§2.5 Elementary Functions

Functions of one complex variable that are formed by compositions, sums, products, and powers of finitely many functions of the following form are known as elementary functions:

1. Power functions including polynomials, rational functions, and their inverses.
2. Trigonometric functions, hyperbolic functions, and their inverses.
3. Exponential functions and their inverses.

Power functions are easily extendable to the complex plane by simply changing the real variable to a complex variable. The other two classes of functions have to be redefined and reinterpreted for the complex plane. It is well known that the exponential function can be expanded as

This is better written as

which is the famous Euler formula. Then for any complex number $𝑧=𝑥+\mathrm{i}𝑦$,

Then trigonometric functions and exponential functions can be written in terms of each other:

Hence, the following relationships are derived:

The complex logarithm, denoted $𝑤=log\left(𝑧\right)$, is the solution to $𝑧={\mathrm{e}}^{𝑤}$. We can then define the inverse trigonometric and hyperbolic functions.

We can also define the power function for non-integer powers with $𝑤={𝑧}^{𝛼}={\mathrm{e}}^{𝛼log\left(𝑧\right)}$. Then power functions can all be written in terms of exponential functions and logarithms. Letting $𝑥=\mathrm{\pi }$ in (2.5.1)$\text{<a href="../../pdf/notes.pdf\#loc-78">(2.5.1)</a>}\text{<a href="\#loc-1">(2.5.1)</a>}$ yields ${\mathrm{e}}^{\mathrm{i}\mathrm{\pi }}=-1$. Furthermore, we can see that exponentiation with an imaginary number is a rotation:

Since all elementary functions can be written in terms of exponential functions and complex logarithms, we will first study the exponential function.

1. The exponential function ${\mathrm{e}}^{𝑧}$ never vanishes as $\left|{\mathrm{e}}^{𝑧}\right|={\mathrm{e}}^{𝑥}>0$.
2. Since ${\mathrm{e}}^{2\mathrm{\pi }\mathrm{i}}=1$, it is periodic over $2\mathrm{\pi }\mathrm{i}$.

3. It is also an entire function with ${\left({\mathrm{e}}^{𝑧}\right)}^{\prime }={\mathrm{e}}^{𝑧}$.

   Write ${\mathrm{e}}^{𝑧}={\mathrm{e}}^{𝑥+\mathrm{i}𝑦}={\mathrm{e}}^{𝑥}\left(cos\left(𝑦\right)+\mathrm{i}sin\left(𝑦\right)\right)$ where $𝑥,𝑦\in ℝ$. Let $𝑢\left(𝑥,𝑦\right)=\Re 𝔢\left({\mathrm{e}}^{𝑧}\right)={\mathrm{e}}^{𝑥}cos\left(𝑦\right)$ and $𝑣\left(𝑥,𝑦\right)=\Im 𝔪\left({\mathrm{e}}^{𝑧}\right)={\mathrm{e}}^{𝑥}sin\left(𝑦\right)$. The first order derivatives are respectively

   $$\frac{𝜕𝑢}{𝜕𝑥}={\mathrm{e}}^{𝑥}cos\left(𝑦\right),\frac{𝜕𝑢}{𝜕𝑦}=-{\mathrm{e}}^{𝑥}sin\left(𝑦\right),$$

   and

   $$\frac{𝜕𝑣}{𝜕𝑥}={\mathrm{e}}^{𝑥}sin\left(𝑦\right),\frac{𝜕𝑣}{𝜕𝑦}={\mathrm{e}}^{𝑥}cos\left(𝑦\right),$$

   and indeed, the condition described by Theorem 2.2.3$\text{<a href="../../pdf/notes.pdf\#loc-52">Theorem 2.2.3</a>}\text{<a href="../complex-differentiation/index.html\#thm-theorem-2-2-3">Theorem 2.2.3</a>}$ is satisfied.

4. For any two complex numbers ${𝑧}_1$ and ${𝑧}_2$, ${\mathrm{e}}^{{𝑧}_1}{\mathrm{e}}^{{𝑧}_2}={\mathrm{e}}^{{𝑧}_1+{𝑧}_2}$.

   In fact, most real exponentiation rules are identical to those in the complex number field. Previously we claimed the periodic properties of ${\mathrm{e}}^{𝑧}$. For $𝑈\subseteq ℂ$, a holomorphic function $𝑓:𝑈\to ℂ$ is univalent over $𝑈$ if it is injective over $𝑈$. This means that the solutions ${𝑧}_1$ and ${𝑧}_2$ satisfying $𝑓\left({𝑧}_1\right)=𝑓\left({𝑧}_2\right)$ will also always satisfy ${𝑧}_1={𝑧}_2$.

