Fungrim home page

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(t)\right|,\;D = \frac{\left|x\right|}{R}
Assumptions:zC  and  xC  and  NZ1  and  RR  and  x<R  and  f(t) is holomorphic on tClosedDisk ⁣(z,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)
\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)
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Sumnf(n)\sum_{n} f(n) Sum
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:
    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