Complex Analysis/Exponentiation and square root

Introduction

This page about Complex_Analysis/Exponentiation_and_square_root can be displayed as Wiki2Reveal slides. Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides. The following aspects of Exponentiation and roots are considered in detail:

• Exponentiation,
• Square root (Radieren) and
• Logarithm

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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