Complex mapping, plotting such functions step by step on paper

Question

I don't want you guys to mention the methods and ways available to plot such functions, I want to plot a complex function on paper, tell me what should I do step by step? Consider the simple function, z = x + iy, w = f(z) = u + iv, I want to show how the x and y values from the z plane are transformed to u and v in w plane. Don't tell me to use a software to do this for me, I want to know how to do it on a paper before diving into software stuff. Thanks in advance

Answer 1

Since you have a function, $u$ and $v$ are presumably given by explicit formulas in $x$ and $y$, e.g., $u = x^3 - 3xy^2$ and $v = 3x^2y - y^3$. For each real number $y_0$, the horizontal line $y = y_0$ maps to the image of the parametric curve $\gamma(x) = u(x, y_0) + iv(x, y_0)$; this curve can be plotted by calculating $\gamma(x)$ at several equally-spaced values of $x$ and connecting the resulting dots in order. Doing this for several equally-spaced values of $y_0$ gives a picture of the image of equally-spaced horizontal grid lines in the $z$ plane. Similarly, fixing $x = x_0$ and varying $y$ plots the images of vertical grid lines. The end result depicts the image in the $w$ plane of a rectangular grid in the $z$ plane.

More algorithmically, suppose you want to draw the image under $f$ of the rectangle with corners $a_1 + ib_1$ and $a_2 + ib_2$ using an $n_1 \times n_2$ grid. Put $\Delta x = (a_2 - a_1)/n_1$ and $\Delta y = (b_2 - b_1)/n_2$. Plot the $(n_1 + 1)(n_2 + 1)$ points $p(k_1, k_2) := f\bigl((a_1 + k_1\, \Delta x) + i(b_1 + k_2\, \Delta y)\bigr)$ for integers $0 \leq k_1 \leq n_1$ and $0 \leq k_2 \leq n_2$, and connect them by joining $p(k_1, k_2)$ to $p(k_1 + 1, k_2)$ for $k_1 < n_1$ (and all $k_2$), and by joining $p(k_1, k_2)$ to $p(k_1, k_2 + 1)$ for $k_2 < n_2$ (and all $k_1$).