Holomorphic equals analytic
Here is the result with no real-variable analogue: every holomorphic function is analytic. If f is holomorphic on a disk |z − a| < R, then f equals its own Taylor series there, f(z) = Σ c_n (z − a)^n, with coefficients c_n = f^(n)(a) / n! given by Cauchy's formula. So holomorphic — a one-derivative condition — secretly means infinitely differentiable and power-series representable. The radius of convergence reaches out to the nearest singularity.
Three kinds of singularity
An isolated [[singularity|singularity]] is a point a where f is holomorphic on a punctured disk 0 < |z − a| < R but not at a itself. There are exactly three flavors. Removable: f stays bounded near a and can be redefined to be holomorphic (e.g. sin(z)/z at 0). [[pole|Pole]] of order m: f blows up like 1/(z − a)^m (e.g. 1/(z − a)^3). [[essential-singularity|Essential]]: wilder than any pole (e.g. e^(1/z) at 0), where by the Casorati–Weierstrass theorem the values come arbitrarily close to every complex number.
Laurent series
To expand around a singularity, allow negative powers. A [[laurent-series|Laurent series]] is f(z) = Σ_{n=−∞}^{∞} c_n (z − a)^n, valid on an annulus r < |z − a| < R. The negative-power part is the “principal part”; it is what detects and classifies the singularity. If the principal part is empty, the singularity is removable; if it stops at −m, you have a pole of order m; if it runs forever, the singularity is essential. The single coefficient c_{−1} is the residue, the hero of the next guide.
Laurent expansion of f(z) = e^z / z^3 around 0.
Start from the Taylor series of e^z (valid for all z):
e^z = 1 + z + z^2/2! + z^3/3! + z^4/4! + ...
Divide every term by z^3:
f(z) = e^z / z^3
= 1/z^3 + 1/z^2 + (1/2)*1/z + 1/3! + (1/4!) z + ...
Read off the structure on the annulus 0 < |z| < infinity:
principal part = 1/z^3 + 1/z^2 + (1/2)/z (stops at -3)
=> 0 is a POLE of order 3.
The residue is the coefficient of 1/z: c_{-1} = 1/2.
Classification check with e^{1/z} (contrast):
e^{1/z} = 1 + 1/z + 1/(2! z^2) + 1/(3! z^3) + ...
principal part has INFINITELY many terms => 0 is ESSENTIAL.