Complex Analysis: Singularities & Laurent Series | KTU Maths GYMAT301 Part 6

The Maclaurin series is a special case of the Taylor series where the expansion is performed around the point z₀ = 0. The formula is: f(z) = Σ(n=0 to ∞) [f^(n)(0)/n!] z^n. This series is named after Colin Maclaurin and is widely used for expanding functions around the origin. The Maclaurin series for common functions are: e^z = 1 + z + z²/2! + z³/3! + ...; sin(z) = z - z³/3! + z⁵/5! - z⁷/7! + ...; cos(z) = 1 - z²/2! + z⁴/4! - z⁶/6! + ...; ln(1+z) = z - z²/2 + z³/3 - z⁴/4 + ...; (1+z)^m = 1 + mz + m(m-1)z²/2! + m(m-1)(m-2)z³/3! + ... The Taylor series expansion of sin(z) around z₀ = 0 is: sin(z) = z - z³/3! + z⁵/5! - z⁷/7! + ... + (-1)^n z^(2n+1)/(2n+1)! + ... containing only odd powers. The nth derivative at z₀ = 0 gives f^(n)(0) = 0 for even n and f^(n)(0) = (-1)^((n-1)/2) for odd n. The Taylor series expansion of cos(z) around z₀ = 0 is: cos(z) = 1 - z²/2! + z⁴/4! - z⁶/6! + ... + (-1)^n z^(2n)/(2n)! + ... containing only even powers. The nth derivative at z₀ = 0 gives f^(n)(0) = 0 for odd n and f^(n)(0) = (-1)^(n/2) for even n.

A complex-valued function f(z) can be expanded into an infinite Taylor series around a point z₀ using the formula: f(z) = f(z₀) + f'(z₀)(z-z₀) + f''(z₀)(z-z₀)²/2! + f'''(z₀)(z-z₀)³/3! + ... This expansion requires computing successive derivatives of the function at z₀. The exponential function f(z) = e^z serves as a classic example, where all derivatives at z₀ = 0 equal 1, producing the series e^z = 1 + z + z²/2! + z³/3! + ... Taylor series can be expanded around any complex number z₀, not just zero, demonstrating the general applicability of this representation for complex functions.

Taylor series expansion for complex analytic functions is given by f(z) = Σₙ₌₀^∞ [f⁽ⁿ⁾(z₀)/n!] (z - z₀)ⁿ, where coefficients are nth derivatives at z₀ divided by n factorial. The Maclaurin series is a special case where z₀ = 0, resulting in f(z) = Σₙ₌₀^∞ [f⁽ⁿ⁾(0)/n!] zⁿ. These series represent functions as infinite power series centered at a specific point, with convergence determined by the radius of convergence.

A power series with nonzero radius of convergence represents an analytic function within its circle of convergence. Taylor's theorem states that any analytic function f(z) in a domain D can be uniquely represented as a power series centered at any point z₀ in D: f(z) = Σ(n=0 to ∞) [f⁽ⁿ⁾(z₀)/n!] (z - z₀)ⁿ, with radius of convergence equal to the distance from z₀ to the nearest boundary point. Using the Cauchy integral formula f(z) = (1/2πi) ∮_C [f(s)/(s - z)] ds, the kernel is expanded as a geometric series, yielding the Taylor series. Taylor's formula with remainder shows the remainder converges to zero as n → ∞. The representation is unique by the identity theorem. Any power series with nonzero radius of convergence is the Taylor series of the function it represents. This characterization holds only for complex analytic functions, not real functions. Practical examples include Maclaurin series for 1/(1 - z) = Σzⁿ, 1/(1 + z) = Σ(-1)ⁿzⁿ, and 1/(1 + z²) = Σ(-1)ⁿz²ⁿ. To find Taylor series for complex functions, use known expansions combined with partial fraction decomposition and algebraic transformations, avoiding computation of infinitely many derivatives.

Taylor series expansion is a method to represent complex functions as infinite power series around a point z₀, given by 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 series converges within a radius determined by the distance to the nearest singularity. Standard expansions include: (1+z)^n = Σ(n choose k) z^k for |z|<1, sin(z) = z - z³/3! + z⁵/5! - ..., and e^z = 1 + z + z²/2! + z³/3! + ..., each valid within their respective convergence regions.

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.

