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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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31 views

Is the Jacobian different for different ${\cal L}^p$ norms?

(I posted this to the math stackexchange, but I've yet to receive an answer so I figured I should post here too, as this forum seems faster to respond and is full of knowledgable people.) Because the ...
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0answers
39 views

What kind of integration is used in different areas of physics? [on hold]

I'm aware that there are multiple types of integration in mathematics, e.g., there's the Riemann integral, the Riemann-Stieljes integral and the Lebesgue integral, but I'm not privy to their ...
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1answer
38 views

Electric field at center of a uniformly charged hemispherical shell [closed]

My attempt : I divided the shell into charged rings by whom I know the electric field at the center by the following relation : $\hskip2in$ Integrating the electric fields by all the rings I get ...
1
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2answers
26 views

Anomalous value of electric field due to a uniformly charged disc at a point on its central axis

Upon trying to find the electric field using integration (as done below) we get the following result : According to me there are two problems with this result : This function is not of odd parity. ...
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0answers
22 views

The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]

I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}...
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1answer
23 views

Kinematics Problem requiring Calculus [closed]

Let the Instantaneous Velocity of a rocket just after launching be given by v={ 3t for 0<= t <2 2t+ 3t^2 for 2<= t<=3 t^3 for t>3 Find the ...
2
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1answer
57 views

How is the complex integration done for the Wigner function in coherent state representation?

$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
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0answers
22 views

Manipulation of momentum factors in Feynman integrals

My current understanding: Consider a massless Feynman graph with propagators of the form $1/p^2$ and some momentum factors in the numerator. Now consider a vertex, which for simplicity of notation is ...
4
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1answer
84 views

Complex Gaussian integral with different source terms

Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is $$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
4
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2answers
168 views

Explicit computation of singular part of two-loop sunrise diagram

For $\phi^4$, there is two-loop self-energy contribution from sunrise (sunset) diagram. The integration is $$ I(p)=\int\frac{d^D p_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D}\frac{1}{(p_1^2+m^2)(p_2+m^2)[...
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2answers
72 views

How is the wave function Lebesgue integrable?

Let's assume we have a plane wave $\psi(x,t)= A_{0}e^{i(kx-wt)}$ in position space. To find the momentum representation of this wave we'd apply the Fourier transform. However, I don't see how this is ...
2
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0answers
46 views

Polylogarithmic integrals

NB: I was sent here from Math.SE, stating that polylog integrals are more common in physics and someone here might have an answer. I have asked in Phys.SE chat whether it was okay to post here but no ...
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0answers
27 views

What is the simplest way of getting the solid angle $\Omega_d$ in a space of $d$ dimensions? [migrated]

It is known that the solid angle in a flat space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$) is given by these formulae: \begin{align}\tag{1} \Omega_{2 n} &= \frac{1}{(n - 1)!} \, 2 \pi^n, \...
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1answer
50 views

How to know what the area under curve represents?

Is there a way to find out what the area under the curve represents? For eg. If i gave you a graph of $v$ with respect to $t$ would you be able to tell me what the area under the curve represents ...
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2answers
26 views

Trying to get moment of inertia of a disc using moment of inertia of a rod

I know how moment of inertia of a disc is calculated using the usual way, but just for fun, I tried this way which is rather giving incorrect answer. I don't know what's the flaw in this and thus ...
2
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0answers
75 views

An involved Feynman integral

Working with QCD, I have found the following integral from Feynman diagrams to solve $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\int\frac{d^4p_2}{(2\pi)^4}\frac{1}{p_2^2-m_0^2} \left(\frac{p\cdot p_1-p_1\cdot ...
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2answers
82 views

How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant?

When writing integrals that look like $$ \int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0) $$ it is often said ...
2
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1answer
62 views

Solving the rocket differential equation

I'm trying to derive the rocket equation. I'm pretty sure that the differential equation for the rocket equation is $$v(t)\delta t =\frac{m(t)\delta t }{m(t)} V_e$$ where $v(t)\delta t$ is the ...
0
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1answer
33 views

Follow up on understanding path integral measures

A while ago I asked the following question: Understanding Measure in Path integrals and got to the conclusion that path integral measures are infinite products of $d\phi(x_i)$ for some scalar field $\...
3
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2answers
156 views

Problem with loop Integral (HQET)

I have come across the Integral: $$ \int_0^{\infty}dx [x^2-ixa+c]^{n-\frac{d}{2}}e^{-bx},$$ where $n = 1,2 ; a,b,c,d \in \mathbb{R}; b,d > 0$. This integral should contain some divergences for $d ...
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0answers
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What is the magic behind Sector Decomposition?

I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177. So far I played around with a toy example and applied the Sector Decomposition method to ...
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1answer
48 views

Problem with converting Integral to Gamma functions (from HQET heavy quark self-energy diagram)

In the calculation of HQET radiative correction, I came across the Equation: $$\int_0^{\infty}d\lambda ~ \lambda^{-\epsilon}(\lambda+\omega)^{-\epsilon} = \frac{1}{2\sqrt{\pi}}\Gamma(\epsilon-\frac{1}{...
1
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1answer
59 views

How to find gravity field of a solid square body?

I was programming gravity simulation and stumbled upon a problem, that Newton's formula for point masses is not enough for me, I need gravity field formula of a solid square body (2D). To simplify ...
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2answers
24 views

Moment of Inertia equation for small volume

Below is the equation of the moment of inertia for small volume elements, $\Delta m$ $$I = \lim_{\Delta m_i \to 0} \sum_{i} r^2_i \Delta m_i = \int r^2 dm$$ Can someone please explain it to me on ...
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1answer
19 views

Scalar field and 2 types of line integrals

Consider the line integral, $\int _ c$f(x,y)$\vec dr$ , where $f(x,y)$ is a scalar field, and it is evaluvated on a curve $c $. After integration we get a vector let it be $\vec I$ . $\int _ c$f(x,...
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1answer
28 views

Meaning of flux 2-integral

Can someone please explain the meaning of flux 2-integral in this sentence: Mass is evaluated as a flux 2-integral at the asymptotic infinity. For asymptotic infinity, I believe it is as ...
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1answer
64 views

Calculating power with force as a function of time [closed]

While doing an AP question in Physics C today I answered the question differently from the professor but I'm not sure what part of my reasoning is incorrect. The question gives the position of a block ...
4
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1answer
40 views

What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?

In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as: $a(t) = a_0$ $v(t) = a_0t + v_0$ $x(t) = \frac{1}{2}...
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0answers
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Deriving moment of inertia of a solid sphere [closed]

I have been trying to calculate it on my own, but the answer I get is different to the one I can find everywhere else, so I have to be wrong. My attempt was a very straightforward one. I used ...
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1answer
47 views

Integral over an area of spacetime [closed]

Is it possible to evaluate this integral in spacetime? $$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$ If it is ...
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1answer
55 views

Integration Using Spherical Coordinates [closed]

So I had to find the moment of inertia of a hollow sphere of mass $M$, radius $R$, and negligible thickness. $dI=R^2 \cdot dm$ where $dm = \dfrac{M}{4\pi R^2}\cdot R^2\sin(\theta)\cdot d\theta\cdot ...
3
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1answer
124 views

The acceleration of the particles by finite difference [closed]

I would like to approximate the acceleration of a molecular dynamics system. I'm following an online tutorial to solve a set of equations for molecular dynamics. I can use $F=ma$ to calculate the ...
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1answer
52 views

Given the parameters of the electrostatics problem, is this integral possible to evaluate analytically? [closed]

A cone with apex at the origin has a height $h$ and a top radius $h$, a uniform charge density with no charge on the top face. I need to find the potential $V$ at a position $z$ on the cone's axis ...
4
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2answers
233 views

Sum to an integral in deriving equipartition theorem

I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum, $${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
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1answer
57 views

Perpendicular weight force on an object that is tipping over [closed]

I'm currently working on a problem I can't seem to find an answer to. I have an object that is hanging over a cliff. This object is exactly 12m in length, and it starts off in equilibrium (6m over the ...
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1answer
56 views

No clue about a term [closed]

