15
votes
Accepted
Closed form of an integral using Mathematica or otherwise
We can substitute $\left\{~ u = 1 - xy,~ v = y ~\right\} $ using IntegrateChangeVariables to get a nice expression for the integral:
...
- 5,042
8
votes
Solving third order DE from fluid dynamics
One can observe that the original equation can be simply integrated twice w.r.t. $x$, this yields: $$y'(x)+y(x)^2-c_2\; x+c_1=0 $$ We could choose arbitrarily signs of constants, now solving it ...
- 57.7k
7
votes
Accepted
An integral using Mathematica or otherwise
Using:
$$\sum _{\alpha =0}^{-A} \binom{-A}{\alpha } (u v w x y)^{\alpha } (1-w)^{-A-\alpha }=(1-w+u v w x y)^{-A}$$
where: A=2 Then:
...
- 14.9k
6
votes
Binomial[-1,-1]
Version 14.1 introduced PascalBinomial, which preserves Pascal's identity for all integer values, and uses a different definition for negative integer $n$ than <...
- 30.6k
6
votes
Accepted
Quantum Field Regularization: Choice between ExpIntegralE and Gamma for Casimir Effect
$Version
(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)
Clear["Global`*"]
w = c Sqrt[kz^2 + q^2];
Use ...
- 161k
6
votes
6
votes
How to transform this combination of $ _2F_1 $?
It can be simplifed more.
Using undocumented command:
...
- 14.9k
5
votes
How to compile inverse error function?
You can port one of the C algorithms for erfinv to Mathematica.
Here, I've used @njuffa's code for erfinvf, which is a single-...
- 30.6k
5
votes
Accepted
What formula does Mathematica use for PolyGamma function of complex order?
For negative integer and negative noninteger Mathematica uses formula:
$$\psi ^{(n)}(x)=\frac{\int_0^x (x-t)^{-n-2} \text{log$\...
- 14.9k
5
votes
Using NDSolve on the Painlevé equations
By the method in this article A method for calculating the Painlevé transcendents, poles of second-order ODE can be passed by auxiliary functions u[x] and ...
- 2,200
5
votes
Accepted
How to compile inverse error function?
You could use OpenCLLink if you need to run this on a huge amount of data and want to use your GPU:
...
- 25.6k
5
votes
Accepted
5
votes
The conversion equation between BesselI and BesselJ
These are not same for all input.
e1 = BesselI[v, x]
e2 = BesselJ[v, I x]/I^v
Assuming[Element[v, Reals] && x > 0, FullSimplify[e1 - e2]]
(* 0 *)
You ...
- 149k
5
votes
Accepted
Are there any commands besides NSolve for solving equations which involve product logarithm?
The first equation only depends on d. Therefore, we solve it first (We need to rationalize to prevent numerical problems):
...
- 54.6k
5
votes
5
votes
Why do I get different results for the products of two identical expressions?
Bug in NProduct maybe. For that second one, rote conversion to logs, then NSum, then Exp ...
- 59.8k
5
votes
Issue in HypergeometricPFQ function:
It's NOT a strange behavior of HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {3/2, 1 - n}, -n]. HypergeometricPFQ works fine.
I made a ...
- 14.9k
5
votes
Accepted
How to verify a FactorialPower identity?
Another way is to use FindSequenceFunction
We start from
FactorialPower[x - i*k, n, m*k]
and compute a few values, say ...
- 16.4k
5
votes
How to port Matlab/Python's multivariate FoxH implementation in Mathematica?
Now it works. The result is consistent with Python/MATLAB.
Any help for accelerating the Mathematica code would be greatly appreciated.
...
- 1,902
5
votes
4
votes
4
votes
Can the Debye functions be implemented using built-in functions?
This only partially answers your question so I'll delete if it is not what you're looking for.
The first type of functions, $D^{(1)}_n (x)$ satisfy the differential equation
$$D^{'(1)}_n (x) = \frac{x^...
- 5,042
4
votes
Numerical integration of MeijerG function in the form MeijerG[{{-2,-8.5,-3.5},{}},{{0},{-1.5}},theta]
You can just use Integrate to evaluate the integral by adding the option PrincipalValue -> True, like
...
- 2,200
4
votes
Accepted
4
votes
Mathematica cannot solve this complicated integration
This integral can be expressed by Leaky Aquifer function or Generalization of the Incomplete Gamma function.
Let's consider:
...
- 14.9k
4
votes
Accepted
4
votes
Accepted
4
votes
Is it possible to express $\text{Li}_2(-\frac18(-i+\sqrt{15})+\text{Li}_2(\frac18 (i+\sqrt{15})$ as an explicit real expression (not numeric)?
The expression can be expressed as infinite sum. All the sums are pretty much the same only in different form and all produce the same terms/summands.
...
- 20.9k
4
votes
Accepted
Is it possible to have the asymptotics of this function?
The following proves your expectations.
...
- 27.7k
4
votes
Is it possible to have the asymptotics of this function?
Another way to get @user64494 result:
...
- 6,767
Only top scored, non community-wiki answers of a minimum length are eligible