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Fungrim entry: 77d6bf

ea+bi=ea(cos ⁣(b)+sin ⁣(b)i){e}^{a + b i} = {e}^{a} \left(\cos\!\left(b\right) + \sin\!\left(b\right) i\right)
Assumptions:aCandbCa \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}
TeX:
{e}^{a + b i} = {e}^{a} \left(\cos\!\left(b\right) + \sin\!\left(b\right) i\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
Sinsin ⁣(z)\sin\!\left(z\right) Sine
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("77d6bf"),
    Formula(Equal(Exp(Add(a, Mul(b, ConstI))), Mul(Exp(a), Add(Cos(b), Mul(Sin(b), ConstI))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

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

2019-08-21 11:44:15.926409 UTC