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}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, \left|x\right|\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}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("1b1ec5"),
    Formula(Equal(f(Add(z, x)), Sum(Mul(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k)), Pow(x, k)), For(k, 0, Infinity)))),
    Variables(f, z, x),
    Assumptions(And(Element(z, CC), Element(x, CC), IsHolomorphic(f(t), ForElement(t, ClosedDisk(z, Abs(x)))))))

Topics using this entry

General analytic functions