# Fungrim entry: 1b1ec5

$f\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right)$
TeX:
f\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right)
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Factorial$n !$ Factorial
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
IsHolomorphic$f(z) \text{ is holomorphic at } z = c$ Holomorphic predicate
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("1b1ec5"),
Formula(Equal(f(Add(z, x)), Sum(Mul(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k)), Pow(x, k)), For(k, 0, Infinity)))),
Variables(f, z, x),
Assumptions(And(Element(z, CC), Element(x, CC), IsHolomorphic(f(t), ForElement(t, ClosedDisk(z, Abs(x)))))))

## 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