$\int_S\int \vec{A}\cdot\hat{n}dS= \int_S\int A cos(\theta)dS= \int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$ I have no clue about the term $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$ ...
2
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1answer
116 views

Integration of the splitting function

I have a problem performing the following integration provided in the paper by Catani and Seymour (arXiv: hep-ph/9605323) page 27. Given is the integral $$ \mathcal{V}=\int_0^1 (z(1-z))^{-\epsilon} \...
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2answers
100 views

How to determine the units of my integral? [closed]

Given that I have some coefficient(i.e. a number) which is to be determined from a radial integral: $$b_{n00} = \frac{(2\pi)^{1/4}}{\sigma^{3/2}} \frac{1}{\sqrt{3}} [C(000|000)]^2 \int^{\infty}_{r = ...
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0answers
40 views

Integration by parts in dimensional regularisation

I have a question concerning integration by parts identities in dimensional regularisation. Appearently, almost every textbook about dimensional regularisation claims that $$\int d^Dl_1...d^Dl_L \...
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0answers
42 views

Explicit integration of the time-dependent Schrodinger equation when eigenvalues are unknown

Let's consider the hydrogen atom Hamiltonian $$H = - \frac{1}{2}\Delta - \frac{1}{r}$$ The solution for the corresponding time-dependent Schrodinger equation is the following: $$\psi = \psi (t = 0){...
4
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1answer
135 views

Time evolution operator in QM

I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator: \begin{align*} i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t) \end{...
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1answer
75 views

Integrating rigid body equations for a game engine simulation

I'm a mechanical engineer who's trying to implement a physics engine for a 3D game simulation, so I apologize for being incorrect or simply ignorant of some aspects of computation. I'm implementing ...
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1answer
31 views

How can I calculate the total rotation a detuned pulse will apply a nuclear spin?

I'm trying to model the effect a radiofrequency pulse will have on a nuclear spin at different detunings. The pulse has a sech lineshape, a pulse area (time integral of the pulse envelope) of $\frac{π}...
0
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1answer
39 views

How to combine limits when integrating in the frequency domain

I want to combine the signal of two separate pulses in the frequency domain in order to calculate their overlap (by multiplying the two signals together and integrating). However, one of these signals ...
1
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1answer
35 views

Integrating Carnahan-Starling Pressure

Given the Carnahan-Starling equation of state for a solution of hard-spheres, $$ Z = \frac{P}{\rho k_BT} = \frac{1 + \eta + \eta^2 - \eta^3}{(1-\eta)^3}$$ where $\rho = N/V$ is the number density and ...
1
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1answer
87 views

Computing integrals for divergent loop amplitudes?

I am trying to compute the cross-section for the diagram below with a divergent triangle loop: $\qquad\qquad\qquad\qquad\qquad$ where $X^0$ and $X^-$ are some fermions with zero and negative charge ...
0
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1answer
33 views

How to find the net electric force exerted on a uniformly charged rod by another, same rod on the x-axis (they don't touch)? [duplicate]

How to find the net electric force exerted on rod 2 by rod 1, both being on the x-axis, both having the same length and constant linear charge density, being some distance apart? More specifically, ...
1
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1answer
48 views

The average velocity of a particle

The Maxwell distribution of velocities is: $$p (v) = (\frac{m}{2\pi K_b T})^{\frac{3}{2}} e^{\frac{-mv^2}{2 k_b T}}$$ I want to understand how to obtain the average value of the velocity. The ...
0
votes
1answer
53 views

About the quadratures method

in the Classical Mechanics (2nd. Ed.) book of Herbert Goldstein, p. 75 it says: "Equations 3-18 and 3-20 are the two remaining integrations, and formally the problem has been reduced to quadratures..."...
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0answers
21 views

Hylleraas Multi-dimensional integral [closed]

Evaluate the hylleraas integral $$\int \int \frac{\exp(-ar_{1}-br_{2}-cr_{12})}{r_{1}r_{2}r_{12}}d^{n}r_{1}d^{n}r_{2} $$ with $r_{1}=|\vec{r_{1}}|$, $r_{2}=|\vec{r_{2}}|$, $r_{12}=|\vec{r_{2}}-\...