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Singularity — Definition, Formula & Examples

A singularity is a point where a mathematical function is not defined or not analytic, often because the function blows up to infinity or oscillates wildly at that point.

A point z0z_0 is a singularity of a function ff if ff is not analytic (or not defined) at z0z_0, yet every neighborhood of z0z_0 contains points where ff is analytic. Singularities are classified as removable, poles, or essential depending on the behavior of ff near z0z_0.

How It Works

To identify a singularity, look for points where the function's formula breaks down — typically where a denominator equals zero or where a logarithm or square root receives an invalid input. In complex analysis, you then classify the singularity by examining the Laurent series of ff around z0z_0. If the series has no negative-power terms, the singularity is removable. If it has finitely many negative-power terms, z0z_0 is a pole. If infinitely many negative-power terms appear, z0z_0 is an essential singularity.

Worked Example

Problem: Identify and classify the singularity of f(z)=1(z−2)3f(z) = \frac{1}{(z - 2)^3} at z=2z = 2.
Step 1: Check whether the function is defined at z=2z = 2. Substituting gives division by zero, so z=2z = 2 is a singularity.
f(2)=1(2−2)3=10(undefined)f(2) = \frac{1}{(2-2)^3} = \frac{1}{0} \quad \text{(undefined)}
Step 2: Determine the type. The Laurent series of ff around z=2z = 2 is simply (z−2)−3(z-2)^{-3}, which has exactly one negative-power term (with exponent −3-3). Since there are finitely many negative powers, this is a pole.
f(z)=(z−2)−3f(z) = (z-2)^{-3}
Step 3: State the order. The most negative exponent is −3-3, so z=2z = 2 is a pole of order 3.
Answer: z=2z = 2 is a pole of order 3.

Why It Matters

Singularities are central to evaluating contour integrals via the residue theorem in complex analysis. In real-variable calculus, recognizing singularities tells you where limits may be infinite or where improper integrals require special treatment. Engineers and physicists encounter singularities when modeling fields, fluid flow, and signal processing.

Common Mistakes

Mistake: Assuming every singularity means the function goes to infinity.
Correction: A removable singularity (e.g., sin⁡zz\frac{\sin z}{z} at z=0z=0) has a finite limit, and an essential singularity (e.g., e1/ze^{1/z} at z=0z=0) oscillates without approaching any single value. Only poles produce infinite limits.