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Fungrim entry: 78bb08

f ⁣(z+x)k=0N1f(k)(z)k!xkCDN1D   where C=suptC,tz=Rf ⁣(t),D=xR\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\!\left(t\right)\right|,\,D = \frac{\left|x\right|}{R}
Assumptions:zCandxCandNZ1andRRandx<RandClosedDisk ⁣(z,R)HolomorphicDomain ⁣(f ⁣(z),z,C)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}}\, \operatorname{ClosedDisk}\!\left(z, R\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\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\!\left(t\right)\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}}\, \operatorname{ClosedDisk}\!\left(z, R\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\right)
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
Factorialn!n ! Factorial
Powab{a}^{b} Power
SupremumsupP(x)f ⁣(x)\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right) 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
Source code for this entry:
Entry(ID("78bb08"),
    Formula(Where(LessEqual(Abs(Sub(f(Add(z, x)), Sum(Mul(Div(Derivative(f(z), Tuple(z, z, k)), Factorial(k)), Pow(x, k)), Tuple(k, 0, Sub(N, 1))))), Div(Mul(C, Pow(D, N)), Sub(1, D))), Equal(C, Supremum(Abs(f(t)), 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), Subset(ClosedDisk(z, R), HolomorphicDomain(f(z), z, CC)))))

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

2019-08-21 11:44:15.926409 UTC