All Questions
Tagged with calculus-and-analysis special-functions
434 questions
8
votes
2
answers
419
views
NIntegrate cannot give high precision result for a well-behaved integral
I want to obtain value of following integral to a high precision (say 30 digits),
NIntegrate[DawsonF[Sqrt[t]]^2/t, {t, 0, Infinity}]
Graph of integrand looks like ...
- 311
0
votes
0
answers
47
views
Ability of Integrate[ ] to try changes of variable on its own
I have a question about Mathematica's ability to try changes of variable when performing symbolic integration. My example is
$$ \int_0^\infty dx\ x^n \exp \left( -h \sqrt{x^2 + a^2} \right) = \frac{2^{...
- 528
0
votes
1
answer
97
views
Converting HurwitzZeta function to PolyGamma function
A generalization produces a result in terms of Zeta[s,a] function, which can be converted to PolyGamma[s-1,a] using the ...
- 645
0
votes
3
answers
126
views
How to compute the Jacobian matrix using Mathematica [duplicate]
Apologies if this is posted to the wrong forum. Have been using Maple and Maxima for 20+ years, and have only recently started using Mathematica (v. 14). Having a heck of a time getting Mathematica to ...
- 101
0
votes
1
answer
86
views
Issue with Reduction of Complete Elliptic Integral of the Second Kind
I am attempting to reduce the following equation:
y ==
I’ve entered it to be reduced as such, where L = 3.95:
By what means may this be properly reduced for y?
Both WolframAlpha and Desmos provide ...
0
votes
1
answer
62
views
Compute integrals in singular integral equation
I'm looking at this paper
https://arxiv.org/abs/2404.07307
and in particular I am interested in eqs. (16) (17), meaning I'd like to check the validity of (16) by insert it in (17).
So I'd like to ...
- 31
2
votes
1
answer
89
views
How to force Mathematica to evaulate some values of LerchPhi function?
Mathematica cannot give LerchPhi[1,0,1]=-1/2, LerchPhi[1,1,1]=0, and LerchPhi[1,1,1/2]=0, as ...
- 645
0
votes
1
answer
92
views
A hypergeometric series function
Maple and Mathematica are very efficient to find closed form of a hypergeometric function as for example:
$$\sum _{k=0}^{\infty } \frac{(-4)^k z^k \Gamma \left(1 k+\frac{1}{6}\right) \Gamma \left(\...
- 15
6
votes
1
answer
238
views
How to force Mathematica to evaulate LerchPhi[1,0,1] to -1/2
Mathematica fails to evaluate LerchPhi[1,0,1] (it gives ComplexInfinity). Based on the relation LerchPhi[1,q,1]=Zeta[q], we ...
- 645
5
votes
0
answers
171
views
Hypergeometric Function Integration Using Mellin-Barnes Representation
I have the following integral:
$$
I=\int_0^1 d \alpha d \beta d \gamma r(\gamma) s(\alpha, \beta) T(\alpha, \beta, \gamma)
$$
where I define
$$r(\gamma)=(\gamma(1-\gamma))^{ -1 / 2+\epsilon / 2}$$
and,...
- 139
1
vote
1
answer
198
views
Integration and expansion of hypergeometric function
I am trying to reproduce the evaluation of the expansion in $\epsilon$ for the function in Appendix B of the paper arxiv.org/abs/0705.0676v3, which involves the hypergeometric function ${}_2F_1$:
\...
- 139
0
votes
2
answers
68
views
Same integral giving different results
I am trying to solve the following integral using Mathematica
$\int_{0}^1 dz \int_{0}^{1} dz' \exp(-k|z-z'|)\cos[\pi p z] \cos[\pi q z]$,
with $p,q\in \mathbb{Z}$. To do so, I am doing the following:
<...
- 585
6
votes
2
answers
367
views
An integral using Mathematica or otherwise
Consider the unit square integral $$I=\int_{(0,1)^5}\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{(1-(1-xyuv)w)^2}\ dxdydudvdw$$
Using Mathematica or otherwise I need a closed form of I, possibly in terms of ...
- 301
1
vote
0
answers
58
views
Problem with Plot3D in Mathematica for plotting functions involving parabolic cylinder, exp, and error functions
I have the following problem to make plots of two functions:
Eq.1
$$
\begin{align*}
A_1(a, \Delta, \omega_0)
&= 1 - \frac{2\Delta^2}{\omega_0} + \frac{a^2}{\omega_0^2}
- \left[1 + \frac{...
