\(\)This introduces the functions with complex arguments. The article Complex Numbers introduced a 2-dimensional number space called the complex-plane (\(\mathbb{C}\)-plane). The arithmetic functions, that we studied since first grade, gracefully extend from the one-dimensional number line onto this new \(\mathbb{C}\)-plane. Here we will introduce functions that operate on these complex numbers.
\(j\)
We refer to the imaginary unit as “\(j\)”, to avoid confusion with electrical engineering, where the variable \(i\) is already used for current.
An overview of the functions is given for reference. We will proof the some of these functions in subsequent paragraphs.
Overview
Consider a complex number \(z\) expressed in either notation style
$$
z = x+jy=r\,(\cos\varphi+j\sin\varphi)=r\,\mathrm{e}^{j\varphi}\nonumber
$$
$$
\begin{align}
\sin z &= \sin x\cosh y + j\,\cos x\sinh y \\[6mu]
\cos z &= \cos x\cosh y + j\,\sin x\sinh y \\[6mu]
\tan z &= \frac{\sin(2 x)}{\cosh(2 y) + \cos(2 x)} + j\,\frac{\sinh(2 y)}{\cosh(2 y) + \cos (2 x)} \\[6mu]
\csc z &= {(\sin z)}^{-1} \\[6mu]
\sec z &= {(\cos z)}^{-1} \\[6mu]
\cot z &= {(\tan z)}^{-1} \\[6mu]
\end{align}
$$
Inverse circular based trigonometry
$$
\DeclareMathOperator{\asin}{asin}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\acos}{acos}
\DeclareMathOperator{\atan}{atan}
\DeclareMathOperator{\acsc}{acsc}
\DeclareMathOperator{\asec}{asec}
\DeclareMathOperator{\acot}{acot}
\begin{align}
\asin z &= \asin b +j\,\sgn(y)\ln\left(a + \sqrt{a^{\mathrm{e}}}-1\right), \quad a\geq1 \land b \text{ in } [\mathrm{rad}]\\[6mu]
\acos z &= \acos b +j \sgn(y) \ln\left(a + \sqrt{a^{\mathrm{e}}}-1\right),\quad a\geq1 \land b \text{ in } [\mathrm{rad}]\\[6mu]
\text{where}\quad
a &= \tfrac{1}{2} \left( \sqrt{(x +1)^{2} + y ^{2} } + \sqrt{ (x -1)^{2} + y^{2}} \right),\nonumber \\[6mu]
b &= \tfrac{1}{2} \left( \sqrt{(x +1)^{2} + y ^{2} } – \sqrt{ (x -1)^{2} + y^{2}} \right),\nonumber \\[6mu]
\sgn(a) &= \begin{cases}-1 & a \lt 0\\[6mu]1 & a \geq 0\end{cases} \nonumber \\[6mu]
\atan z &= \tfrac{1}{2}\left(\pi – \atan\left(\frac{1+ y}{x}\right) -\atan\left(\frac{1-y}{x}\right)\right) \\
&\quad +j\,\tfrac{1}{4}\,\ln\left( \frac{\left(\frac{1+y}{x}\right)^2 +1}{\left(\frac{1-y}{x}\right)^2 +1} \right) \\[6mu]
\acsc z &= \asin(z^{-1}) \\[6mu]
\asec z &= \acos(z^{-1}) \\[6mu]
\acot z &= \atan(z^{-1}) \\[6mu]
\end{align}
$$
Hyperbolic based trigonometry
$$
\DeclareMathOperator{\csch}{csch}
\DeclareMathOperator{\sech}{sech}
\begin{align}
\sinh z &= \cos y \sinh x + j\,\sin y\cosh x \\[6mu]
\cosh z &= \cos y \cosh x + j\,\sin y\sinh x \\[6mu]
\tanh z &= \frac{\sinh(2y)}{\cosh(2x)} +j\,\frac{\sin(2 y)}{\cosh(2 x) + \cos(2y)}\\[6mu]
\csch z &= {(\sinh z)}^{-1} \\[6mu]
\sech z &= {(\cosh z)}^{-1} \\[6mu]
\coth z &= {(\tanh z)}^{-1} \\[6mu]
\end{align}
$$
Inverse hyperbolic based trigonometry
$$
\DeclareMathOperator{\asin}{asin}
\DeclareMathOperator{\acos}{acos}
\DeclareMathOperator{\atan}{atan}
\DeclareMathOperator{\acsc}{acsc}
\DeclareMathOperator{\asec}{asec}
\DeclareMathOperator{\acot}{acot}
\DeclareMathOperator{\csch}{csch}
\DeclareMathOperator{\asinh}{asinh}
\DeclareMathOperator{\acosh}{acosh}
\DeclareMathOperator{\atanh}{atanh}
\DeclareMathOperator{\acsch}{acsch}