5. The function ${\mathrm{e}}^{𝑧}$ is univalent over any horizontal strip of height $2\mathrm{\pi }$.

   Let ${𝑧}_1={𝑥}_1+\mathrm{i}{𝑦}_1$ and ${𝑧}_2={𝑥}_2+\mathrm{i}{𝑦}_2$, with ${𝑥}_1,{𝑦}_1,{𝑥}_2,{𝑦}_2\in ℝ$, and assume ${\mathrm{e}}^{{𝑧}_1}={\mathrm{e}}^{{𝑧}_2}$. Then

   $${\mathrm{e}}^{{𝑥}_1}{\mathrm{e}}^{\mathrm{i}{𝑦}_1}={\mathrm{e}}^{{𝑥}_2}{\mathrm{e}}^{\mathrm{i}{𝑦}_2}.$$

   The moduli are equal, and therefore ${𝑥}_1={𝑥}_2$. By the periodic nature of exponentiation of imaginary numbers, ${𝑦}_1-{𝑦}_2=2\mathrm{\pi }𝑘$, where $𝑘\in \mathbb{Z}$. To satisfy univalence over a region $𝑈$, we must exclude distinct points whose imaginary parts differ by an integer multiple of $2\mathrm{\pi }$. Thus, we may select $𝑈$ to be any horizontal strip

   $$2\mathrm{\pi }𝑘\le \Im 𝔪\left(𝑧\right)<2\mathrm{\pi }\left(𝑘+1\right)$$

   or

   $$2\mathrm{\pi }𝑘<\Im 𝔪\left(𝑧\right)\le 2\mathrm{\pi }\left(𝑘+1\right).$$

   Similar to the exponential function, any belt region with thickness $2\mathrm{\pi }$ is a region over which $log$ is univalent.

Next we examine the complex logarithm.

1. From the periodicity of $𝑧={\mathrm{e}}^{𝑤}$, $log$ is a multi-valued function.

2. Let $𝑧=𝑟{\mathrm{e}}^{\mathrm{i}𝜃}$ and $𝑤=𝑢+\mathrm{i}𝑣$, where $𝑟,𝜃,𝑢,𝑣\in ℝ$. Then

   $$𝑟{\mathrm{e}}^{\mathrm{i}𝜃}={\mathrm{e}}^{𝑢+\mathrm{i}𝑣},$$

   and ${\mathrm{e}}^{𝑢}=𝑟$, meaning that $𝑢=log\left(𝑟\right)$ and $𝑣=𝜃+2\mathrm{\pi }𝑘$, where $𝑘\in \mathbb{Z}$. Then

   $$𝑤=log\left(𝑟\right)+\mathrm{i}\left(𝜃+2\mathrm{\pi }𝑘\right),$$

   and using modulus-argument notation,

   $$log\left(𝑧\right)=log\left|𝑧\right|+\mathrm{i}arg\left(𝑧\right),$$

   where $arg\left(𝑧\right)$ is the multi-valued argument function. We denote the principal branch of the argument function by

   $$Arg:ℂ\setminus \left\{0\right\}\to \left(-\mathrm{\pi },\mathrm{\pi }\right].$$
   The principal branch of $log\left(𝑧\right)$, or $Log\left(𝑧\right)$, can be defined such that $\Im 𝔪\left(Log\left(𝑧\right)\right)\in \left(-\mathrm{\pi },\mathrm{\pi }\right]$.

The functions $sin$ and $cos$, through their exponential form, still satisfy the familiar properties such as their derivatives, periodicity of $2\mathrm{\pi }$, parity, sum and difference formulas, and the fundamental identities

However, due to the extension, some properties do not hold. For instance, $sin\left(𝑧\right)$ and $cos\left(𝑧\right)$ are unbounded, as along the imaginary axis they resemble their hyperbolic counterparts, which are unbounded along the real line.

We now examine the regions over which they are univalent. Consider

Define the auxiliary functions

Then

$𝜉$ is clearly univalent on $ℂ$, as it is a linear map, specifically, a rotation by $\frac{\mathrm{\pi }}{2}$ radians followed by scaling. The function $𝜁$ is univalent on any domain $𝑈\subseteq ℂ$ such that for all ${𝜉}_1,{𝜉}_2\in 𝑈$, ${𝜉}_1-{𝜉}_2\ne 2\mathrm{\pi }\mathrm{i}𝑘$ for any $𝑘\in \mathbb{Z}$. If ${𝜉}_1=\mathrm{i}{𝑧}_1$ and ${𝜉}_2=\mathrm{i}{𝑧}_2$, then this translates to ${𝑧}_1-{𝑧}_2\ne 2\mathrm{\pi }𝑘$ for $𝑘\in \mathbb{Z}$. The function

is univalent on regions excluding pairs $\left({𝜁}_1,{𝜁}_2\right)$ such that ${𝜁}_1=\frac{1}{{𝜁}_2}$. In terms of $𝑧$, this condition becomes ${\mathrm{e}}^{\mathrm{i}{𝑧}_1}{\mathrm{e}}^{\mathrm{i}{𝑧}_2}\ne 1$, or equivalently, ${𝑧}_1+{𝑧}_2\ne 2\mathrm{\pi }𝑘$ for any $𝑘\in \mathbb{Z}$.

Combining these constraints, we conclude that $cos\left(𝑧\right)$ is univalent on any vertical strip in the complex plane of width $\mathrm{\pi }$, such as a region of the form

Let us now consider the specific region

and analyze how it is mapped under $cos\left(𝑧\right)$.

