Laurent Series Explained | Complex Analysis

Analytic functions can be expanded as Taylor series around a point z₀: f(z) = Σ [f⁽ⁿ⁾(z₀)/n!] (z - z₀)ⁿ. This expansion is valid within a disk centered at z₀ with radius R, called the radius of convergence. The radius is found using the ratio test: R = lim |aₙ/aₙ₊₁| or root test: R = lim |aₙ|^(1/n). If the limit is zero, R = ∞ (converges everywhere); if infinite, R = 0 (converges only at center).

To find the radius of convergence for a Taylor series, identify the nearest singularity to the expansion center and calculate the distance between them. The radius of convergence equals this distance. For example, if expanding about z = -2 and the nearest singularity is at z = 3, the radius of convergence is 5 units. The series will converge for all points within this circle and diverge for points outside it.

Power series are infinite series of the form ∑c_n(x-a)^n, where c_n are coefficients and a is the center point. The radius of convergence R determines where the series converges: R = 1/L where L is found using ratio test (lim|c_n/c_{n+1}|) or root test (lim|c_n|^(1/n)). The series converges absolutely for |x-a| < R and diverges for |x-a| > R. At endpoints, convergence must be tested separately. Taylor series approximate functions near a point: f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^(n)(a)(x-a)^n/n!. The error is bounded by Lagrange remainder R_n(x) = f^(n+1)(c)(x-a)^(n+1)/(n+1)!. The binomial series for (1+x)^α is ∑[α choose n]x^n, where [α choose n] = α(α-1)...(α-n+1)/n!. Functions can be composed with Taylor series by substitution. To find radius of convergence, apply ratio test: L = lim|c_n/c_{n+1}|, then R = 1/L. For ∑(7^n/9^n)(x+7/9)^n, c_n = 7^n/9^n, so L = 9/7, R = 7/9. For ∑(1/n!)(x/3)^n, c_n = 1/(n!3^n), L = 0, R = ∞ (converges everywhere).

A Taylor series is a power series representation of an analytic function about a point z₀. The series is f(z) = Σ[n=0 to ∞] [f^(n)(z₀)/n!](z - z₀)^n, where f^(n)(z₀) is the nth derivative of f at z₀. The radius of convergence R is the distance from z₀ to the nearest singularity (non-analytic point). For example, for f(z) = 1/(z-5) with singularity at z = 5, if we expand about z₀ = 1, the radius of convergence is |5 - 1| = 4. The region of convergence is |z - z₀| < R.

A power series is ∑c_n(x-a)^n with constant coefficients c_n and real variable x. The interval of convergence is where the series converges, and the radius of convergence R is found via the ratio test: R = lim|c_n/c_{n+1}|. For ∑(x-2)^n/n^2, R=1, interval [1,3]. The exponential series ∑x^n/n! has R=∞. Taylor series expand functions as ∑f^(n)(a)(x-a)^n/n!. For e^x at a=0: ∑x^n/n!. For sin(x) at a=0: ∑(-1)^n x^(2n+1)/(2n+1)!. For cos(x) at a=0: ∑(-1)^n x^(2n)/(2n)!. These series converge for all real x.

Cauchy's Integral Theorem states that if a function is analytic inside and on a closed contour, the integral around that contour is zero. Cauchy's Integral Formula provides the value of the integral when singular points are enclosed: for simple poles (no z^n term), the integral equals 2πi × f(z₀); for higher-order poles (with z^n term), the integral equals 2πi × f^(n-1)(z₀)/(n-1)!. The key steps to solve such problems are: (1) identify the singular points by setting the denominator to zero, (2) determine which points lie inside the contour by comparing their magnitudes with the radius, (3) apply the appropriate formula based on whether points are inside or outside, and (4) for higher-order poles, compute the necessary derivatives.

This section establishes the foundational framework for evaluating complex integrals using Cauchy's Integral Formula. The core principle states that if f is holomorphic inside and on a simple closed curve C, and a is a point inside C, then ∮_C f(z)/(z-a) dz = 2πi f(a). Three detailed examples demonstrate practical application: (1) ∮|z|=2 e^z/(z-1) dz = 2πe, where z=1 lies inside the circle; (2) ∮|z-2|=3 e^z/[z(z-6)] dz = -πi/3, where only the pole at z=0 contributes since z=6 lies outside; (3) ∮|z|=2 cos(πz)/[(z+1)(z-5)] dz = πi/3, where z=-1 is inside while z=5 is outside. These examples illustrate how to identify pole locations relative to integration contours and apply the formula systematically.

