Fungrim entry: 1b1ec5

f\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\right)

TeX:

f\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right) \subset \operatorname{HolomorphicDomain}\!\left(f\!\left(z\right), z, \mathbb{C}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("1b1ec5"),
    Formula(Equal(f(Add(z, x)), Sum(Mul(Div(Derivative(f(z), Tuple(z, z, k)), Factorial(k)), Pow(x, k)), Tuple(k, 0, Infinity)))),
    Variables(f, z, x),
    Assumptions(And(Element(z, CC), Element(x, CC), Subset(ClosedDisk(z, Abs(x)), HolomorphicDomain(f(z), z, CC)))))

Topics using this entry

General analytic functions