# Fungrim entry: 78bb08

$\left|f\!\left(z + x\right) - \sum_{k=0}^{N - 1} \frac{{f}^{(k)}(z)}{k !} {x}^{k}\right| \le \frac{C {D}^{N}}{1 - D}\; \text{ where } C = \mathop{\operatorname{sup}}\limits_{t \in \mathbb{C},\,\left|t - z\right| = R} \left|f(t)\right|,\;D = \frac{\left|x\right|}{R}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| < R \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, R\right)$
TeX:
\left|f\!\left(z + x\right) - \sum_{k=0}^{N - 1} \frac{{f}^{(k)}(z)}{k !} {x}^{k}\right| \le \frac{C {D}^{N}}{1 - D}\; \text{ where } C = \mathop{\operatorname{sup}}\limits_{t \in \mathbb{C},\,\left|t - z\right| = R} \left|f(t)\right|,\;D = \frac{\left|x\right|}{R}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| < R \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, R\right)
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Sum$\sum_{n} f(n)$ Sum
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Factorial$n !$ Factorial
Pow${a}^{b}$ Power
Supremum$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$ Supremum of a set or function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
RR$\mathbb{R}$ Real numbers
IsHolomorphic$f(z) \text{ is holomorphic at } z = c$ Holomorphic predicate
Source code for this entry:
Entry(ID("78bb08"),
Formula(Where(LessEqual(Abs(Sub(f(Add(z, x)), Sum(Mul(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k)), Pow(x, k)), For(k, 0, Sub(N, 1))))), Div(Mul(C, Pow(D, N)), Sub(1, D))), Equal(C, Supremum(Abs(f(t)), For(t), And(Element(t, CC), Equal(Abs(Sub(t, z)), R)))), Equal(D, Div(Abs(x), R)))),
Variables(f, z, x, N, R),
Assumptions(And(Element(z, CC), Element(x, CC), Element(N, ZZGreaterEqual(1)), Element(R, RR), Less(Abs(x), R), IsHolomorphic(f(t), ForElement(t, Subset(ClosedDisk(z, R)))))))

## 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