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Fungrim entry: 1b1ec5

f ⁣(z+x)=k=0f(k)(z)k!xkf\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}
Assumptions:zCandxCandClosedDisk ⁣(z,x)HolomorphicDomain ⁣(f ⁣(z),z,C)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
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
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\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)))))

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

2019-08-17 11:32:46.829430 UTC