List in the function exponent meaning

Question

I get a weird exponential inside the function that appears as the result of an integral in Mathematica. Namely, the function

$Hypergeometric1F1^{(1,0,0)}[1, 1, -x^2]$

has the 3-component list in the exponential. I have searched everywhere to find the explanation of the $(1,0,0)$ exponent, but couldn't find it. Does anyone know its meaning?

It seems that number of terms in the exponent must match the number of arguments.

Thanks in advance.

Raymond Manzoni · Answer 1 · 2018-03-21T14:08:42.037

1

$\operatorname{Hypergeometric1F1}^{(1,0,0)}[1, 1, -x^2]\;$ is the partial derivative of $\;{}_1F_1(x_1;x_2;x_3)\,$ relatively to the first parameter : $\quad\displaystyle\frac{\partial }{\partial x_1}\; {}_1F_1(x_1;x_2;x_3)\;$ taken at $\;x_1=1,\,x_2=1,\,x_3=-x^2$.

$\operatorname{Hypergeometric1F1}^{(0,2,0)}[1, 1, -x^2]\;$ means $\;\;\left.\displaystyle\frac{\partial^2 }{\partial (x_2)^2}\;{}_1F_1(x_1;x_2;x_3)\right|_{x_1=1,\,x_2=1,\,x_3=-x^2}$

and so on...

The same notation is used for other multi-parameters functions.

Wolfram's Derivative operator $D$ is described here with some examples including :

Input : $\quad D[g[x, y], y]$
Output : $\quad g^{(0,1)}[x,y]$

edited Mar 21 '18 at 14:08

answered Mar 21 '18 at 13:51

Raymond Manzoni

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Thanks a lot! Couldn't find it anywhere really. – Mihailo Mar 22 '18 at 16:41
Glad it helped @Mihailo! This rather convenient notation was probably inspired by the classical $;f^{(1)}(x)=f'(x),;f^{(2)}(x)=f''(x),\cdots,$ which can be generalized to multiple integrals as in this paper or even to fractional derivatives. Excellent continuation, – Raymond Manzoni Mar 23 '18 at 23:08

List in the function exponent meaning

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