For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
Questions tagged [integration]
1474 questions
62
votes
3 answers
How is the Saddle point approximation used in physics?
I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative description of the method and why it's used and for…
BeauGeste
- 1,741
40
votes
5 answers
What does an integral symbol with a circle mean?
I have frequently seen this symbol used in advanced books in physics:
$$\oint$$
What does the circle over the integral symbol mean? What kind of integral does it denote?
user11543
32
votes
3 answers
Why aren't Runge-Kutta methods used for molecular dynamics simulations?
One of the most used schemes for solving ordinary differential equations numerically
is the fourth-order Runge-Kutta method. Why isn't it used to integrate the equation of motion of particles in molecular dynamics?
WedgeAntilles
- 481
32
votes
5 answers
Why are the number of magnetic field lines finite in a particular area?
One can draw/imagine as many unique (curved/straight) lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line). Then how can the number of field lines in a particular area be a…
Tim Crosby
- 1,373
32
votes
3 answers
When is Lebesgue integration useful over Riemann integration in physics?
Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesgue integration can be more powerfully used even in physical situations that can be…
Larry Harson
- 5,456
32
votes
2 answers
Gaussian integral with imaginary coefficients and Wick rotation
Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking for about 40 minutes, which leads me to believe…
Adomas Baliuka
- 1,814
32
votes
4 answers
Complex integration by shifting the contour
In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is:
$$\int_{-\infty}^\infty dk_0 \frac{e^{-ik_0z_0}}{k_0^2-\kappa^2}$$
where $k$ and…
user2582713
- 623
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- 13
28
votes
4 answers
Why is the functional integral of a functional derivative zero?
I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e.
$$\int [d\phi ]\frac{\delta…
user22208
- 493
28
votes
3 answers
Heisenberg's uncertainty principle for mean deviation?
The Heisenberg uncertainty principle states that
$$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$
However, this is only for the standard deviation. What is the inequality if the mean deviation, defined as
$$\bar \sigma_x=\int_{-\infty}^{\infty} \lvert…
Zach466920
- 1,137
23
votes
4 answers
Non-unique zero function in the function space (Hilbert space)
I have just started studying about quantum mechanics, and I was studying the definition of the inner product for functions; I am also quite new to linear algebra. While studying I think I encountered a contradiction on the definition of the inner…
Soroush khoubyarian
- 651
- 4
- 19
22
votes
1 answer
Derivation of KLT relations
The KLT relations (Kawai, Lewellen, Tye) relate a closed string amplitude to a product of open ones. While I get the physics behind this I don't really understand the derivation in the original paper (see https://doi.org/10.1016/0550-3213(86)90362-7…
A friendly helper
- 1,077
- 8
- 18
20
votes
3 answers
Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
Mark
- 341
20
votes
5 answers
How to get distance when acceleration is not constant?
I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question.
The equation for distance of an accelerating object with constant acceleration is:
$$d=ut +\frac{1}{2}at^2$$
which…
ben
- 1,547
19
votes
4 answers
Is dimensional analysis valid for integrals
Can we apply dimensional analysis for variables inside integrals? Ex: if we have integral $$\int \frac{\text{d}x}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right),$$ the LHS has no dimensions, while the RHS has dimensions of…
rathankar
- 307
18
votes
5 answers
How to turn a sum into an integral?
In, An Introduction to Thermal Physics, page 235, Schroder wants to evaluate the partition function
$$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$
in the limit that $kT\gg\epsilon$, thus he writes
$$Z_{tot}\approx\int_0^\infty…
GedankenExperimentalist
- 1,081