The Cauchy-Riemann equations: u_x = v_y, u_y = -v_x

A function \(f = u + iv\) is complex differentiable exactly where its real and imaginary parts satisfy the Cauchy-Riemann equations. The residual \(\max(|u_x - v_y|, |u_y + v_x|)\) vanishes everywhere for \(z^2\) and \(e^z\), but not for \(\bar z\), \(|z|^2\), or \(\operatorname{Re} z\). \(|z|^2\) is the interesting case: it satisfies the equations only at \(z = 0\).

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.complex_analysis import cauchy_riemann, complex_derivative, complex_grid

Holomorphic vs. non-holomorphic functions

functions = {
    "z^2": lambda z: z**2,
    "exp(z)": np.exp,
    "conj(z)": np.conj,
    "|z|^2": lambda z: abs(z) ** 2,
    "Re(z)": lambda z: z.real,
}
z0 = 0.7 - 0.4j
for name, f in functions.items():
    r = cauchy_riemann(f, z0)
    print(f"{name:8s} u_x={r.u_x:+.3f} v_y={r.v_y:+.3f} u_y={r.u_y:+.3f} v_x={r.v_x:+.3f}  residual={r.residual:.1e}")
print("d/dz z^2 at z0:", complex_derivative(lambda z: z**2, z0), "= 2 z0 =", 2 * z0)

z^2      u_x=+1.400 v_y=+1.400 u_y=+0.800 v_x=-0.800  residual=5.6e-11
exp(z)   u_x=+1.855 v_y=+1.855 u_y=+0.784 v_x=-0.784  residual=5.6e-11
conj(z)  u_x=+1.000 v_y=-1.000 u_y=+0.000 v_x=+0.000  residual=2.0e+00
|z|^2    u_x=+1.400 v_y=+0.000 u_y=-0.800 v_x=+0.000  residual=1.4e+00
Re(z)    u_x=+1.000 v_y=+0.000 u_y=+0.000 v_x=+0.000  residual=1.0e+00
d/dz z^2 at z0: (1.400000000040258-0.8000000000230045j) = 2 z0 = (1.4-0.8j)

Where does the squared modulus satisfy the equations?

z = complex_grid((-1, 1), (-1, 1), 41)
residual = np.vectorize(lambda p: cauchy_riemann(lambda w: abs(w) ** 2, p).residual)(z)
fig, ax = plt.subplots()
image = ax.imshow(residual, origin="lower", extent=(-1, 1, -1, 1), cmap="viridis")
fig.colorbar(image, ax=ax, label="Cauchy-Riemann residual")
ax.set_title(r"$|z|^2$ is complex differentiable only at $z = 0$")
ax.set_xlabel("Re z")
ax.set_ylabel("Im z")

Text(48.922222222222274, 0.5, 'Im z')