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Fungrim entry: 0aac97

(a+bi)c+di=Mcedθ(cos ⁣(cθ+dlog(M))+isin ⁣(cθ+dlog(M)))   where M=a+bi,  θ=arg ⁣(a+bi){\left(a + b i\right)}^{c + d i} = {M}^{c} {e}^{-d \theta} \left(\cos\!\left(c \theta + d \log(M)\right) + i \sin\!\left(c \theta + d \log(M)\right)\right)\; \text{ where } M = \left|a + b i\right|,\;\theta = \arg\!\left(a + b i\right)
Assumptions:aR  and  bR  and  cR  and  dR  and  a+bi0a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; c \in \mathbb{R} \;\mathbin{\operatorname{and}}\; d \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a + b i \ne 0
TeX:
{\left(a + b i\right)}^{c + d i} = {M}^{c} {e}^{-d \theta} \left(\cos\!\left(c \theta + d \log(M)\right) + i \sin\!\left(c \theta + d \log(M)\right)\right)\; \text{ where } M = \left|a + b i\right|,\;\theta = \arg\!\left(a + b i\right)

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; c \in \mathbb{R} \;\mathbin{\operatorname{and}}\; d \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a + b i \ne 0
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Coscos(z)\cos(z) Cosine
Loglog(z)\log(z) Natural logarithm
Sinsin(z)\sin(z) Sine
Absz\left|z\right| Absolute value
Argarg(z)\arg(z) Complex argument
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("0aac97"),
    Formula(Equal(Pow(Add(a, Mul(b, ConstI)), Add(c, Mul(d, ConstI))), Where(Mul(Mul(Pow(M, c), Exp(Neg(Mul(d, theta)))), Add(Cos(Add(Mul(c, theta), Mul(d, Log(M)))), Mul(ConstI, Sin(Add(Mul(c, theta), Mul(d, Log(M))))))), Equal(M, Abs(Add(a, Mul(b, ConstI)))), Equal(theta, Arg(Add(a, Mul(b, ConstI))))))),
    Variables(a, b, c, d),
    Assumptions(And(Element(a, RR), Element(b, RR), Element(c, RR), Element(d, RR), NotEqual(Add(a, Mul(b, ConstI)), 0))))

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

2020-04-08 16:14:44.404316 UTC