- 11
6
votes
1
answer
277
views
Closed form of an integral using Mathematica or otherwise
Define $$I=-\int_0^1\int_0^1 \frac{x^2(1-x)y^2(1-y)(2(1-xy)+(1+xy)\log(xy))^3}{(1-xy)^7}\ dxdy $$
Now using Wolfram Alpha
$I\approx 0.00133186$.
Using Mathematica or otherwise, I need to find a ...
- 301
3
votes
1
answer
102
views
Many contradictory results for a single integral
I am interested in solving the integral $$
\int_{\mathbb{R}} dx \frac{e^{i a x^2}}{(x - b - ic)^d(x - b + ic)^d}
$$
for $a,b,c \in \mathbb{R}$ and $d \in \mathbb{N}$. As an early step, I have asked ...
- 131
1
vote
2
answers
174
views
Confused about the output of `CosIntegral`
I am interested in the properties of the cosine integral function, in this case the anti-derivative of Cos[Pi*x]/x, which Mathematica evaluates to ...
- 2,335
5
votes
5
answers
294
views
Integration involving Piecewise function and DiracDelta function
I want to calculate an integration, which reads
where $\delta\left(q_{23}^{01}\right)=\delta\left(1+q_1-q_2-q_3\right)$ and $\mathrm{min}(1,q_1,q_2,q_3)$ means the minimum of $(1,q_1,q_2,q_3)$.
What ...
- 495
3
votes
3
answers
172
views
Issue in HypergeometricPFQ function:
I have a solution from integral:
A = Integrate[x^n*(1 + x)^n*Exp[-n*x^2] /. n -> 1, {x, 0, Infinity}] //Expand
(*1/2 + Sqrt[π]/4*)
%//N
(*0.943113*)
Then I ...
- 16.6k
2
votes
2
answers
205
views
How to calculate this improper integral?
How to calculate the Integral involving MarcumQ function whose Integral interval is (0, +Infinity)?
Any help (code or reference) would be greatly appreciated.
...
- 2,067
2
votes
1
answer
81
views
Bugs with `Integrate` when dealing with complex situation?
I'm trying to calculate $\int_0^{\infty } \exp \left(t \left(-x^3+(1+i) x\right)\right) dx$ with mathematica, and different assumptions on t give different results:
...
- 2,295
1
vote
1
answer
56
views
Laplace transform of special function
The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{zt}t^{a-...
- 111
3
votes
2
answers
226
views
How to calculate the PDF of product of two random variables from generalized gamma distributions?
Namely, we want to find the explicit formula of PDF of double Generalized Gamma distribution.
...
- 2,067
1
vote
1
answer
62
views
Finding value of a function at limit zero
I want to evaluate the below function at limit zero. I tried with a function which has double differentiation for which it is giving me a finite value. For triple differentiation it is consistently ...
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:
$$\omega = c\ \sqrt{q^2+k_z^2}$$
$$ E = \sum_{k_z=n\pi/a} 2 \int\...
- 307
1
vote
0
answers
104
views
Differentiation of Parabolic Cylinder Functions and Hermite Polynomials w.r.t to Order
When I evaluated (ver. 12.3) parabolic functions below in the vanishing order limit: $u \rightarrow 0$, I got the following:
...
- 420
3
votes
2
answers
75
views
Using FindSequenceFunction to solve the integral of the product of Legendre polynomials and power functions
The integral of the product of Legendre polynomials and power functions:
$I=\int_{-1}^1 x^n \mathrm{P}_l(x) \mathrm{d} x$
The calculation result from the textbook is:
$$
\begin{aligned}
I
& =0
\...
- 2,425
1
vote
1
answer
102
views
How to integrate Legendre polynomials with parameters?
The orthogonality of Legendre polynomials:
$\int_{-1}^1 \mathrm{P}_l(x) \mathrm{P}_k(x) \mathrm{d} x=0, \quad k \neq l$
But
...
- 2,425
1
vote
1
answer
127
views
Find the range of Legendre polynomials
The range of Legendre polynomials in the Reals domain is [-1, 1]. How can we calculate it using FunctionRange or other MMA code?
...
- 2,425
1
vote
2
answers
155
views
Mathematica cannot solve this complicated integration
first-time here. I am trying to use Mathematica to evaluate a solution from Duhamel's principle, the integration looks like
$\int^t_0\frac{e^{-ks-\frac{r^2}{2(2Ds+\sigma^2+2D_pt)}}}{2Ds+\sigma^2+2D_pt}...
- 11
2
votes
0
answers
100
views
Why does HeavisideTheta give different value at the point 0 and 0.?
