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\!\left(t\right)\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 \gt 0 \,\mathbin{\operatorname{and}}\, \operatorname{ClosedDisk}\!\left(z, R\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\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\!\left(t\right)\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 \gt 0 \,\mathbin{\operatorname{and}}\, \operatorname{ClosedDisk}\!\left(z, R\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)$	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

Source code for this entry:

Entry(ID("b6582a"),
    Formula(Where(LessEqual(Abs(Div(Derivative(f(z), Tuple(z, z, k)), Factorial(k))), Div(C, Pow(R, k))), Equal(C, Supremum(Abs(f(t)), 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), Subset(ClosedDisk(z, R), HolomorphicDomain(f(z), z, CC)))))

Topics using this entry

General analytic functions