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Fungrim entry: b6582a

f(k)(z)k!CRk   where C=suptC,tz=Rf(t)\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:zCandkZ0andRRandR>0andf(t) is holomorphic on tClosedDisk ⁣(z,R)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
Absz\left|z\right| Absolute value
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Factorialn!n ! Factorial
Powab{a}^{b} Power
SupremumsupxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
RRR\mathbb{R} Real numbers
IsHolomorphicf(z) is holomorphic at z=cf(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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2019-10-05 13:11:19.856591 UTC