Cauchy's Integral Formula states that for a function f(z) analytic inside and on a simple closed contour C, and a point a inside C: ∮_C f(z)/(z-a)ⁿ dz = (2πi)/(n-1)! × f^(n-1)(a). To evaluate integrals: (1) Identify poles of the integrand, (2) Determine which poles lie inside the contour, (3) Decompose the integrand into partial fractions, (4) Apply the formula to each pole inside the contour. For the integral ∮_|z|=3/2 (4-3z)/[z(z-1)(z-2)] dz, poles at z=0 and z=1 are inside (|z|<1.5), while z=2 is outside. The integral equals 2πi times the sum of residues at interior poles.

This comprehensive section covers the fundamental principles of complex integration. Key topics include: (1) Basic properties - linearity, direction reversal, constant multiplication, and path additivity; (2) Cauchy's Theorem - if f(z) is analytic and f'(z) is continuous within a domain D bounded by a closed contour C, then ∮C f(z) dz = 0; (3) Cauchy's Integral Formula - ∮C dz/(z-α) = 0 if α is exterior, and = 2πi if α is interior; (4) Path independence in simply connected domains; (5) Cauchy's Residue Theorem for nested contours. These principles allow direct evaluation of complex integrals without performing the integration.

This section introduces Cauchy's fundamental theorem in complex analysis. The theorem states that if a function f(z) is analytic at all points inside and on a simple closed curve C, and its derivative f'(z) is continuous, then the contour integral ∫C f(z)dz equals zero. The closed region D is defined as consisting of all points interior to C together with the boundary points on C. The complex variable z = x + iy has differential dz = dx + i dy, and the complex function f(z) = u + iv decomposes into real and imaginary parts for integration purposes.

An isolated singularity of a complex function f at z₀ occurs when f is holomorphic in a punctured disk D(z₀, r) = {z : 0 < |z - z₀| < r} but undefined at z₀. The three types are: (1) Removable singularities - where f can be extended to be holomorphic at z₀; (2) Poles - where |f(z)| → ∞ as z → z₀; (3) Essential singularities - where f exhibits unpredictable behavior. For f(z) = z on ℂ\{0}, z=0 is removable (extend by f(0)=0). For f(z) = 1/z, z=0 is a pole. For f(z) = e^(1/z), z=0 is essential (approaching from different directions gives different limits).

Complex functions with isolated singularities can be expressed using Laurent series: f(z) = Σₖ₋∞^∞ aₖ(z - z₀)ᵏ, comprising a principal part (negative powers) and analytic part (positive powers). Singularities are classified by examining the principal part: removable singularities have no negative terms (e.g., sin(z)/z at z=0), poles have finite negative terms (order equals highest negative power), and essential singularities have infinitely many negative terms (e.g., e^(1/z) at z=0). A function has a zero of order m at z₀ if f(z₀) = f'(z₀) = ... = f^(m-1)(z₀) = 0 but f^(m)(z₀) ≠ 0. Equivalently, f(z) = (z - z₀)ᵐg(z) where g(z) is analytic and g(z₀) ≠ 0. For rational functions f(z) = h(z)/g(z), f(z) has a pole of order m at z₀ if and only if g(z) has a zero of order m at z₀, provided h(z) is analytic and h(z₀) ≠ 0.

A singular point (or singularity) of a function f(z) is a point z = a where f(z) is not analytic. This occurs when the function fails to be differentiable or cannot be expressed as a convergent power series at that point. For example, f(z) = 1/z has a singular point at z = 0 because the function is undefined there.

A singular function in complex analysis is a point where the function is not analytic (not differentiable), which can be either isolated or non-isolated; isolated singularities are points where the function fails to be analytic but can be surrounded by a neighborhood where the function is analytic everywhere else, while non-isolated singularities occur when such neighborhoods do not exist.

A singular point (or singularity) of a complex-valued function is a point where the function fails to be analytic. This occurs when the derivative does not exist at that point, even though it may exist in the neighborhood around it. For example, the function f(z) = 1/z has a singular point at z=0 because the derivative does not exist there, while for all z ≠ 0, the derivative exists and equals -1/z². The key distinction is that singular points represent isolated locations where differentiability breaks down, despite the function behaving normally in the surrounding area.