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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59 views

Can I use dimensional regularization with this integral?

I would like to extract the divergence of this integral in 4d Euclidean space: $$\int d^4z \frac{1}{(x-z)^4}\tag{1}$$ This divergence is expected to cancel with other divergences, which I got using ...
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51 views

Why is this equation correct?

This problem seems to be solved the exact same way in multiple solution books, so I'm certain that the way it is done is correct and that I'm just rusty when it comes to calculus due to multiple years ...
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1answer
95 views

When can I set $d=4$ in dimensional regularization?

I am using dimensional regularization to extract the divergence of some complicated integral. I work in $d=2\omega$ dimensions, with $\omega\approx 2$. After I extract the divergence, I have an ...
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85 views

How to solve integrals with $3$ Feynman parameters? [migrated]

I would like to evaluate integrals of the following type (in position space): $$\int \frac{d^{2\omega}z}{\left[(x_1-z)^2 (x_2-z)^2 (x_3-z)^2 \right]^A} \tag{1}$$ I can introduce three Feynman ...
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1answer
39 views

Work done to compress a liquid in container [closed]

Into a compressed container containing water with pressure p and volume V we want to pump additional water. What is the work done? Unlike in the ideal gas, the work cannot be simply found out using ...
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42 views

Integrating a product of Gaussian distributions

I'm stuck at this relatively easy looking integral where I have gaussian distributions \begin{equation} \sigma(x,y)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2+y^2}{4\sigma^2}} \end{equation} and the ...
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4answers
100 views

Confusion about this simple electrostatics line integral

Suppose we want to find the electrostatic potential $\phi_{0}$, with reference to infinity, at $r_{0}$ resulting from a positive charge $q$ located at the origin. For simplicity, let us assume we are ...
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2answers
60 views

Evaluation of contour integration [closed]

Consider the integral $\int^{\infty}_{-\infty}\frac{q\exp(iqR)}{q^2-k^2}dq$. This kind of equation appears in evaluation of Green's function in scattering theory.We use contour integration to evaluate ...
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3answers
68 views

Find velocity from acceleration equation [closed]

Suppose the acceleration of a particle is a function of $x$, where $$a(x) = (2.2 s^{-2})x$$ (a) If the velocity is zero when x = 1.0 m, what is the speed when x = 3.4 m? (b) How long does it take ...
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71 views

What am I not understanding about this double integration of acceleration to get position?

Brilliant.org has a module on classical mechanics and I'm having difficulty with a mathematical step. They want you to represent position in terms of acceleration and then to solve the double integral ...
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323 views

Help on a hard integral [migrated]

So, I'm doing an extensive homework of electromagnetism and we are searching for the total electromagnetic angular momentum of the Thomson dipole. In the end, there is one integral we cannot solve. By ...
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14 views

“Euclid’s test” , Negative Pressure and Measure Theory

I don't understand what does it mean for 'Euclid's Test' when they talk about negative pressure Using Euclid’s test for a hypothesis of examining its implications, one finds that negative pressure ...
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46 views

Monte Carlo integration - convergence

I have a 5D integral being calculated with a Monte Carlo uniform random sampling. The issue is that the region of integration is very small and for 100000 points I get only around 20-30 points every ...
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1answer
96 views

A triple integral in Spherical coordinates from Jackson's book on Electrodynamics [closed]

I have been trying a solution for the following integral from Jackson but i do not seem to go anywhere. Please help. The problem is to compute interaction energy due to 2 charges. Compute following ...
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1answer
50 views

Oscillator integral for frequency

If, for a (not necessarily simple harmonic) oscillator I have that $$\frac{dx}{dt} = G(x)$$ then I can express the period of motion as $$\int_{0}^{T/4} dt = \int_{0}^{X_{max}} \frac{dx}{G(x)}.$$ What ...
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2answers
53 views

Area under a velocity graph

If I took the definite integral of a velocity graph from 0 to 10 seconds, the answer would be the change in position over those 10 seconds correct? I am told by my teacher the area is change in ...
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52 views

Gaussian oscillatory integral evaluation using regularization

To evaluate the Gaussian integral $$ \int_{-\infty}^\infty dx e^{iax^2} = \sqrt{\frac{\pi i}{a}}, $$ one can use an appropriate contour as here, or use the method of "regularization", contained for ...
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1answer
45 views

