factors of the form 1/(EH-E+i*eta), where a small eta was added ...
flg's user avatar
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Lorentz-invariant phase space integral

Consider the following Lorentz invariant integral associated to a $2\to 2$ scattering: \begin{equation*} I = \int \frac{d^3\mathbf{p_3}}{(2\pi)^3 2E_3} \int \frac{d^3\mathbf{p_4}}{(2\pi)^3 2E_4} \...
Spectree's user avatar
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2 answers
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Some integrals in QED Renormalisation

I am currently leaning the renormalisation of QED and I have met some tricky integral that seems unsolvable. The integrals are shown in Quantum Field Theory and the Standard Model by Schwartz, page ...
quantumology's user avatar
1 vote
1 answer
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How to derive the magnetic field at a distance $x$ (in the same plane) from the centre of a circular loop with current $I$ and radius $R$? [closed]

how to get the expression for magnetic field at any point in a circular loop (plane) with current $I$ and radius $R$, I can't seems to find it anywhere and all I'm getting is the derivation for ...
Farhan's user avatar
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Density of States, Photoemission and Integrating the Number of States

I'm reading Fowler's theory on photoemission. I'm stuck on a part which Fowler helpfully identifies as "obvious". Fowler sets up the free electron model, suggesting that electrons need a ...
Tomi's user avatar
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3 votes
2 answers
326 views

Proving a Grassmann integral identity

How to prove the following identity $$ \begin{align} \int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\...
Faber Bosch's user avatar
2 votes
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Closed form expression of 2D CFT integral

I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is $$ \int d^2z \...
NoName's user avatar
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Integration over the complex plane and the completeness relation of the coherent states [duplicate]

I am studying some of the properties of coherent states using the book "Introductory Quantum Optics" by C. Gerry & L. Knight. (C. Gerry & L. Knight, Chapter 3, Section 5) And when I ...
Uriel Casco D's user avatar
-2 votes
0 answers
52 views

Fermi-Dirac integrals [closed]

I accounted a problem when deriving the Fermi-Dirac integrals shown below Could someone point me out why the lower limit of integtal could subsitutite from $Ec$ to $0$? 20240410 edit I have derived ...
Jack Huang's user avatar
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Integration by parts to switch sign on anti-Hermitian Louivillian

I am self studying non-equilibrium stat mech and a common theme is that I am unable to reproduce the "this follow from integration by parts" parts of derivations. Here is the most recent: In ...
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Integral identity for thermal states

Context: In this paper, the authors establish a relationship between quantum Fisher information and the imaginary part of the dynamic susceptibility of a system. To do so they utilize the identity: $$...
sicklerick's user avatar
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Centre of Mass of Slice of Sphere

What is the position of the centre of mass of a slice $\theta$ wide, taken from a hollow sphere of radius R? Say surface mass density = $\sigma$ It is in the shape of a "watermelon slice". (...
magneticMono_Poal247's user avatar
3 votes
1 answer
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What is the homogeneous charge density (vs inhomogeneous)?

Books/exams often write something like "a homogeneously charged sphere"/"a homogeneously charged rod" - I am unsure what exactly is meant. Take the example of a sphere with radius $...
F L's user avatar
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Integral of Theta function [closed]

I'm trying to compute the following integral, useful to calculate Amplitudes in String theory \begin{equation} \int \frac{d^2z}{\tau_2} \;\partial^2_z \log \vartheta_1\left(z\right) = - \frac{\pi}{\...
Roddy 's user avatar
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Residue theorem application

First I want to provide a little bit of context: I finished my undergrad degree in physics in 2008 and after that I moved into strategic consulting and into the financial world. Right now, at 41 years ...
ateixeira82's user avatar
3 votes
2 answers
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Hooke's law and derivation of work done by spring confusion

For simplicity I'll do this in 1 direction so I can let the sign dictate direction and ditch the vector notation. The force done by a spring can be written as $$F = -k\Delta x , $$however this ...
Bumbling Astronomer's user avatar
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How to know the position of an object when calculating the center of mass, without using integrals? [closed]

For example, if I have a 1/4 piece of pizza, what is the position?
asd20234's user avatar
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1 answer
77 views

Exponential decay of propagator outside lightcone

In Tong's lecture notes (http://www.damtp.cam.ac.uk/user/tong/qft.html) page 38, he calculates the following propagator: $$D(x-y) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{2E_\vec{p}} e^{-ip \cdot (x-y)}....
Stallmp's user avatar
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1 answer
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Electric field at a point created by a charged object (derivation/integration process)

I was hoping someone can help me understand the math behind the electric field (electrostatics). I have gaps in my knowledge about integrals and derivatives (university moves very quickly and it has ...
1899DVX's user avatar
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Movement of a mass on a spring in damped SHM

Suppose a ball is connected to a spring attached to a wall, and they are in space, i.e. assume no gravity. The ball is put into a fluid with Stoke's drag and oscillates backwards and forwards relative ...
Yitian Chen's user avatar
2 votes
1 answer
62 views

Integral Quantity Interpretation

Is there any natural interpretation for the following quantity? $$\int_{\vec{r}(t)} \nabla(\vec{v} \cdot \vec{A})dt \ .$$ Where: $$\vec{v} = \frac{d \vec{r}(t)}{dt} \ ,$$ is the velocity of the path ...
Jbag1212's user avatar
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Why is the differential form of Gauss's Law equivalent to the integral form?

