# Fungrim entry: 18873d

$\operatorname{Im}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \sin\!\left(c \theta + d \log(M)\right)\; \text{ where } M = \left|a + b i\right|,\;\theta = \arg\!\left(a + b i\right)$
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(M)\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$\operatorname{Im}(z)$ Imaginary part
Pow${a}^{b}$ Power
ConstI$i$ Imaginary unit
Exp${e}^{z}$ Exponential function
Sin$\sin(z)$ Sine
Log$\log(z)$ Natural logarithm
Abs$\left|z\right|$ Absolute value
Arg$\arg(z)$ Complex argument
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("18873d"),
Assumptions(And(Element(a, RR), Element(b, RR), Element(c, RR), Element(d, RR), NotEqual(Add(a, Mul(b, ConstI)), 0))))