Gauss's law for magnetism : double integral

Gauss's law for magnetism is stated as followed with the beautiful closed surface double integral (by wikipidia): $$ \mathop{\vcenter{ \huge\unicode{x222F}\, }}_{S} \mathbf{B} \cdot \text d\...
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82 views

Field of a uniformly charged disk: integration question

In my book (University Physics by Young and Freedman), during solving the common example of finding the electric field along the x-axis from a uniformly charged disk, they arrive at this differential ...
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2answers
51 views

Integral evaluation

I am reading this paper where I have encountered the following integral: $$ I_2 = \lim_{a\rightarrow \infty}\int_0^\infty e^{-\beta k^2} \frac{\cos(2ka)}{\kappa^2 + k^2}dk.$$ where $\beta>0$ is ...
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7 views

Angular positions of points on a pole figure

I have to integrate a certain physical property of a crystal within a section of orientation space, Say between the space bound by directions <100>, <110> and <111> and so on. How can I ...
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1answer
38 views

How to take the Fourier transform of the propagator of a vector field?

In the paper Wilson Loops in N=4 Supersymmetric Yang--Mills Theory, the authors give the following generalized Fourier transform for a propagator in $d=2\omega$ dimensions: $$\int \frac{d^{2\omega}p}{...
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55 views

Crystal lattice Fourier coefficient

In solid state physics we have a periodic function over the lattice as $$f(\boldsymbol{r})=\sum_{\boldsymbol{G}}f_{\boldsymbol{G}}\mathrm{e}^{2\pi i\boldsymbol{G}\cdot\boldsymbol{r}}$$ where $$\...
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3answers
45 views

On moment of inertia calculation of a triangle

I have seen multiple videos on YouTube which start off the process of calculating the moment of inertia of a triangle about it's base by writing it equal to  $\int(y^2\ dA )$ where $y$ is the ...
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1answer
28 views

Inconsistency regarding two charge elements on a uniformly charged disk

Let us say we have the following uniformly and positively charged thin disk: Suppose also that the total charge on the disk equas $Q$, with which we could define a constant charge density $\sigma$ ...
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1answer
80 views

Simple question on quantization of EM field

This is part of the lecture note of QFT by Tong: I have a question on the last part of (6.27). Why does it hold? Actually this question is not about QFT and is about just integral. The typical way to ...
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52 views

Proving an equivalent [duplicate]

Can anyone help me prove the following equivalent found in Peskin & Schroeder on page 27. $$ \frac{1}{4\pi^2} \int_{m}^{+\infty} dE \sqrt{E^2-m^2} e^{-iEt} \sim_{t \to \infty} e^{-imt}$$ The ...
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1answer
50 views

Momentum replacement in the axial anomaly calculation in dimensional regularisation (‘t Hooft prescription)

I have been studying the axial anomaly and everywhere I see the calculation of the triangle loop using dimensional regularisation (see for example pages 661-664 of section 19.2 of Peskin). In the ‘t ...
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1answer
49 views

What does this description about kinematics mean?

In my $Theoretical \ Mechanics$ book I recently read, it gives the following description about "the path" in a certain time interval: $$\vec{r}(t)=\vec{r}(t_0)+\vec{v}(t_0)(t-t_0)+\int_{t_0}^tdt^{\...
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1answer
88 views

Why $\int_m \left( y^2 + z^2 \right)dm \neq \frac{1}{3}mL^2$?

for a cylinder, let's say with constant density with radius $3$ and height $10$ so $$ \rho(r, \theta, y)=1 $$ so $$ dm = \rho\left(r, \theta, y\right)r \, dr \, d\theta \, dy = r \, dr \, d\theta \, ...
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36 views

Neutron mass fraction evolution: approximation

In Mukhanov's "Physical foundations of cosmology" on page 102 the author considers an equation for the evolution of the neutron mass fraction $X_{n}\equiv n_{n}/(n_{p}+n_{n})$: $$ \tag 1 \dot{X}_{n}(t)...
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1answer
88 views

Gaussian wave packet with a step potential

In principle of quantum mechanics by Shankaar on page 170, while doing transmission and reflection index for a step potential for a Gaussian wave packet moving to the right. We come to this ...
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2answers
44 views

Find the work done against the force

If there is a force $\mathbf F(x)=ax\hat i$ and an object is moving from $x_2>0$ to $x_1>0$ in the opposite direction of force. Then work could be calculated as follows $$\int_{x_2}^{x_1} -F(x)\...
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1answer
89 views

Why does the integral symbol disappear when applying a functional derivative?

it is known that variation is defined by following: but could anyone tell me why the integral symbol disappears after following functional derivative?
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2answers
133 views

Are the infinitesimal lengths in integrals bounded by the Planck length? [closed]

When we integrate something say work, $\int F\cdot ds $ then we will get work but what exactly is $ds$? how much is ds? Is it the Planck length? Are we just pretending there exists some infinitesimals ...
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4answers
34 views

Is work equal to area between graph and $x$-axis in a graph of force vs displacement?

