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 ...
- 245k
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 ...
- 4,628
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 ...
- 50k
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
$$ \int_0^\infty dx\ x^n \exp \left( -h \sqrt{x^2 + a^2} \right) = \frac{2^{...
- 528
0
votes
3
answers
126
views
How to compute the Jacobian matrix using Mathematica [duplicate]
Apologies if this is posted to the wrong forum. Have been using Maple and Maxima for 20+ years, and have only recently started using Mathematica (v. 14). Having a heck of a time getting Mathematica to ...
- 59.6k
0
votes
1
answer
62
views
Compute integrals in singular integral equation
I'm looking at this paper
https://arxiv.org/abs/2404.07307
and in particular I am interested in eqs. (16) (17), meaning I'd like to check the validity of (16) by insert it in (17).
So I'd like to ...
- 56.9k
8
votes
4
answers
622
views
Inverse Laplace Transform of Hypergeometric Function
Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral
...
- 29.1k
2
votes
1
answer
89
views
How to force Mathematica to evaulate some values of LerchPhi function?
Mathematica cannot give LerchPhi[1,0,1]=-1/2, LerchPhi[1,1,1]=0, and LerchPhi[1,1,1/2]=0, as ...
- 17.1k
6
votes
4
answers
510
views
Approximation of the Fabius function with a quotient of exponentials
Approximation of the Fabius function $f(x) = \text{FabiusF}[x+1]\cdot \text{HeavisideTheta}[1-x^2]$ - FabiusF[x] doesn't work in Wolfram-Alpha
I am looking to figure out how well the displaced version ...
- 175
0
votes
1
answer
92
views
A hypergeometric series function
Maple and Mathematica are very efficient to find closed form of a hypergeometric function as for example:
$$\sum _{k=0}^{\infty } \frac{(-4)^k z^k \Gamma \left(1 k+\frac{1}{6}\right) \Gamma \left(\...
- 163k
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
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 ...
- 245k
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....
- 245k
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 ${}_2F_1$:
\...
- 16.6k
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
$$
\begin{align*}
A_1(a, \Delta, \omega_0)
&= 1 - \frac{2\Delta^2}{\omega_0} + \frac{a^2}{\omega_0^2}
- \left[1 + \frac{...
- 2,590
1
vote
3
answers
1k
views
Calculation of an integral with a Bessel function
How can I calculate below integral? Is it possible to calculate it in Mathematica? If yes, how? Am I doing something wrong?
...
0
votes
2
answers
68
views
Same integral giving different results
I am trying to solve the following integral using Mathematica
$\int_{0}^1 dz \int_{0}^{1} dz' \exp(-k|z-z'|)\cos[\pi p z] \cos[\pi q z]$,
with $p,q\in \mathbb{Z}$. To do so, I am doing the following:
<...
- 5,620
6
votes
2
answers
367
views
An integral using Mathematica or otherwise
Consider the unit square integral $$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$$
Using Mathematica or otherwise I need a closed form of I, possibly in terms of ...
- 16.6k
3
votes
1
answer
102
views
Many contradictory results for a single integral
I am interested in solving the integral $$
\int_{\mathbb{R}} dx \frac{e^{i a x^2}}{(x - b - ic)^d(x - b + ic)^d}
$$
for $a,b,c \in \mathbb{R}$ and $d \in \mathbb{N}$. As an early step, I have asked ...
CommunityBot
- 1
6
votes
1
answer
277
views
Closed form of an integral using Mathematica or otherwise
Define $$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 ...
- 6,944
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.
...
- 16.6k
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 ...
- 245k
8
votes
1
answer
560
views
Finding Euler spiral convergence point for powers other than 2
I'm playing around with Euler spirals, using a variable power to see how the plot changes,
$
\left\{\begin{matrix}
x = \int_0^s \cos \left ( s^n \right ) ds \\
y = \int_0^s \sin \left ( s^n \right ) ...
- 57.9k
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 ...
- 5,124
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 ...
- 163k
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
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:
...
- 29.1k
3
votes
1
answer
140
views
How reliable is AsymptoticIntegrate?
In new mathematica 12 there is a new function AsymptoticIntegrate. However, it seems that it gives me incorrect results in some cases. To be completely honest, it ...
- 16.6k
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-...
- 163k
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?
...
- 3,689
1
vote
1
answer
175
views
How to find analytical solution for integral from product of Bessel functions of second kind with different order?
I need help with analytical solution for the following integral:
...
- 3,689
6
votes
2
answers
342
views
Why do these identical limits give different results?
Bug introduced in 12.0.0.0. and persisting through 14.0.0.0.
I wanted to calculate this limit:
...
- 3,369
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:
$$\omega = c\ \sqrt{q^2+k_z^2}$$
$$ E = \sum_{k_z=n\pi/a} 2 \int\...
- 307
1
vote
1
answer
62
views
Finding value of a function at limit zero
I want to evaluate the below function at limit zero. I tried with a function which has double differentiation for which it is giving me a finite value. For triple differentiation it is consistently ...
- 151k
6
votes
1
answer
339
views
DSolve: unable to solve the conditions
I have a second order differential equation with $2$ boundary conditions and want to use the following code to solve it:
...
- 57.9k
1
vote
0
answers
104
views
Differentiation of Parabolic Cylinder Functions and Hermite Polynomials w.r.t to Order
When I evaluated (ver. 12.3) parabolic functions below in the vanishing order limit: $u \rightarrow 0$, I got the following:
...
- 29.1k
3
votes
2
answers
75
views
Using FindSequenceFunction to solve the integral of the product of Legendre polynomials and power functions
The integral of the product of Legendre polynomials and power functions:
$I=\int_{-1}^1 x^n \mathrm{P}_l(x) \mathrm{d} x$
The calculation result from the textbook is:
$$
\begin{aligned}
I
& =0
\...
- 163k
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
...
- 5,820
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}...
- 16.6k
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
$$\boxed{...
- 40.7k
1
vote
1
answer
147
views
Limit of Hypergeometric Functions
Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following
...
- 29.1k
1
vote
2
answers
259
views
Error complex function ERFI(X): looking for alternative function representations?
I have some analytical results from a physics problem, where the Mathematica gives the results in terms of complex error function. I would like to explore another function representation using ...
CommunityBot
- 1
4
votes
1
answer
143
views
Strange result simplifying higher order BesselJ [duplicate]
Consider the following integral:
$$\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....
11
votes
1
answer
1k
views
Expanding derivatives of hypergeometric functions
Sometimes Mathematica expresses results of integration or summation in terms of symbolic derivatives of Hypergeometric2F1 function, and cannot further simplify ...
- 1,907
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 $n = 1, 2, 3$ (...
- 6,944
0
votes
1
answer
108
views
Dirac Delta does not converge
I have trouble evaluating a simple integral in Mathematica. I have the code:
...
- 56.9k
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)
...
