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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:zCandxCandf(t) is holomorphic on tClosedDisk ⁣(z,x)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
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
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
Absz\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)))))))

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

2019-10-05 13:11:19.856591 UTC