# Tag Info

## Hot answers tagged special-functions

7

Unfortunately, there is no single approach that will lead to robust, accurate, and high-performance implementations across the large universe of special functions. Often, two or more methods must be used for different parts of the input domain, and the necessary research and implementation work may take weeks for elementary functions and months for higher ...

3

$F_{2l}(x)$ can be calculated in closed form for $\text{Im}\ x<0$ using the integral from Gradsteyn and Ryzhik $$\int_0^1u^\lambda P_{2l}(u)du=\frac{(-1)^l\Gamma\left(l-\frac{\lambda}{2}\right)\Gamma\left(\frac{1}{2}+\frac{\lambda}{2}\right)}{2\Gamma\left(-\frac{\lambda}{2}\right)\Gamma\left(l+\frac{3}{2}+\frac{\lambda}{2}\right)},\quad \text{Re}\ ... 2 Notice that: \Gamma(m) = \lim\limits_{n \to \infty} \frac{n! \; n^m}{m \times (m+1) \times \dots \times (m+n)} = \lim\limits_{n \to \infty} \frac{(m-1)! \; n! \; n^m}{(m + n)!} = (m - 1)! \times \lim\limits_{n \to \infty} \frac{n! \; n^m}{(m + n)!}. Now let's show that \lim\limits_{n \to \infty} \frac{n! \; n^m}{(m + n)!} = 1: \lim\limits_{n \to ... 2 I suspect there's a well defined special function related to this series. Is it so ? Not really. Basically, \text{Exl}(x)=-F'(1), where F(k)=\displaystyle\sum_{n\ge0}\frac{x^n}{n!^k}~.~ The only known values of F are F(0)=\dfrac1{1-x},~F(1)=e^x, and F(2)=I_0\big(2\sqrt x\big). See Bessel function for more information. Neither F, let alone ... 1 With some help from Mathematica:$$G_{2,3}^{3,1}\left(z\left| \begin{smallmatrix} 0,1 \\ 0,0,0 \\ \end{smallmatrix} \right.\right) = -2 z \, {_3F_3}\left(\begin{smallmatrix}1,1,1\\2,2,2\end{smallmatrix}\middle|\,z\right)-\frac{1}{2} \frac{\partial^2}{\partial t^2}\left.{_1F_1}\left(\begin{smallmatrix}t\\1\end{smallmatrix}\middle|\,z\right)\right|_{t=0}+ ...

1

One way to describe what makes translating sine waves special is the following: sine waves $f(t)$ have the property that the subspace of the space of functions spanned by their translates $f(t + t_0), t_0 \in \mathbb{R}$ is $2$-dimensional, spanned by $\cos t, \sin t$. What other functions have this property? (I'm not going to require periodicity yet; this ...

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