4.1: Complex Algebra

Any complex number \(z\) can be written as \[z = x + i y,\] where \(x\) and \(y\) are real numbers that are respectively called the real part and the imaginary part of \(z\). The real and imaginary parts are also denoted as \(\mathrm{Re}(z)\) and \(\mathrm{Im}(z)\), where \(\mathrm{Re}\) and \(\mathrm{Im}\) can be regarded as functions mapping a complex number to a real number.

The set of complex numbers is denoted by \(\mathbb{C}\). We can define algebraic operations on complex numbers (addition, subtraction, products, etc.) by following the usual rules of algebra and setting \(i^2 = -1\) whenever it shows up.

Example \(\PageIndex{1}\)

Let \(z = x + i y\), where \(x, y \in \mathbb{R}\).
What are the real and imaginary parts of \(z^2\)? \[\begin{align} z^2 &= (x+iy)^2 \\ &= x^2 + 2x(iy) + (iy)^2 \\ &= x^2 - y^2 + 2ixy \end{align}\] Hence, \[\mathrm{Re}(z^2) = x^2 -y^2, \;\;\; \mathrm{Im}(z^2) = 2xy.\]

We can also perform power operations on complex numbers, with one caveat: for now, we’ll only consider integer powers like \(z^2\) or \(z^{-1} = 1/z\). Non-integer powers, such as \(z^{1/3}\), introduce vexatious complications which we’ll postpone for now (we will figure out how to deal with them when studying branch points and branch cuts in Chapter 7).

Another useful fact: real coefficients (and only real coefficients) can be freely moved into or out of \(\textrm{Re}(\cdots)\) and \(\textrm{Im}(\cdots)\) operations: \[\left\{\begin{array}{l} \mathrm{Re}(\alpha z + \beta z') = \alpha \, \mathrm{Re}(z) + \beta\, \mathrm{Re}(z')\\ \mathrm{Im}(\alpha z + \beta z') = \alpha \, \mathrm{Im}(z) + \beta\, \mathrm{Im}(z')\end{array}\right.\qquad\mathrm{for}\;\alpha, \beta \in \mathbb{R}.\]

As a consequence, if we have a complex function of a real variable, the derivative of that function can be calculated from the derivatives of the real and imaginary parts, as shown in the following example:

Example \(\PageIndex{2}\)

If \(z(t)\) is a complex function of a real input \(t\), then \[\mathrm{Re}\left[\frac{dz}{dt}\right] = \frac{d}{dt} \mathrm{Re}\left[z(t)\right], \;\;\textrm{and}\;\;\; \mathrm{Im}\left[\frac{dz}{dt}\right] = \frac{d}{dt} \mathrm{Im}\left[z(t)\right].\] This can be proven using the definition of the derivative: \[\begin{align} \mathrm{Re}\left[\frac{dz}{dt}\right] &= \;\; \mathrm{Re}\left[\lim_{\delta t \rightarrow 0} \frac{z(t+\delta t) - z(t)}{\delta t}\right] \\ &= \lim_{\delta t \rightarrow 0} \left[\frac{\mathrm{Re}[z(t+\delta t)] - \mathrm{Re}[z(t)]}{\delta t}\right] \\ &= \frac{d}{dt} \mathrm{Re}\left[z(t)\right]. \end{align}\] The \(\mathrm{Im}[\cdots]\) case works out similarly. Note that the infinitesimal quantity \(\delta t\) is real; otherwise, this wouldn’t work.