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Complex Analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of a complex variable of complex numbers. It is helpful in many branches of mathematics, including real analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory.
The polynomial map z↦z3z\mapsto z^3 cubes the distance from the origin and triples the argument.
A complex polynomial is a function of the form
p(z)=a0+a1z+⋯+anzn,\begin{equation*}p(z)=a_0+a_1z+\cdots+a_nz^n,\end{equation*}
where a0,…,an∈Ca_0,\ldots,a_n\in\mathbb{C}. Even the simple monomial z3z^3 has a clear geometric effect: if z=reiθz=re^{i\theta}, then
z3=r3ei3θ.\begin{equation*}z^3=r^3e^{i3\theta}.\end{equation*}
The exponential map z=x+iy↦ez=exeiyz=x+iy\mapsto e^z=e^xe^{iy} sends vertical strips to annuli and horizontal strips to sectors.
For z=x+iyz=x+iy, the complex exponential separates into a radial and angular part:
ez=exeiy.\begin{equation*}e^z=e^xe^{iy}.\end{equation*}
Thus the real part xx controls the radius, while the imaginary part yy controls the argument.

References

  • [Needham2023]Needham, Tristan. Visual Complex Analysis. 25th Anniversary Edition. Oxford University Press, 2023. First edition published in 1997.