All Questions
12 questions
5
votes
1
answer
271
views
Quantum Field Regularization: Choice between ExpIntegralE and Gamma for Casimir Effect
For the Casimir effect with exponential regularization, we compute the vacuum energy between the plates (somewhat simplified) with:
$$\omega = c\ \sqrt{q^2+k_z^2}$$
$$ E = \sum_{k_z=n\pi/a} 2 \int\...
- 307
1
vote
1
answer
242
views
Discrepancy with Hurwitz Zeta function
I've come across an issue while using Wolfram Mathematica that I don't quite understand.
Consider the following symbolic computation:
...
- 27
0
votes
1
answer
85
views
Finite result under "Integrate" while infinite under "NIntegrate" of a complicated integral with "NSum"
I have been trying to compute the following complicated integral along with summation, details codes/function of which is given below:
...
- 127
4
votes
1
answer
196
views
How to calculate the sum of the series of Hermite polynomial?
I want to calculate the infinite sum of Hermite polynomials, which works fine with older versions of Mathematica, but with version 13 it doesn't.
The infinite sum is:
...
- 43
0
votes
0
answers
53
views
Evaluation of a double summation invovlving hypergeometric and exponential functions
I am trying to compute the following double summation over the indices, $m$ and $n$, which involves the hypergeometric function, ${}_2 F_1$, an exponential function and, factorials as a part of a ...
- 85
1
vote
1
answer
158
views
How does Mathematica evaluate these sum and integral?
How does Mathematica internally evaluate the following (interrelated) sum and integral, and how does it do the subsequent simplification? (I mean, based on what mathematical facts?)
...
- 855
4
votes
5
answers
350
views
Calculating the Dottie number using an infinite series
The Dottie number is the solution to the equation
$\cos(x) = x$
It is approximately equal to $0.739085133215160641655312.$
This number can be expressed analytically in the following form (see this ...
- 311
0
votes
2
answers
491
views
Sum a certain hypergeometric-function-based expression pertaining to an integration problem
I would like to sum over the index $h$ from 3 to $\infty$, the expression
...
- 2,385
8
votes
2
answers
392
views
Calculating sum of BesselJ[n, x]
My friend has a sum in his research paper that looks like this
$$
\sum_{n=-\infty}^{\infty}\frac{J_n^2(x)}{n-\kappa}.
$$
He was able to calculate this sum analytically, by substituting the ...
- 215
9
votes
2
answers
292
views
Infinite sum not evaluated unless split into even and odd terms
This sum
s = Sum[Gamma[k/2]/(2 k!), {k, 1, ∞}]
$\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}\right)}{2 k!}$
is returned unevaluated (version 10.1.0).
...
- 13.1k
13
votes
2
answers
661
views
Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$
I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$
I tried
Limit[n*Sum[...
- 1,119
13
votes
3
answers
544
views
Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$
I have an expression
2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}]
...
- 1,119