Complex numbers as ordered pairs: How to define exponentiation?

Question

Forgive me if this question has already been asked, however, I could not find it.

In Section 1.8 (Complex Numbers and Functions) of their book, "Mathematical Methods for Physicists", by Arfken, Harris, Weber, Jurgen, the authors state that "A complex number is nothing more than an ordered pair of two real number" and later on, they define $i = (0,1)$ and state, "... introduction of the symbol i is not necessary here, but it is convenient"

I was wondering, in the same spirit of defining complex numbers as ordered pair of real numbers, how can we define exponentiation of a number to an ordered pair?

How would we define $e^{(1,0)}$, which is the real number $e^1$ and is equal to $e$. How would we define $e^{(0,1)} = e^i$?

Answer 1

Since we have defined addition and multiplication for complex numbers, we can form powers and write series expansions. $$e^z=1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots $$ In a typical class of real analysis, this will also be the definition of $\exp(x)$. Then we define $e=\exp(1)$, and prove $\exp(x)= e^x$.

Answer 2

In the following, I will try to outline how the definition of a complex exponential function in terms of pairs of real numbers makes sense, but not how to calculate it explicitly.

Before asking ourselves what the definition of the complex exponential function is it can be useful to recall one common way of defining the exponential function $\exp$ over the reals as a series: $$ \label{series}\tag{1} \exp:\mathbb{R}\to\mathbb{R}_+,\ x\mapsto \sum_{k=0}^{\infty} \frac{x^k}{k!} $$ The right hand side of the equation is a series, i. e. the limit of a sequence of partial sums (in this case: $\lim_{n\to\infty}\sum_{k=0}^{n} \frac{x^k}{k!}$). In order for a limiting procedure to make sense, we need at least the notion of a norm ($||\cdot||$). Check out the link for more details. Over the reals, the norm is just the absolute value: $||x||=|x|$ for any $x\in\mathbb{R}$. Lastly, in the above definition, raising a real number $x$ to the $k$-th power is just multiplying $x$ exactly $k$ times itself.

Now we can try to figure out an appropriate way of extending this concept to the complex numbers. The authors of your book correctly state that a complex number is nothing more than a pair of real numbers, however, this can be a bit misleading as the field of complex numbers $\mathbb{C}$ is also equipped with an addition as well as a multiplication: Letting $a, b\in\mathbb{C}$ with $a = (a_1, a_2), b = (b_1, b_2)$, where $a_1,a_2,b_1,b_2\in\mathbb{R}$, we can define \begin{align} \label{addition}\tag{2a} a + b &:= (a_1+b_1, a_2 + b_2)\\\text{and}\quad \label{multiplication}\tag{2b} a \cdot b &:= (a_1b_1 - a_2b_2, a_2 b_1 + a_1 b_2). \end{align} Note that by definition of $\mathbb{C}$, the right hand side is well defined and lies in $\mathbb{C}$. In order for a limit to make sense, we can define a norm on the complex numbers as well: Letting $z=(z_1,z_2)\in\mathbb{C}$, we can define $$ \label{limit}\tag{3} ||z|| := \sqrt{z_1^2+z_2^2}. $$ It is not difficult to check that this definition satisfies the properties of a norm. Now that we have a notion of addition, multiplication and a limiting procedure (the latter can be defined through the norm), we can extend the definition $(\ref{series})$ as follows: $$ \exp:\mathbb{C}\to\mathbb{C},\ x\mapsto \sum_{k=0}^{\infty} \frac{x^k}{k!} $$ Again, the right hand side is defined to be a limit $(\ref{limit})$ of partial sums. The sums on the other hand are given as repeated addition $(\ref{addition})$ of complex numbers, i. e. pairs of real numbers. Lastly, raising a complex number $(a_1, a_2)$ to the $k$-th power can be defined as repeated multiplication as per $(\ref{multiplication})$.

As for the notation, one can define $e^x$ to be $\exp{x}$ as well as $e = \exp{(1, 0)}$. One can then continue to show that this notation of a power corresponds directly to the process of repeated multiplication.

From this definition one can continue to look at the properties (e. g. $\exp{(a+b)}=\exp{a}\cdot \exp{b}$ for any $a,b\in\mathbb{C}$) in order to actually calculate the exponential of a complex number, i. e. in your case figuring out the limit of $$ e^i = e^{(0, 1)} = \sum_{k=0}^{\infty} \frac{(0, 1)^k}{k!} = \lim_{n\to\infty} \sum_{k=0}^{n} \frac{(0, 1)^k}{k!}, $$ where $(0, 1)^k$ is just repeated application of $(\ref{multiplication})$.