I can understand the Differential form of Gauss's Law ∇⋅𝐄= $\frac{ρ}{ɛ_0}$ as saying that the source of electric field vectors or flow disperse(The divergence of the electric field) is equal to the ...
244529's user avatar
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1 answer
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Problem with Deriving work done by gravitational force and gravitational potential energy from the first principles

Suppose we have a system with Two point masses of mass $M$ and mass $m$. And we want to derive Work done. Lets say M is fixed or $M>>m$. Initially assume mass m is at rest at a distance of $a$ ...
evmorfia's user avatar
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Integral from Feynman & Hibbs Appendix - solvable with Cauchy principal value [migrated]

In Appendix "Some Useful Definite Integrals" of Feynman & Hibbs "Quantum Mechanics and Path Integrals" they use some integral formulas that I'm struggling to derive. The ...
Felipe's user avatar
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1 vote
2 answers
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Sign Problem When Calculating the Potential of an Infinite Line Charge [duplicate]

Similar questions have been asked before in the forum regarding the sign problems when integrating the electric field, but they do not fully clarify the problem for me, so I just want to ask for some ...
ZZZ's user avatar
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1 answer
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What's the path of charged particle around a circular magnetic field?

I'm trying to simulate electromagnetic fields. At the moment I'm exploring only magnetic fields (I know you can't have it without electric charge). Anyway, I created a current carrying wire (just a ...
NatGazer's user avatar
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1 answer
38 views

Trying to evaluate integral using cylindrical basis vectors [closed]

To better explain my question, I'll describe the problem that gave me the idea, using that the eletric field for a linear distribution of charge is: $$\vec E = \int_L{\frac{k \cdot \rho_{l} \cdot dl}{|...
Fernando Henrique Vaz Mello's user avatar
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1 answer
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Elliptical orbit, work done by gravity

Consider two points (A & B) on one half of an elliptical orbit. Satellite moves from point A to B. I want to calculate the work done by the gravitational force, but I DO NOT WANT to use energy ...
Tony Duarte's user avatar
6 votes
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89 views

Fourier transform of Feynman Integral

In Nastase, Introduction to AdS/CFT, the first chapter talks a little about the star-triangle duality. In fact, it was claimed that the Fourier Transform of a Feynman-like diagram in position space in ...
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Green's function of screened Coulomb interaction using partial Fourier transform

The solution of the differential equation (DGL) $$ (-\epsilon_0\nabla^2 + l^{-2} )G(\vec{r},\vec{r}') = \delta(\vec{r},\vec{r}') $$ is given by a screened Coulomb potential $$ G(\vec{r},\vec{r}') = \...
user203417's user avatar
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2 answers
97 views

Problem with understanding the definition of electric potential [closed]

Here is the definition I know: $V(\vec r)-V(\vec r_0)=\int_{\vec r_0}^{\vec r} -\vec E \cdot\vec dr$ I have lots of problems with this topic. How can I choose the starting point? and does it matter ...
Dor's user avatar
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5 answers
106 views

Physical meaning of line integral involving force

I'm curious about the physical meaning of the following equation: \begin{equation} \oint \mathbf{F} \cdot d \mathbf{s} = 0 \end{equation} What does this physically mean? I think is has something to do ...
Jan Oreel's user avatar
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Second-order variation of an integral

I was reading the paper SISAR Imaging for Space Debris based on Nanosatellites, in which the Fresnel-Kirchoff diffraction formula is applied for a scenario in which the receiver, transmitter and ...
DaDSPGuy's user avatar
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1 answer
86 views

Sign choice for line element while finding potential due to point charge [closed]

The canonical derivation for the $1/r$ potential due to a point charge is as follows: We consider an electric field of the form $$\mathbf{E}=\frac{q}{4\pi\epsilon_0 r^2}\hat{\mathbf{r}},$$ and ...
user1394273's user avatar
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1 answer
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Why does taking average pressure and multiplying with area given total hydrostatic force

I am not able to wrap my head around this. Why does taking average pressure and multiplying it with the area given total hydrostatic force? In many books, the reason was because pressure is linearly ...
Hammock's user avatar
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1 answer
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Calculating the Second Virial Coefficient

I have seen many different derivations of the second virial coefficient, but none of them explains how they turn a double integral into a single integral. The second virial coefficient is: $$ B_2=-\...
Kubrik's user avatar
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-2 votes
2 answers
58 views

Can the different differentiation notations be equated and do they have an integral definition? [closed]

