Fungrim entry: b6582a

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

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; R > 0 \;\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
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("b6582a"),
Formula(Where(LessEqual(Abs(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k))), Div(C, Pow(R, k))), Equal(C, Supremum(Abs(f(t)), For(t), And(Element(t, CC), Equal(Abs(Sub(t, z)), R)))))),
Variables(f, z, k, R),
Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)), Element(R, RR), Greater(R, 0), IsHolomorphic(f(t), ForElement(t, Subset(ClosedDisk(z, R)))))))

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

2020-03-29 16:01:42.585089 UTC