Complex Functions & Holomorphicity

A function $f:\mathbb C\to\mathbb C$ is also a map $\mathbb R^2\to\mathbb R^2$, but complex differentiability is much stronger than ordinary two-variable differentiability. The difference quotient must approach the same complex number while the increment approaches zero from every direction in the plane. That directional agreement forces a special Jacobian, the Cauchy–Riemann equations, harmonic real and imaginary parts, and eventually power-series behavior.

What holomorphicity buys in an applied model #

A generic differentiable map of the plane may stretch two directions by different amounts or shear a right angle. A holomorphic map with nonzero derivative has only one local scale and one rotation. That rigidity lets it change geometry while preserving small angles.

In a homogeneous isotropic medium with constant material coefficients and no volumetric heat source or free charge, steady temperature or electrostatic potential is harmonic. Velocity potentials for ideal incompressible, irrotational two-dimensional flow are harmonic as well. When a suitable conformal map exists, it can turn a difficult boundary into a circle or line, solve the simpler boundary-value problem there, and map the result back. The physical conclusion comes from those model assumptions; complex numbers are the compact language that exposes the structure.

The derivative must be direction-independent #

At $z_0$, the complex derivative is

$$f'(z_0)=\lim_{h\to0,\ h\in\mathbb C}\frac{f(z_0+h)-f(z_0)}{h}.$$

The phrase $h\to0$ in $\mathbb C$ is load-bearing. Testing only real increments checks one path, not the limit.

For $f(z)=z^2$,

$$\frac{(z+h)^2-z^2}{h}=2z+h\to2z,$$

independently of the direction of $h$. For $f(z)=\overline z$ at $0$,

$$\frac{\overline h}{h}$$

equals $1$ along the real axis and $-1$ along the imaginary axis, so the derivative does not exist. This is the fastest diagnostic for many candidates: compare two approach directions.

Complex differentiability implies continuity. Holomorphicity is even more rigid: all higher derivatives exist and the function has a local convergent power series.

Cauchy–Riemann equations encode the allowed Jacobian #

Write

$$f(x+iy)=u(x,y)+iv(x,y).$$

Approaching through a real increment gives

$$f'(z)=u_x+iv_x.$$

Approaching through an imaginary increment gives

$$f'(z)=v_y-iu_y.$$

The two must agree, so

$$u_x=v_y, \qquad u_y=-v_x.$$

These are necessary wherever $f'$ exists. A standard sufficient condition is that the four first partial derivatives exist and are continuous in a neighborhood and satisfy the equations there.

The real Jacobian is

$$Df= \begin{pmatrix} u_x & u_y\\ v_x & v_y \end{pmatrix} = \begin{pmatrix} a & -b\\ b & a \end{pmatrix}, \qquad f'(z)=a+ib.$$

Multiplication by $a+ib$ is a rotation by $\arg f'(z)$ followed by scaling by $|f'(z)|$. When $f'(z)\ne0$, a holomorphic map is therefore locally angle-preserving and orientation-preserving. “Conformal” is a local statement; zeros of $f'$ are exceptional.

Real and imaginary parts are harmonic #

If $u,v$ have continuous second partials and satisfy Cauchy–Riemann, then

$$u_{xx}+u_{yy}=v_{yx}-v_{xy}=0,$$

and similarly $v_{xx}+v_{yy}=0$. Both are harmonic.

Given a harmonic $u$ on a simply connected domain, a harmonic conjugate $v$ can be recovered, up to an additive constant, from

$$v_y=u_x, \qquad v_x=-u_y.$$

Path independence is a domain question. A local conjugate may exist while no single-valued global conjugate exists on a domain with a hole.

Elementary entire functions #

The exponential power series converges for every $z$:

$$e^z=\sum_{n=0}^{\infty}\frac{z^n}{n!}.$$

For $z=x+iy$,

$$e^z=e^x(\cos y+i\sin y).$$

It is entire, never zero, and has period $2\pi i$. Complex sine and cosine are defined by exponentials,

$$\cos z=\frac{e^{iz}+e^{-iz}}2, \qquad \sin z=\frac{e^{iz}-e^{-iz}}{2i},$$

and are also entire. Unlike their real restrictions, they are unbounded in the imaginary direction.

Arguments, logarithms, and powers require branches #

For $z\ne0$, write $z=re^{i\theta}$ with $r=|z|$. The angle is not unique:

$$\arg z=\theta+2\pi k,\qquad k\in\mathbb Z.$$

Consequently the multivalued logarithm is

$$\log z=\ln|z|+i(\theta+2\pi k).$$

A branch chooses one continuous argument on a domain. The principal branch

$$\mathop{\mathrm{Log}}z=\ln|z|+i\mathop{\mathrm{Arg}}z, \qquad -\pi<\mathop{\mathrm{Arg}}z<\pi,$$

is holomorphic on $\mathbb C\setminus(-\infty,0]$. It is not continuous across its branch cut and is undefined at $0$.

No holomorphic logarithm exists on all of $\mathbb C\setminus\{0\}$: going once around the origin changes a continuous argument by $2\pi$.

For complex $\alpha$, a single-valued power must inherit a branch:

$$z^\alpha=\exp\bigl(\alpha\mathop{\mathrm{Log}}z\bigr).$$

Using the multivalued logarithm instead gives a set of values. Familiar algebraic laws such as $\mathop{\mathrm{Log}}(zw)=\mathop{\mathrm{Log}}z+\mathop{\mathrm{Log}}w$ may differ by $2\pi i$ and are not globally valid without compatible branch and domain choices.

Zeros and local structure #

If a holomorphic function is not identically zero and $f(z_0)=0$, then for some integer $m\ge1$,

$$f(z)=(z-z_0)^m g(z), \qquad g(z_0)\ne0.$$

The integer $m$ is the zero's multiplicity. Zeros are isolated unless the function vanishes identically on the connected domain. This local factorization is the bridge to poles, Laurent series, residues, and the argument principle.

Diagnostic workflow #

1. State the open domain; do not call a function holomorphic “at” a boundary point without a neighborhood.
2. Test the derivative limit along incompatible directions when nonexistence is suspected.
3. For $f=u+iv$, compute all four first partials and check both Cauchy–Riemann equations.
4. To use the converse, verify continuity of the partials in a neighborhood.
5. For logarithms, roots, inverse trigonometric functions, and noninteger powers, name the branch and domain.
6. Separate local and global claims: local derivatives may exist even when topology prevents a global primitive or logarithm.