4.1: Complex Algebra
- Page ID
- 34533
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.