Fungrim entry: 18873d

\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:

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

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
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Sin`	$\sin\!\left(z\right)$	Sine
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg\!\left(z\right)$	Complex argument
`RR`	$\mathbb{R}$	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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