Complex Analysis/Exponentiation and square root

Introduction

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Exponentiation

Let ${\textstyle z\in \mathbb {C} }$ and we consider the exponentiation ${\textstyle z^{n}\in \mathbb {C} }$ of the complex number $z$ as repeated multiplication of $z$ . Among other things, the polar coordinates support for the geometric interpretation of the operation "exponentiation".

Natural exponents - Polar coordinates

For natural numbers ${\textstyle n}$ the exponent ${\textstyle n}$ in the polar form is calculated ${\textstyle z=r\mathrm {e} ^{\mathrm {i} \varphi }}$

z^{n}=r^{n}\cdot \mathrm {e} ^{\mathrm {i} n\varphi }=r^{n}\cdot (\cos n\varphi +\mathrm {i} \cdot \sin n\varphi )

(see also De Moivre's Theorem)

Natural exponents - algebraic representation

For algebraic form ${\textstyle z=a+b\,\mathrm {i} }$ using the binomial law

z^{n}=\sum _{k=0, \atop k{\text{ gerade}}}^{n}{\binom {n}{k}}(-1)^{\frac {k}{2}}a^{n-k}b^{k}+\mathrm {i} \sum _{k=1, \atop k{\text{ ungerade}}}^{n}{\binom {n}{k}}(-1)^{\frac {k-1}{2}}a^{n-k}b^{k}.

Roots of a complex number

The roots can be represented in the following form:

{\textstyle z_{k}={\sqrt[{n}]{|z|}}\cdot \exp \left({\frac {\mathrm {i} \varphi }{n}}+k\cdot {\frac {2\pi \mathrm {i} }{n}}\right)\quad (k=0,1,\dots ,n-1)}

Remark to roots

The exponentiation of the expression ${\textstyle k\cdot {\frac {2\pi \mathrm {i} }{n}}}$ generates a multiple of ${\textstyle 2\pi }$ . The Term ${\textstyle {\frac {\mathrm {i} \varphi }{n}}}$ generates exactly the desired angle of ${\textstyle n}$ injection of ${\textstyle \varphi }$ - see also roots of complex numbers.

Logarithms

The complex natural logarithm is ambiguous (other than the logarith in the real values). A complex number ${\textstyle w}$ is called logarithm of the complex number ${\textstyle z}$

\mathrm {e} ^{w}=z.

Periodicity of the exponential function

With ${\textstyle w}$ being the logarithm of $z$ , each number ${\textstyle w+2m\pi \mathrm {i} }$ with any ${\textstyle m\in \mathbb {Z} }$ is also a logarithm of $z$ . It is therefore possible to work with Branch of the Logarithm, i.e. with values of a specific area of the complex plane.

Main branch of logarithm

The main branch of the natural logarithm of the complex number

z=re^{\mathrm {i} \phi }

with ${\textstyle r>0}$ and ${\textstyle -\pi <\phi \leq \pi }$ is

\ln z=\ln r+\mathrm {i} \phi .

Note - Main branch

The main branch of the natural logarithm of the complex number ${\textstyle z}$ is

\ln z=\ln |z|+\mathrm {i} \,\operatorname {Arg} (z),

where ${\textstyle \operatorname {Arg} (z)}$ is the main branch of the Arguments of ${\textstyle z}$ .

The finite subgroups

All elements of a finite subgroup of the multiplicative group of units ${\textstyle \mathbb {C} ^{\times }=\mathbb {C} \setminus \{0\}}$ are roots of unity. Among all order of element in group theory is maximum natural number, for example ${\textstyle n\in \mathbb {N} }$ . Since ${\textstyle \mathbb {C} }$ is commutative, an element with this maximum order then also generates the group, so that the group is cyclic and is exactly generated by the elements

\exp \left({2\pi \mathrm {i} k \over n}\right),\quad k=0,1,\dotsc ,n-1

there. All elements are located on the unit circle.

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