1. $𝜉\left(𝑧\right)=\mathrm{i}𝑧$ maps the region $\left\{𝑧\in ℂ:0<\Re 𝔢\left(𝑧\right)<\mathrm{\pi }\right\}$ to $\left\{𝜉\in ℂ:0<\Im 𝔪\left(𝜉\right)<\mathrm{\pi }\right\}$.
2. $𝜁\left(𝜉\right)={\mathrm{e}}^{𝜉}$ maps this region to the upper half-plane $\Im 𝔪\left(𝜁\right)>0$ since $0<Arg\left(𝜁\right)<\mathrm{\pi }$ and $0<\left|𝜁\right|$.
3. $𝑤\left(𝜁\right)=\frac{𝜁+\frac{1}{𝜁}}{2}$ maps $\Im 𝔪\left(𝜁\right)>0$ to $ℂ\setminus \left(\left(-\infty ,-1\right]\cup \left[1,\infty \right)\right)$.

Thus, the composition $cos\left(𝑧\right)=𝑤\circ 𝜁\circ 𝜉$ is univalent on the strip

and the image of this strip under $cos$ is

We will now analyze the inverse cosine function, denoted $arccos\left(𝑧\right)$. Consider

Then

Then $arccos$ is also a multi-valued function. We can also define

Lastly, we will examine the power function. Let $𝛼=𝑢+\mathrm{i}𝑣$ where $𝑢,𝑣\in ℝ$. Then

and in polar form,

Let

and

Then ${𝑧}^{𝛼}={𝑟}_{𝑘}{\mathrm{e}}^{\mathrm{i}{𝜃}_{𝑘}}$, where $𝑘\in \mathbb{Z}$. Analyzing the coefficient of $𝑣$ in the exponent of ${𝑟}_{𝑘}$, ${𝑧}^{𝛼}$ is multi-valued if $𝑣\ne 0$.

Then assuming $𝑣=0$, we have

Doing casework on $𝛼$,

1. If $𝛼=𝑢\in \mathbb{Z}$, then $𝑢$ can be absorbed into $𝑘$, and ${𝑧}^{𝛼}$ is single-valued.

2. If $𝛼=𝑢\in \mathbb{Q}$ with reduced fractional form $\frac{𝑝}{𝑞}$, where $𝑝,𝑞\in \mathbb{Z}$, $𝑞>0$, and $gcd\left(𝑝,𝑞\right)=1$, then the multi-valued function ${𝑧}^{𝛼}$ is given by

   $$\begin{array}{cc}{𝑧}^{𝛼}& ={\left|𝑧\right|}^{\frac{𝑝}{𝑞}}exp\left(\mathrm{i}\frac{𝑝}{𝑞}\left(Arg\left(𝑧\right)+2\mathrm{\pi }𝑘\right)\right)\\ & ={\left|𝑧\right|}^{\frac{𝑝}{𝑞}}exp\left(\mathrm{i}\frac{𝑝}{𝑞}Arg\left(𝑧\right)\right)exp\left(2\mathrm{\pi }\mathrm{i}\frac{𝑝}{𝑞}𝑘\right),\end{array}$$

   for $𝑘\in \mathbb{Z}$. These values are periodic with period $𝑞$, since

   $$exp\left(2\mathrm{\pi }\mathrm{i}\frac{𝑝}{𝑞}\left(𝑘+𝑞\right)\right)=exp\left(2\mathrm{\pi }\mathrm{i}\frac{𝑝}{𝑞}𝑘\right),$$

   as $exp\left(2\mathrm{\pi }\mathrm{i}𝑝\right)=1$ for integer $𝑝$. To prove there are exactly $𝑞$ distinct values, consider $𝑘=0,1,2,\dots ,𝑞-1$. The exponential factors are $exp\left(2\mathrm{\pi }\mathrm{i}\left(\frac{𝑝}{𝑞}\right)𝑘\right)$. These are distinct if, for $0\le 𝑗<𝑘\le 𝑞-1$,

   $$exp\left(2\mathrm{\pi }\mathrm{i}\frac{𝑝}{𝑞}𝑗\right)\ne exp\left(2\mathrm{\pi }\mathrm{i}\frac{𝑝}{𝑞}𝑘\right),$$
   which holds unless $\left(\frac{𝑝}{𝑞}\right)\left(𝑘-𝑗\right)\in \mathbb{Z}$, or equivalently unless $𝑞$ divides $𝑝\left(𝑘-𝑗\right)$. Since $gcd\left(𝑝,𝑞\right)=1$, $𝑞$ must divide $𝑘-𝑗$, but $\left|𝑘-𝑗\right|<𝑞$ and $𝑘-𝑗\ne 0$, a contradiction. Thus, ${𝑧}^{𝛼}$ has exactly $𝑞$ distinct values.
3. If $𝛼=𝑢\in ℝ\setminus \mathbb{Q}$, then ${𝑧}^{𝛼}$ is infinite-valued.

Lastly, there exist series representations of functions using power functions, namely Taylor series, and trigonometric functions, namely Fourier series. There does not exist another distinct representation using exponential functions, as trigonometric functions can be written in terms of them.