Analytic function

A function $f$ is analytic at $z_0$ if it is complex differentiable on some open disk around $z_0$ — not just at the single point $z_0$, but on a whole neighborhood. $f$ is analytic on a set $D$ if it is analytic at every point of $D$. Synonyms: holomorphic, regular.

A function analytic on all of $\mathbb{C}$ is called entire.

The distinction from “merely differentiable”

A function can be complex differentiable at a single point yet fail to be analytic there. Example: $f(z)=|z{|}^2+iz$ is complex differentiable only at $z=0$ (a check via C-R gives $u=x^2+y^2-y$, $v=x$, with C-R forcing $x=0$ and $y=0$ simultaneously). A single point isn’t an open set, so $f$ is analytic nowhere. The open-neighborhood requirement matters.

In practice, the dichotomy is sharp: most functions are either analytic on a whole open region (or fail to be analytic anywhere). The intermediate case (differentiable on isolated points or curves) is rare and never used for real work.

Examples of analytic / entire functions

• Polynomials: entire.
• Rational functions $p(z)/q(z)$: analytic on $\mathbb{C}$ minus the zeros of $q$.
• $e^z$, $\sin z$, $\cos z$: entire.
• $\sinh z$, $\cosh z$: entire.
• Sums, products, compositions of analytic functions: analytic on the intersection of their domains.
• Quotients: analytic where the denominator is nonzero.
• $\mathrm{Log}z$: analytic on $\mathbb{C}\setminus (-\infty ,0]$ (the slit plane). See Complex logarithm.

Examples that are not analytic anywhere

$\bar{z}$, $|z|$, $\mathrm{Re}(z)$, $\mathrm{Im}(z)$ — all continuous, all infinitely differentiable as functions of $(x,y)$, none complex differentiable. The Cauchy-Riemann equations fail at every point.

The remarkable properties of analytic functions

The single condition “complex differentiable on an open set” turns out to imply a long list of striking consequences. None of these have real-variable analogs.

1. Infinitely differentiable. Analytic functions have derivatives of all orders — once differentiable is enough. By contrast, real functions can be differentiable but not twice differentiable.

2. Equal to their Taylor series. On any disk of analyticity around $z_0$, $f(z)=\sum f^{(n)}(z_0)(z-z_0{)}^n/n!$ converges to $f$. See Taylor series.

3. Boundary values determine interior values. The Cauchy integral formula expresses $f$ at any interior point of a contour from its values on the contour. In real analysis you can change interior values of a function on $[0,1]$ without changing boundary values; in complex analysis you cannot.

4. Zero closed-loop integrals. Cauchy’s theorem: if $f$ is analytic inside and on a simple closed contour, $\oint fdz=0$.

5. Bounded entire $\Rightarrow$ constant. Liouville’s theorem. A consequence: the Fundamental theorem of algebra.

6. Maximum on the boundary. Maximum modulus principle: $|f|$ on a bounded region attains its max on the boundary, not interior.

7. Real and imaginary parts are harmonic. Both $u$ and $v$ solve Laplace’s equation, $u_{xx}+u_{yy}=0$. See Harmonic function.

8. Conformal where derivative is nonzero. Angles between curves are preserved. See Conformality.

All of these flow from the single definition. The Cauchy integral formula is the lynchpin — once you have it, everything else falls out in 1–3 line proofs.

Practical test for analyticity

$f=u+iv$ is analytic on an open set $D$ iff (i) $u,v,u_x,u_y,v_x,v_y$ are continuous on $D$ and (ii) the Cauchy-Riemann equations $u_x=v_y$, $u_y=-v_x$ hold throughout $D$.

For functions built from $z$ (polynomials, $e^z$, trig, log, compositions), C-R is automatic and analyticity is easy. For functions built from $z$ and $\bar{z}$, only those that effectively don’t depend on $\bar{z}$ are analytic — a slogan: analytic functions are functions of $z$ alone, not of $\bar{z}$.

Why this matters

The pivotal tools of complex analysis — contour integrals, the residue theorem, conformal mapping — all rest on analyticity. Beyond mathematics, analyticity is the structural reason why so many physical fields are governed by Laplace’s equation, why steady-state electrical and fluid problems admit elegant complex-function solutions, and why the transfer functions in control theory are rational analytic functions on the $s$-plane.