Are these all equivalent and is there an extension of this to other notation? Does anyone have a clear and concise chart equating the different notation dialects? I am also curious if there are more ...
Kenneth Mikolaichik's user avatar
1 vote
0 answers
69 views

Complex gaussian integral with a complex action and different source terms [duplicate]

I am trying to use the following Gaussian path integral identity $$\int D[\phi_1,\phi_1^*,\cdots,\phi_n,\phi_n^*] \exp(i\int z^\dagger D z+i\int f^\dagger z+z^\dagger g) = \det{D}^{-1}\exp(-i\int f^\...
user1830663's user avatar
0 votes
1 answer
59 views

Difference between general formula for cosmological comoving distance compared to Carroll's one

I came across two different formulas for the transverse comoving distance in cosmology from GR based on Friedmann's solution in the FLRW metric for an expanding space homogenoeus and isotropic: a) .....
Ennio's user avatar
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3 votes
0 answers
86 views

Questions about derivation of Fano resonance

In Fano's original paper about Fano resonance [https://doi.org/10.1103/PhysRev.124.1866], starting from equation (3b) $$V_{E'}a+E'b_{E'}=Eb_{E'}\tag{3b}$$ one gets an expression for $b_{E'}$ $$b_{E'}=\...
Norman's user avatar
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1 vote
2 answers
74 views

$\vec{ds}$ vector of work formula

I know $W= \int_{x_1}^{x_2}\vec{F}\,.\vec{ds}\ $ I am working with frictional force. Here $x_2>x_1$, $\vec{F}$= - $\mu mgcos\theta \hat{i}$ As $x_2>x_1$, $\vec{ds}= \hat{i}dx $ So the ...
Himalayan's user avatar
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0 answers
59 views

Solution for the scale factor for a curved universe containing only matter

In my textbook, Introduction to cosmology by Barbara Ryden, the author gives directly the solution for the following integral if $\Omega_0 > 1$: $$H_ot = \int^a_0 \frac{da}{[\Omega_0/a + (1-\...
merlinbluepickle's user avatar
3 votes
1 answer
100 views

Another Fourier transformation but with a $\sqrt{\mathbf{q}^2 + m^2}$ term now

Just as the title proclaims, I have a Fourier transformation I am trying to determine. Here is the Fourier transformation in its full form: \begin{equation} \int\frac{d^3q}{(2\pi)^3}e^{i\mathbf{q}\...
MathZilla's user avatar
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1 vote
0 answers
57 views

Peskin and Schroeder page 201, solving integral over angular part

They make the following steps: $$\int\frac{\Omega_\bf{k}}{4\pi}\ \frac{1}{(\hat{k}\cdot p')(\hat{k}\cdot p)} = \int_0^1d\xi\int\frac{\Omega_\bf{k}}{4\pi}\frac{1}{(\xi\ \hat{k}\cdot p'+(1-\xi)\ \hat{k}\...
Jens Wagemaker's user avatar
1 vote
1 answer
66 views

Peskin and Schröeder page 195, IR divergence in the electron vertex function, rewriting integral

Peskin and Schröeder make the following statement $$\int_0^1dxdydz\ \delta(x+y+z-1)\frac{1-4z+z^2}{\Delta(q^2=0)}=\int_0^1dz\int_0^{1-z}dy\frac{-2+(1-z)(3-z)}{m^2(1-z)^2},\tag{p.195}$$ where $$\Delta =...
Jens Wagemaker's user avatar
1 vote
1 answer
64 views

Integration to find force exerted by liquid on a tube

I was browsing PhysicsForums and I came across an interesting fluids question. I understood the method suggested to solve it, but I wanted to see if I could solve it through integral calculus. [...
zxen's user avatar
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-1 votes
5 answers
392 views

(Not a flat-earther) The mathematics of an infinite flat earth using gauss' law for gravity

On the flat earth website, they prove that the gravitational pull of an infinite flat earth is finite. Is their proof correct?. I'm not that good at physics and can't determine if they're correct ...
Oggy Bob's user avatar
2 votes
1 answer
83 views

Where does the absolute value sign come from for this integral on Peskin&Schroeder page 14?

On page 14 of Peskin&Schroeder, the authors are calculating the amplitute of a free particle propagating between states $|\vec{x_0}\rangle$ and $|\vec{x}\rangle$ using relativistic quantum ...
Rescy_'s user avatar
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2 votes
1 answer
48 views

Derivation of conserved current

Can someone give me some steps on showing the last line. From the line, $$-\int d^3x \space \epsilon_{abc} \space [ (\nabla^2\phi_b) \phi_c - m^2\phi_b\phi_c ] $$ I cannot actually see how could this ...
King Meruem's user avatar
4 votes
1 answer
240 views

Branch cut of a one-loop bubble diagram after cutting a single propagator

I am trying to understand Cutkosky cutting rules and generalized unitarity. Consider the article https://arxiv.org/abs/0808.1446 by Arkani-Hamed, Cachazo & Kaplan. In chapter 5.1 equation 133, the ...
Andrea's user avatar
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