Is this true even when displacement is not in direction of force? $$W = \int (F\cos\theta)\text dx$$ and area is $\int F \text dx$. In my book that area is given as one of the definitions ...
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2answers
106 views

Integral as summation in quantum mechanics

I have just started QM and one thing that keeps bugging me is that whenever we have a continuous summation we take it as an integral (like in the formula below)... So why can we do this why summation ...
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1answer
27 views

How to recover the potential field from Green's function and Poisson's equation for a point charge

I first find the Green's function for the following PDE in $n=3$ dimensions, where $k:=|k|^2$. $$\nabla^2G(x,x')=\delta^3(x-x')$$ Upon Fourier transforming both sides, and inverting, I find that $$G(x,...
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32 views

Constant Acceleration and Displacement

How can I conduct an experiment to show that the area under a velocity-time graph equals the displacement when the velocity is changing at a constant rate? I've tried to measure free falling objects, ...
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1answer
28 views

Finding suitable element to perform integration upon [closed]

Is there any precise (proper) method or technique to specify the element on which integration will be performed. Is it the same for all properties like moment of inertia, gravitational potential, ...
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0answers
29 views

Total amount of diffusely reflected light off of a sphere?

I have a numerical simulation that uses ray tracing to calculate the total amount of light picked up by a sensor, after diffusely reflecting off of an object. To validate this simulation, I'd like to ...
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1answer
227 views

Showing $I=\int d^3k\int dk^0\frac{1}{k^4}$ to be logarithmically divergent

Consider a momentum integral of the form $$I=\int d^3k\int dk^0\frac{1}{k^4}$$ where $k^2=(k^0)^2-(\vec{k})^2$ and the integral over $k^0$ runs from $-\infty$ to $+\infty$. This integral is common in ...
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23 views

Units after integrating a ratio metric: Point Source Transfer Mobility

I want to calculate the so called Line Source Transfer Mobility (LSTM) from 10 equispaced Point Source Transfer Mobilities (PSTM). The PSTM measures the response to an excitation, and it is defined ...
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1answer
61 views

Fractional Fourier Transform and Fresnel Propagation

I am currently trying to wrap my head around Fresnel propagation, and I understand it is mathematically linked to the Fractional Fourier Transform, but I'm having a hard time with the units and the ...
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1answer
61 views

Direction of integration and boundary limits in electromagnetism?

I have encountered several problems regarding the choice of direction of integration and the boundary limits, this semester in electromagnetism. Is there some rule, so I don't do it wrong again. In ...
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2answers
85 views

Contour for integration in 1D scattering problem

A plane wave scattered by a 1D potential can be described by, $$\psi(x) = \begin{cases} e^{ikx} + R e^{-ikx}, & x<0\\ T e^{ikx}, & x>0 \end{cases}$$ where $R$ is the reflection ...
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1answer
75 views

Delta function and potential step [closed]

I have a potential consisting of an attractive delta funtion well located at the origin and a superimposed with a potential step at the origin, just like: With $$V(x)=-\lambda \delta (x) +V_0 ...
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3answers
76 views

Change of variable in 4-dimensional integral

If I have a measure $d^4 x$ and I want to perform a conformal transformation $x^\mu \rightarrow \frac{x^\mu}{x^2}$, how do I get that the transformed measure is $\frac{d^4 x}{x^8}$? I started by ...
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2answers
30 views

Intuition of Distance covered when accelerating [duplicate]

When you're moving at $5$ m/s for $1$ second, you have traveled $5$ m. When you're moving at $5$ m/s (initial velocity) and you accelerate $2$ m/s for $1$ second, you have traveled $5$ m + $1$ m (...
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52 views

Computation of Wigner Functions

The Wigner function can be computed as the Fourier transform of the Weyl-ordered characteristic function: $$ W(\alpha) = \frac{1}{\pi^2} \int e^{\lambda^* \alpha - \lambda \alpha^*} C_W(\lambda) d^2\...