Holomorphic functions

Complex differentiability is far stronger than real differentiability; this section sets up the definitions and first consequences.

1. Complex differentiability and holomorphic functions (Def. 1.1), with the usual rules (Prop. 1.2).
2. The Cauchy–Riemann equations are necessary (Thm 1.3) and, with continuous partials, sufficient (Thm 1.4).
3. Real and imaginary parts are harmonic (Cor. 1.5); harmonic conjugates (Example 1.6).
4. Power series are holomorphic inside their disc of convergence (Thm 1.7).

Let $U\subseteq\mathbb{C}$ be open. $f: U\to\mathbb{C}$ is complex differentiable at $a\in U$ if

exists. $f$ is holomorphic on $U$ if it is complex differentiable at every point of $U$; entire if holomorphic on $\mathbb{C}$.

Complex differentiable functions are continuous. Sums, products, quotients (where the denominator is nonzero) and compositions of holomorphic functions are holomorphic, with the usual formulas. Polynomials are entire and rational functions are holomorphic away from the zeros of the denominator.

If $f = u + iv$ is complex differentiable at $a = x_0 + iy_0$, then the partial derivatives of $u, v$ exist at $(x_0, y_0)$ and

with $f'(a) = u_x + iv_x = v_y - iu_y$.

If $u, v$ have continuous first partial derivatives on $U$ satisfying the Cauchy–Riemann equations, then $f = u + iv$ is holomorphic on $U$.

If $f = u + iv$ is holomorphic and $u, v$ are $C^2$ (true automatically, see Section 4), then $\nabla^2u = u_{xx} + u_{yy} = 0$ and $\nabla^2v = 0$. A pair $u, v$ satisfying Cauchy–Riemann are harmonic conjugates.

$u = x^3 - 3xy^2$ is harmonic. From $v_y = u_x = 3x^2 - 3y^2$, $v = 3x^2y - y^3 + g(x)$; then $v_x = 6xy + g'(x) = -u_y = 6xy$, so $g$ is constant. Hence $f = u + iv = z^3 + ic$.

If $\sum a_n(z - c)^n$ has radius of convergence $R > 0$ (given by $\frac1R = \limsup|a_n|^{1/n}$), its sum $f$ is holomorphic on $D(c, R)$ with $f'(z) = \sum na_n(z - c)^{n-1}$, which has the same radius. Hence $f$ is infinitely complex differentiable there and $a_n = \frac{f^{(n)}(c)}{n!}$.

• Built from $z$ by algebra, $\exp$, power series: holomorphic where defined.
• Involves $\bar z$, $|z|$, $\operatorname{Re}z$ explicitly: test Cauchy–Riemann; usually fails.
• Differentiable at isolated points or along a curve is not holomorphic (needs an open set).
• Check $\nabla^2u = 0$ before looking for a conjugate.