Fungrim entry: 78bb08

\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:

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| \lt 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| \lt 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
`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("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)))))

Topics using this entry

General analytic functions