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Fungrim entry: bc4d0a

(a+bi)c+di=Mcedθ   where M=a+bi,  θ=arg ⁣(a+bi)\left|{\left(a + b i\right)}^{c + d i}\right| = {M}^{c} {e}^{-d \theta}\; \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|{\left(a + b i\right)}^{c + d i}\right| = {M}^{c} {e}^{-d \theta}\; \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
Absz\left|z\right| Absolute value
Powab{a}^{b} Power
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Argarg(z)\arg(z) Complex argument
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("bc4d0a"),
    Formula(Equal(Abs(Pow(Add(a, Mul(b, ConstI)), Add(c, Mul(d, ConstI)))), Where(Mul(Pow(M, c), Exp(Neg(Mul(d, theta)))), 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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2020-04-08 16:14:44.404316 UTC