Fungrim entry: 0aac97

{\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:

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:

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

Topics using this entry

Powers