Complex Numbers and Complex Exponentials

Definition A complex number is an expression $z = a + bi$ with $a$ and $b$ real numbers, where $i$ is a symbol with $i^2 = -1$. Two complex numbers are equal when both their parts are equal. The real number $a$ is the real part, written $\operatorname{Re} z$, and the real number $b$ is the imaginary part, written $\operatorname{Im} z$. The set of all complex numbers is $\mathbb{C}$, and the real numbers are the complex numbers with $b = 0$.

Definition The complex plane identifies $z = a + bi$ with the point $(a, b)$ of $\mathbb{R}^2$, or with the vector $\langle a, b \rangle$ from the origin. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

Definition For $z = a + bi$ and $w = c + di$,

The product rule comes from expanding and using $i^2 = -1$. With these operations $\mathbb{C}$ obeys the same commutative, associative and distributive laws as $\mathbb{R}$, and every $w \ne 0$ has a reciprocal, so division by nonzero numbers is defined. Unlike $\mathbb{R}$, there is no ordering of $\mathbb{C}$ compatible with arithmetic, so inequalities like $z < w$ have no meaning.

Definition The conjugate of $z = a + bi$ is $\overline{z} = a - bi$. For all $z$ and $w$,

and $z \overline{z} = a^2 + b^2$, a nonnegative real number that is zero only for $z = 0$. Also $\operatorname{Re} z = \frac{1}{2}(z + \overline{z})$, and $z$ is real exactly when $\overline{z} = z$. For $w \ne 0$, division is done by multiplying top and bottom by $\overline{w}$,

Definition The modulus of $z = a + bi$ is

the distance from $0$ to $z$ in the complex plane. It satisfies $\lvert zw \rvert = \lvert z \rvert \lvert w \rvert$. For $z \ne 0$, an argument of $z$ is any real $\theta$, in radians, with

Arguments of the same $z$ differ by integer multiples of $2\pi$. The principal argument $\operatorname{Arg} z$ is the one in $(-\pi, \pi]$. The number $0$ has no argument.

Definition Every $z \ne 0$ can be written in polar form

with $\theta$ an argument of $z$. By the angle addition formulas, if $w = s(\cos \varphi + i \sin \varphi)$ then

and by induction $z^n = r^n(\cos n\theta + i \sin n\theta)$ for every positive integer $n$. The same holds for $n = 0$ and for negative $n$, because $1/z = r^{-1}(\cos \theta - i \sin \theta)$. For a positive integer $n$, a number $w = R(\cos \varphi + i \sin \varphi)$ with $R > 0$ has exactly $n$ distinct $n$th roots,

where $R^{1/n}$ is the positive real root.

Definition For a complex $z$ define

The series converges absolutely for every $z$, by comparison with the real series for $e^{\lvert z \rvert}$. For real $\theta$, the powers of $i\theta$ cycle through $1, i, -1, -i$ times $\theta^n$, and absolute convergence allows the even and odd terms to be collected separately,

This is Euler’s formula, with $\theta$ in radians.

Definition The series above satisfies $e^{z + w} = e^z e^w$ for all complex $z$ and $w$. This is a standard fact, proved by multiplying the two absolutely convergent series term by term, and here it is used without proof. Combined with Euler’s formula, for real $a$ and $b$,

So $\lvert e^{a + bi} \rvert = e^a > 0$, which means $e^z$ is never $0$, and $e^{z + 2\pi i} = e^z$. For a complex constant $r$ and a real variable $t$, differentiating real and imaginary parts separately gives

Definition Let $p$ and $q$ be real constants and let $z(t) = u(t) + i v(t)$ be a complex valued solution of $y'' + p y' + q y = 0$ on an interval, with $u$ and $v$ real valued. Then $u$ and $v$ are each real solutions. The reason is that

and both brackets are real because $p$ and $q$ are, so both must be $0$. In particular, if $r = \alpha + i\beta$ with $\beta \ne 0$ solves $r^2 + p r + q = 0$, then $e^{\alpha t} \cos \beta t$ and $e^{\alpha t} \sin \beta t$ are real solutions.