As the following figure described, why does HeavisideTheta give different value at the point 0 and 0.? Especially at the coordinate origin, one gives the value 1 and another 0 when HeavisideTheta[0]=0....
- 693
0
votes
0
answers
104
views
Speed up the integration of HeavisideTheta over the range from zero to infinity
The problem is described in the following figure. Mathematica version 14.0, OS windows 10 LTSC.
The integration interval is [0,infinity), and the value of HeavisideTheta in the point 0 is 0. we can ...
- 693
1
vote
1
answer
147
views
Limit of Hypergeometric Functions
Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following
...
- 471
5
votes
1
answer
374
views
Solving third order DE from fluid dynamics
I am trying to use DSolve to solve a differential equation from G. Batchelor: An Introduction to Fluid Dynamics, eq. 5.12.4:
[...] equation now reduces to
$$\boxed{...
- 47
4
votes
1
answer
143
views
Strange result simplifying higher order BesselJ [duplicate]
Consider the following integral:
$$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$.
The solution of this should contain a single Bessel function:
$$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
1
vote
1
answer
144
views
Can the Debye functions be implemented using built-in functions?
It is claimed in the comments here that the Debye functions can be implemented using built-in special functions. This is clearly true for some Debye functions, e.g., $D_n^{(1)}(x)$ for $n = 1, 2, 3$ (...
- 1,061
1
vote
2
answers
147
views
Integral convergent or divergent?
How can I find out for what values of r (both lower and upper limits), is this integral convergent/divergent?
...
- 413
0
votes
1
answer
108
views
Dirac Delta does not converge
I have trouble evaluating a simple integral in Mathematica. I have the code:
...
- 790
2
votes
2
answers
191
views
0
votes
0
answers
57
views
Manipulating Dirichlet characters and L functions
I read some basic knowledge about characters and L functions, and would like to play around with them in MMA.
I tried to do the following things, but ending in minor success. (MMA notation used)
...
- 153
1
vote
4
answers
186
views
Evaluation of an integral using Mathematica or otherwise
I need to find a closed form (in terms of known functions) using Mathematica or otherwise of $$\Re\left(\int_{\frac{1}{2}}^{1}\frac{\tan^{-1}\left(\frac
{1-x}{\sqrt{-i-x^2}}\right)}{\sqrt{-i-x^2}}\ dx\...
- 301
3
votes
1
answer
110
views
0
votes
2
answers
152
views
Complex integral with branch cuts
I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in....
- 479
1
vote
1
answer
132
views
Calculate $\lim_{n\to\infty} \left(\frac{e}{3}\right)^{3n} a_n^4$
I need to calculate the limit $$\lim_{n\to\infty} \left(\frac{e}{3}\right)^{3n} a_n^4 $$ where $a_n=\sum_{r=0}^{n}\left(\binom{n}{r}\binom{n+r}{r}\right)^2$ and $e$ is the natural base of logarithm.
...
- 301
0
votes
0
answers
81
views
Is EllipticK defined correctly in Mathematica? [duplicate]
I needed to calculate
F[a]=Integrate[1/(Sqrt[x^2+1]Sqrt[x^2+a^2]),{x,0,Infinity}]
so I fed it to Mathematica. The result I got was:
...
2
votes
0
answers
85
views
Mathematica 13.3 forgot integral defining Bessel functions [duplicate]
I installed the newest version of Mathematica 13.3.0.0 on Mac. It looks like it forgot how to compute a simple integral
Integrate[Cos[n t - z Sin[t]], {t, 0, Pi}]
...
- 21
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
2
votes
0
answers
161
views
Closed form for a sum involving Bernoulli numbers
I need a closed form for the sum $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(2^{2n}-1)\pi^{2n}B_{2n} {2n+4 \choose m}x^{2n+4}}{(2n)!}$$
where $0<x<1$, $B_{2n}$ denotes Bernoulli numbers , $m\in\mathbb{...
- 301
6
votes
3
answers
332
views
How to prove an identity involving a hypergeometric function?
How to prove with the help of Mathematica the following statement?
$$ {}_2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-1\right)=\frac{\pi -3 \sqrt{3} \log \left(\sqrt[3]{2}-1\right)-6 \tan^{-1}\left(\...
- 19.8k
3
votes
1
answer
468
views
Evaluating ${}_5F_4\left(1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right)$
Using Mathematica, how can I find a closed-form expression (in terms of elementary functions) of $$ {}_5F_4\left ( 1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right ),$$ where ${}_5F_4$ represents the ...
- 301