\DeclareMathOperator{\asech}{asech}
\DeclareMathOperator{\acoth}{acoth}
\begin{align}
\asinh z &= -j \asin(jz) \\[6mu]
\acosh z &= j \acos z \\[6mu]
\atanh z &= -j \atan(jz) \\[6mu]
\acsch z &= j \acsc(jz) \\[6mu]
\asech z &= -j \asec z \\[6mu]
\acoth z &= j \acot(jz) \\[6mu]
\end{align}
$$
This can be visualized dividing the angles by \(n\) and taking the \(n\)th root of the length of the vector. The other vectors will be separated by \(\frac{2\pi}{n}\) radians.
Visualization of complex root with real exponent
Wait a minute
Depending on how we measure the angle \(\varphi\), we get different answers? Correct, because adding \(2k\pi\) to \(\varphi\) still maps to the same complex number, but may give a different function value.
In comparison, the functions that we saw described do not produce different results when adding extra rotations to the angle. Other multivalued functions are \(\log{z}\), \(\mathrm{arcsin}z\) and \(\mathrm{arccos}z\).
In general:
the \(n\)th root has \(n\) values, because when we add \(2k\pi\) to the angle \(\varphi\), for \(k\in\mathbb{Z}\), we may get different results.
The big question becomes: how do we define the angle \(\varphi\)?
Different ways of measuring \(\varphi\)
Multi-valued
Even real valued functions can have multiple values. Remember \(\sqrt{1}=\{-1,1\}\)? Using equation \(\eqref{eq:root}\), we find the function values that we are familiar with.
all roots have magnitude \(1\), but their angles \(\varphi\) are \(\pi\) apart.
Similarly, the cube root \(\sqrt[3]{1}\) has three roots, two of which are complex. All roots have magnitude \(1\), but their angles \(\varphi\) are \(\frac{2\pi}{3}\) apart.
For real valued arguments, we conventionally choose \(\varphi\) in the range \([0,2\pi)\) where the function is single-valued and where we find a positive function value. This default single-value is called the principal value.
Besides that the function \(\sqrt[n]{z}\) is not differentiable at \(0\), it has no discontinuities. To make the function single-valued, we can limit the range of \(\varphi\) similar to what we usually do for real valued arguments. The technical term for this is branch cut. We then only express \(\varphi\) so that it doesn’t cross the branch cut. Some common branch cuts are shown in the table below. In the table \(\mathbb{R}^-\) stands for the negative real axis.
Example branch cuts and their effect on the function value
Branch cut
Range for \(\varphi\)
Effect
Consistent with
just under \(\mathbb{R}^-\)
\((-\pi,\pi]\)
\(\Re(z)\geq0\)
Sqrt of real numbers
just under \(\mathbb{R}^+\)
\([0,2\pi)\)
\(\Im(z)\geq0\)
Phase shift in waves
No matter where you define the branch cut, when \(z\) approaches a point on the branch cut from opposite sides, either the real or imaginary part of the function value abruptly changes signs. In practice, the best place for the branch cut depends on the application. For instance, it there is already a discontinuity at the point \(-1\), you may as well put the branch cut just under \(\mathbb{R}^-\).
Real and Imaginary part of \(\sqrt{z}\) as function of \(\varphi\)
We will use the \(\mathbb{C}\)-plane extensively as we explore the physic fields of electronics and domain transforms.
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