Holomorphic Functions and the Cauchy–Riemann Equations

A function of a complex variable that is differentiable once is automatically differentiable infinitely often, equal to its own Taylor series, and rigid enough that its values on a tiny disc determine it everywhere. This astonishing rigidity — absent in real analysis — is what makes complex analysis so powerful in physics. This page defines holomorphic functions, derives the Cauchy–Riemann equations, and previews the multivalued functions that force branch cuts. It follows complex-plane.md — which constructs $\mathbb{C}$, its metric, and the complex limit — and assumes the real analysis spine.

Complex differentiability

A function $f:U\to \mathbb{C}$ on an open set $U\subseteq \mathbb{C}$ is complex-differentiable (holomorphic) at $z_0$ if the limit $$f^{\prime }(z_0)=\underset{h\to 0}{\lim }\frac{f(z_0+h)-f(z_0)}{h}$$ exists — where $h\in \mathbb{C}$ approaches $0$ from any direction. This last clause is the crux: unlike a real derivative (two directions), the complex difference quotient must converge to the same value along every path in the plane. That is a severe constraint, and everything special about holomorphic functions flows from it. A function holomorphic on all of $\mathbb{C}$ is called entire.

The Cauchy–Riemann equations

Write $z=x+iy$ and $f=u(x,y)+iv(x,y)$. Demanding that the difference quotient agree along the real direction ($h\to 0$ real) and the imaginary direction ($h\to 0$ imaginary) gives $f^{\prime }(z)=u_x+i{v}_x=v_y-i{u}_y$, hence the Cauchy–Riemann (CR) equations $$\boxed{u_x=v_y,u_y=-v_x}$$

Theorem. $f$ is holomorphic on $U$ iff $u,v$ have continuous first partials satisfying the CR equations there. Then $f^{\prime }(z)=u_x+i{v}_x$.

The CR equations say the Jacobian of $(x,y)\mapsto (u,v)$ is a rotation–scaling (the matrix $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$ of multiplication by $a+ib$) — holomorphic maps are exactly those that are locally conformal (angle-preserving) with a complex-linear derivative, the theme of conformal-mapping.md.

Harmonic conjugates

Differentiating the CR equations and adding shows both parts are harmonic: $$u_{xx}+u_{yy}=0,v_{xx}+v_{yy}=0.$$ So the real and imaginary parts of a holomorphic function solve Laplace's equation; $v$ is the harmonic conjugate of $u$ (recovered on a simply connected domain by integrating the CR equations — an exactness statement, cf. de Rham). This is the entry point to potential theory — the Poisson integral, the Dirichlet problem, and the conformal solution of 2D electrostatics, ideal fluid flow, and steady heat conduction — developed in harmonic-functions.md.

Power series and analyticity

A function is analytic at $z_0$ if it equals a convergent power series ${\sum }_{n\ge 0}{a}_n(z-z_0{)}^n$ on a neighbourhood. Power series are holomorphic term-by-term inside their disc of convergence (radius $R=1/\mathrm{limsup}|a_n{|}^{1/n}$), and — the central miracle, proved via Cauchy's formula —

Holomorphic = analytic. Every holomorphic function is analytic: it is infinitely differentiable and equals its Taylor series $\sum \frac{f^{(n)}(z_0)}{n!}(z-z_0{)}^n$ on the largest disc avoiding a singularity.

The two words are used interchangeably. The radius of convergence reaches exactly to the nearest singularity — which is why a real series like $\frac{1}{1+x^2}=\sum (-1{)}^n{x}^{2n}$ mysteriously diverges at $|x|>1$: the singularities at $z=\pm i$ are invisible on the real line but cap the complex disc.

Elementary and multivalued functions

The standard functions extend to the complex plane by their power series: $$e^z=\sum \frac{z^n}{n!},\cos z,\sin z(\text{giving }{e}^{iz}=\cos z+i\sin z),\text{polynomials, rational functions}.$$ $e^z$ is entire and periodic with period $2\pi i$. Its inverse, the logarithm $$\log z=\ln |z|+i\arg z,$$ is therefore multivalued ($\arg z$ defined only mod $2\pi$), as are $z^{\alpha }=e^{\alpha \log z}$ and $\sqrt{z}$. Making them single-valued requires a branch cut (a curve removed from the plane) or a Riemann surface (riemann-surfaces.md); the ambiguity around a branch point ($z=0$ for $\log$) is the source of the physical thresholds and cuts of scattering amplitudes. The winding of $\arg z$ around $0$ is the same $H^1(\mathbb{C}\setminus \{0\})$ nontriviality met in de-rham.md.

References

• Ahlfors, Complex Analysis — the classic.
• Stein & Shakarchi, Complex Analysis.
• Needham, Visual Complex Analysis — geometric intuition for conformality.