# Fungrim entry: 65c610

${e}^{x + y} = \sum_{k=0}^{\infty} \sum_{n=0}^{\infty} {n + k \choose k} \frac{{x}^{k} {y}^{n}}{\left(n + k\right)!}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}$
TeX:
{e}^{x + y} = \sum_{k=0}^{\infty} \sum_{n=0}^{\infty} {n + k \choose k} \frac{{x}^{k} {y}^{n}}{\left(n + k\right)!}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Exp${e}^{z}$ Exponential function
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("65c610"),
Formula(Equal(Exp(Add(x, y)), Sum(Sum(Mul(Binomial(Add(n, k), k), Div(Mul(Pow(x, k), Pow(y, n)), Factorial(Add(n, k)))), For(n, 0, Infinity)), For(k, 0, Infinity)))),
Variables(x, y),
Assumptions(And(Element(x, CC), Element(y, CC))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC