Differentiability at isolated points, like $|z|^2$ at $0$, is not much use. The useful notion asks for a derivative on a whole open set, and it turns out to be extremely strong: such functions have derivatives of every order and equal their power series. This lesson names them and shows how to find where a formula is analytic, using the Cauchy–Riemann equations and the chain rule.
Key idea
$f$ is analytic (also called holomorphic) on an open set $U$ if it is complex differentiable at every point of $U$; it is analytic at a point if it is analytic on some disk around it. A function analytic on the whole plane is entire. Polynomials, $e^z$, $\sin z$ and $\cos z$ are entire, and sums, products and compositions of analytic functions are analytic; a quotient $\frac{g}{h}$ is analytic wherever $g$ and $h$ are and $h \ne 0$.
Why it works
The derivative rules prove the closure statements: if $g$ and $h$ have derivatives at every point of $U$, so do $g + h$, $gh$, and $\frac{g}{h}$ (where $h \ne 0$), with the familiar formulas. For a composition, the chain rule $(g \circ h)' = g'(h(z))\,h'(z)$ works exactly as in real calculus. The word "open" matters: analytic at $z_0$ asks for a whole disk around $z_0$, which is why $|z|^2$, differentiable at the single point $0$, is analytic nowhere.
Worked examples
Worked example: Where is it analytic?
Where is $f(z) = \dfrac{z + 1}{z^2 + 1}$ analytic?
- The top and the bottom are polynomials, analytic everywhere.
- The bottom is $0$ when $z^2 = -1$, that is $z = i$ or $z = -i$.
- So $f$ is analytic on the plane with the two points $\pm i$ removed. Note that $z^2 + 1$ has no real zeros, which is exactly why these points are easy to miss.
Answer: everywhere except $\pm i$
Worked example: Zeros of a cubic
Where is $g(z) = \dfrac{e^z}{z^3 - 4z}$ not analytic?
- $z^3 - 4z = z(z - 2)(z + 2)$.
- $g$ fails to be analytic at $0$, $2$ and $-2$, and is analytic everywhere else.
Answer: $0$, $2$, $-2$
Worked example: A derivative by the rules
Find $f'(i)$ for $f(z) = (z^2 + 3)^4$.
- Chain rule: $f'(z) = 4(z^2 + 3)^3 \cdot 2z$.
- At $z = i$: $z^2 + 3 = 2$, so $f'(i) = 4 \cdot 8 \cdot 2i = 64i$.
Answer: $64i$
Worked example: Zeros of an exponential
Where is $\dfrac{1}{e^z - 1}$ not analytic?
- Where $e^z = 1$. Writing $z = x + iy$: $|e^z| = e^x = 1$ forces $x = 0$, and $e^{iy} = 1$ forces $y = 2\pi k$.
- So at $z = 2\pi ik$ for every integer $k$: infinitely many points on the imaginary axis.
Answer: $z = 2\pi i k$
Common mistake
Forgetting the complex zeros of a denominator. $\frac{z + 1}{z^2 + 1}$ looks harmless if you only think of real $z$, but it is not analytic at $\pm i$. When deciding where a quotient is analytic, solve "bottom $= 0$" over the complex numbers, including non-real solutions.