Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,537 questions
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How to express Meijer G-function into a form involving well-known functions?
I'm working on MAT 14.3, I am attempting to evaluate the integral int[n] for n=0,1,2,3 in symbolic form :
...
- 1,038
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ComplexExpand on BesselJ
I know ComplexExpand assumes its variables are real by default. However, when applied to BesselJ function, it is different. For ...
Qiang Lu
3
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2
answers
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Converting hypergeometric function to Struve form
In my question on MathOverflow, I was looking for a closed form result of the following sum:
$$\sum _{k=0}^n \frac{(-1)^{n-k} x^{2 k} (2 (n-k)-1)\text{!!}}{(2 k)\text{!!}}.$$
Someone suggested me to ...
1
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1
answer
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How do we find an asymptotic approximation of the function $\mathbf{P}(r)=\left(t_1+\prod\limits_{k=1}^{r}(t_2+k^{s})\right)^n$?
Suppose we have the following function, where $s\in\mathbb{R}$ and $t_1,t_2,n\in\mathbb{N}\cup\{0\}$ are constants:
$$\mathbf{P}(r)=\left(t_1+\prod_{k=1}^{r}(t_2+k^{s})\right)^n$$
Question: What is ...
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Convergence question about HypergeometricPFQ
Update in response to some questions and comments:
The book I mentioned is
Special Functions (Encyclopedia of Mathematics and its Applications, Series Number 71) by George Andrews , Richard Askey ...
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2D $q$-Racah kernel $\tilde{K}$, the pre-limit barcode kernel $K_{L}^{\text{barcode}}$, and the conjectual limit $\mathcal{K}^{\text{barcode}}(s,t)$
Looking for Mathematica implementations of 2D $q$-Racah kernel $\tilde{K}$, the pre-limit barcode kernel $K_{L}^{\text{barcode}}$, and the conjectual limit $\mathcal{K}^{\text{barcode}}(s,t)$.
Does ...
- 2,364
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Behaviour of Integrate with GenerateConditions flag
I am trying the following integral with Mathematica 13.3
...
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Series expansion involving incomplete Beta function
I am trying to expand the following around u=1/2.
...
- 1,957
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1
answer
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Help validating the solution to a nonlinear ODE: Does it solves other non-linear ODEs too? [closed]
I am working right now in this another question by doing examples and I got stuck in proving the results I found through Wolfram-Alpha are right.
The main differential equation is the following ODE:
$$...
- 187
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Expanding the HypergeometricPFQ function gives complex values
I have a function
HypergeometricPFQ[{}, {2, 2}, -Log[n]]
which for real n > 0 gives a real values, as can be seen on the ...
- 3,471
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How to improve the Mathematica implementations of q-Bessel polynomial?
I know Mathematica has implemented some QFunctions.
I have found this literature: q-Bessel polynomials, q-Laguerre polynomials, q-Charlier polynomials, etc.?
I have implemented q-Bessel polynomials in ...
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Using `JacobiP` to approximate `BetaDistribution[a,b]`
I'm looking at generalized moment expansions of BetaDistribution. Below is an example using ChebychevT, I'm looking for help extending this to cover the general ...
- 7,681
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Integrating Hypergeometric2F1 with large arguments
I am trying to integrate a function which contains two hypergeometric functions ${}_2F_{1}(a,b,c,z)$. In my particular case,
\begin{align}
a &= 1 + \ell - n,\\
b &= 1 + \ell + i \lambda x,\\
...
- 413
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Demonstrating integral result involving Bessel function
On my laptop Mathematica does not take very long to determine that $\int_0^{\infty } \frac{\left(1-\exp \left(-a x^2\right)\right) J_0(b x)}{x} \, dx$ is equivalent to $\frac{1}{2} \Gamma \left(0,\...
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How to speed up evaluation of a computationally expensive function?
The evaluation of the following code takes ages to finish:
...
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Seek Mathematica implementation of q-analogs of Special functions
I know Mathematica has implemented some QFunctions.
What about Mathematica implementations of Elliptic gamma function, Hahn–Exton q-Bessel function, Jackson q-Bessel function,
q-exponential, q-gamma ...
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Plotting spherical harmonics
I would like to recreate the following figure of spherical harmonics from Wikipedia.
I used the codes from Plotting real spherical harmonics with SphericalPlot3D, but the resulting spherical ...
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134
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How to properly use DirichletL?
I'm trying to construct in Mathematica the analytic continuation of the L-function associated with the number field Q(i). This was done, though opaquely, in the captivating YouTube series "RH ...
- 713
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Obtaining symbolic series expansion of $(W_0(-z/e)+1)^{-1}$
Is there a way to massage Mathematica to give a symbolic expression for $s$th series coefficient of $(W_0(-z/e)+1)^{-1}$
...
- 7,681
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FoxH function not working
I want to calculate the following Fox H-function:
$$
H^{1,1}_{1,2}
\left(
0.2
\,
\middle|
\begin{array}
\,
(-1, 1) \\
(-1, 1), \, (0, 0.5)
\end{array}
\right)
$$
So I tried ...
