Residue (complex analysis)

Residue in complex analysis refers to a complex number that describes the behavior of a complex function near an isolated singularity. More formally, it quantifies the contribution of a singularity to the integral of the function along a closed curve encompassing the singularity. This concept is fundamental to the residue theorem, a powerful tool for evaluating contour integrals.

Definition

Let ''f''(z) be a complex function with an isolated singularity at z₀. The residue of ''f''(z) at z₀, denoted Res(f, z₀), is defined as the coefficient of the (z − z₀)⁻¹ term in the Laurent series expansion of ''f''(z) around z₀. That is:

''f''(z) = Σ_n=−∞^∞ a_n(z − z₀)ⁿ

where Res(f, z₀) = a₋₁.

Calculation

The method for calculating the residue depends on the nature of the singularity at z₀.

Simple Pole

If z₀ is a simple pole (a pole of order 1), the residue can be calculated using the formula:

Res(f, z₀) = lim_z→z₀ (z − z₀) ''f''(z)

This formula is particularly useful and often the simplest method for calculating the residue at a simple pole.

Pole of Order m

For a pole of order ''m'' (m > 1), the residue is given by:

Res(f, z₀) = lim_z→z₀ (1/((m-1)!) * d^m−1/dz^m−1 [(z − z₀)^m ''f''(z)])

This involves taking the (m-1)-th derivative of a modified function.

Essential Singularity

Calculating the residue at an essential singularity is generally more challenging and often requires finding the Laurent series expansion directly. There is no single, simple formula for this case.

Residue Theorem

The residue theorem states that the contour integral of a complex function around a closed curve is equal to 2πi times the sum of the residues of the function at the singularities enclosed by the curve. This provides a powerful method for evaluating complex integrals, often avoiding lengthy direct integration methods.

Applications

The concept of residues and the residue theorem find broad applications in various fields, including:

Evaluation of definite integrals
Solving differential equations
Physics and engineering problems involving complex analysis

📖 WIPIVERSE