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Fungrim entry: 18873d

Im ⁣((a+bi)c+di)=Mcedθsin ⁣(cθ+dlog ⁣(M))   where M=a+bi,θ=arg ⁣(a+bi)\operatorname{Im}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \sin\!\left(c \theta + d \log\!\left(M\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)
Assumptions:aRandbRandcRanddRanda+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:
\operatorname{Im}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \sin\!\left(c \theta + d \log\!\left(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
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
Powab{a}^{b} Power
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
Absz\left|z\right| Absolute value
Argarg ⁣(z)\arg\!\left(z\right) Complex argument
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("18873d"),
    Formula(Equal(Im(Pow(Add(a, Mul(b, ConstI)), Add(c, Mul(d, ConstI)))), Where(Mul(Mul(Pow(M, c), Exp(Neg(Mul(d, theta)))), 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), Unequal(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.

2019-08-17 11:32:46.829430 UTC