- 181
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Conical functions with Imaginary argument
I am playing with a differential equation whose solutions are conical functions (Legendre functions with a complex order) with a pure imaginary argument
$
P_{i\nu -1/2}(i \sinh \xi)
$
or $Q_{i\nu -...
- 211
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Why is Mathematica thinking that 1 is a variable?
I'm trying to solve an ODE and upon executing:
...
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Simplifying expression with Erf and Erfc
I'm trying to validate the solution of an integral in a research paper, but I'm getting a different result. I get two terms of Erf and two terms of ...
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Why the series expansion of the complete elliptic integral of the third kind fails?
I am trying to expand the complete elliptic integral of the third kind:
...
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2
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216
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I want to solve integrals in ellipsoidal coordinates?
I want to see the analytical expressions for these two integrals where s is ellipsoidal coordinates with "a" as a major axis and s>-c^2, where c is the minor axis for an ellipsoidal.
<...
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Integral in terms of 2F1(a,b;c;z) gives indeterminate answer, known well behaved solution. How do I get it to give well behaved answer? Report Issue?
I'm doing various definite integrals that I want analytic answers for (even in terms of special nonelementary functions), and I keep running into issues. Mathematica will give me an answer, but it is ...
- 75
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Numerical Computation of Multivariate Fox-H function using given codes
This PDF is the expression of SNR of practical RIS-aided system: https://ieeexplore.ieee.org/document/9387559 (expression A.8)
Authors mentioned some sources of computation of multivariate Fox-H type ...
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Derivative of QPochhammer
Related
Taylor expansion of a function containing QPochhammer[q, q, n]
My question
...
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Double Integral Which is Clearly Convergent Not Converging
Mathematica has successfully computed this integral: $$\int_0^{\infty}\int_0^\infty{}dqdl(c+(q+l)^2)^{-s}$$ assuming $s>2,c>0$. The integrand comes from a first order approximation of Bessel ...
- 31
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Did the computation of Bessel functions slow down in 14.2?
This is taking much longer to plot than I would expect:
...
- 7,554
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How to get the correct result of the difference of two large numbers? [duplicate]
I found that when the variables of the Airy functions are complex numbers such as a=-36.3 - 63.1 I and b=-37.7- 65.4 I in the ...
- 35
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3
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Numerical integration involving Bessel function $e^{-x}I_n(x)$ with large $n$
I want to numerically integrate expressions involving $e^{-x}I_n(x)$, say
$$
\int_0^\infty \frac{1}{x\sqrt x}\Bigl(e^{-x}I_n(x)\Bigr)\Bigl(e^{-x}I_0(x)\Bigr)dx,
$$
with $I_n(x)$ the modified Bessel ...
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Changing the variables changes the formula result [closed]
I am intrigued why just changing the variables in an infinite series changes the result. Actually, the two different results are equivalent, but why is it affected by a mere change in variables?
The ...
user86788
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2
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Issue with Hypergeometric Summation in Mathematica: Discrepancy Between Sum and Analytical Expression
I am working on a symbolic summation in Mathematica involving a Hypergeometric function. Specifically, I define the following term:
...
5
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2
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411
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Numerical approximation of implicit functions
I have an expression $t \equiv t(x)$,
\begin{equation}
t = 2^{-2+\frac{a}{2}} c^{1-\frac{a}{2}} \Gamma\left(-1+\frac{a}{2}, \frac{c}{2} x^2\right)
\end{equation}
where $\Gamma\left(-1+\frac{a}{2}, \...
- 787
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votes
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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
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How do I efficiently generate, store and read in Gaunt(-like) coefficients from a file?
I'm calculating the mode-mixing between various spin-weighted spherical harmonics (SWSHs) $_sY_{l}^{m}(\theta,\varphi)$. In what follows the indices can take the following values, $l_{max}$ being some ...
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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^{...
- 568
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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 ...
- 667
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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 ...
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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
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1
answer
118
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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
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2
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259
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Solving ODE with DiracDelta on the right-hand side
I want to use Mathematica to validate the solution to this differential equation
$ (E+ \frac{\hbar^2}{2m} \frac{d^2}{dx^2}-\kappa x)G(x,y) = i \hbar \delta(x-y) $
The solution is supposed to be
$$ ...
2
votes
1
answer
127
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How to force Mathematica to evaulate some values of LerchPhi function?
Mathematica cannot give LerchPhi[1,0,1]=-1/2, LerchPhi[1,1,1]=0, and LerchPhi[1,1,1/2]=0, as ...
- 667
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A hypergeometric series function
Maple and Mathematica are very efficient to find closed form of a hypergeometric function as for example:
$$\sum _{k=0}^{\infty } \frac{(-4)^k z^k \Gamma \left(1 k+\frac{1}{6}\right) \Gamma \left(\...
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