Complex integration along a closed curve is called complex integration or line integral. Cauchy's Integral Theorem states that if f(z) is analytic within and on a closed curve C, then ∮_C f(z) dz = 0. A function is analytic at a point if differentiable in its neighborhood; if non-analytic at z=a, then z=a is a singular point. Cauchy's Integral Formula gives f(z₀) = (1/2πi) ∮_C f(z)/(z - z₀) dz for z₀ inside C. For higher derivatives, f^(n)(z₀) = (n!/(2πi)) ∮_C f(z)/(z - z₀)^(n+1) dz. Cauchy's Extension Theorem handles multiple singular points by summing individual contributions.

A complex line integral is defined as ∫F(z)dz along a curve gamma, where F(z) = U + iV is a complex-valued function. This decomposes into real line integrals ∫(Udx - Vdy) + i∫(Vdx + Udy). A function is analytic at a point if it is differentiable in some neighborhood, and if differentiable once in a region, it is infinitely differentiable. Cauchy's theorem states that if F(z) is analytic in a domain G and gamma is a closed contour in G, then ∫F(z)dz = 0. More generally, the integral is path-independent within G. Cauchy's Integral Formula states that if F(z) is analytic in a simply connected domain G and gamma is a closed contour in G, then for any point z inside gamma, F(z) = (1/(2πi)) ∫F(ζ)/(ζ - z) dζ. The proof involves deforming the contour to a circle centered at z, which is possible because the integral depends only on the homotopy class of the contour within the domain of analyticity. The integral ∫dζ/(ζ - z) around a circle centered at z evaluates to 2πi.

Cauchy's Theorem states that if a function f(z) is analytic within and on a closed contour C, then the integral of f(z) around C equals zero. Cauchy's Integral Formula states that if f(z) is analytic within and on a closed contour C, and a is any point inside C, then the integral of f(z)/(z-a) around C equals 2πi times f(a). These theorems form the foundation for complex integration, where the key steps involve identifying singular points (where the function is undefined), determining whether these points lie inside or outside the integration contour, and applying the appropriate formula to evaluate the integral.

Cauchy's Integral Formula states that for a function f(z) analytic in a domain D and on its boundary curve C, and for any point z₀ inside C, the integral ∮_C f(z)/(z - z₀)^(n+1) dz = (2πi/n!) f^(n)(z₀), where f^(n)(z₀) is the nth derivative of f evaluated at z₀. This theorem allows evaluation of complex line integrals without direct integration by identifying singular points (where denominators equal zero), determining which lie inside the integration curve, and applying the formula with appropriate derivatives. The formula generalizes to higher powers in the denominator, where n+1 represents the power of the singularity.

In topology on the complex plane, a circle consists only of boundary points (|z - z₀| = r), an open disk includes all interior points but excludes the boundary (|z - z₀| < r), a closed disk includes both interior and boundary points (|z - z₀| ≤ r), an annulus is the region between two concentric circles (R₁ < |z - z₀| < R₂), a neighborhood of a point z₀ is the set of all points within some distance ε from z₀ (excluding the boundary), and a deleted neighborhood is a neighborhood with the central point z₀ removed (0 < |z - z₀| < ε).

In topology, a topology on the complex plane is defined as a collection of subsets satisfying three axioms: containing the empty set and the whole plane, being closed under finite intersections, and being closed under arbitrary unions; the usual topology on the complex numbers is generated by open disks (sets of points within a fixed distance from a center), where a set is open if every point has an open disk around it contained entirely within the set, and key concepts include interior points (points with disks entirely inside the set), boundary points (points where every disk contains both interior and exterior points), accumulation points (points where every disk contains other points of the set), and isolated points (points with disks containing no other set points); a domain is defined as an open and connected subset of the complex plane.

A neighborhood of z₀ is the set of points within distance r from z₀. A deleted neighborhood excludes z₀ itself. A point is an interior point if some neighborhood around it lies entirely within the set. A point is an exterior point if some neighborhood around it contains no points of the set. A boundary point has neighborhoods containing both points in the set and points not in the set. A set is open if all its points are interior points (contains no boundary points).

The topology of the complex plane involves fundamental concepts including open disks (|z-z₀|<r), closed disks (|z-z₀|≤r), circles (|z-z₀|=r), and annuli (r₁<|z-z₀|<r₂), along with key point classifications: interior points (surrounded by set points), exterior points (surrounded by complement points), boundary points (every neighborhood contains both set and complement points), and limit points (every neighborhood contains at least one set point other than itself). Sets are classified as open (all points are interior), closed (contain all limit points), bounded (fit within some disk), compact (closed and bounded), connected (any two points can be joined by a path within the set), domains (non-empty open connected sets), and regions (domains plus some or all boundary points).

This section introduces topological concepts for the complex plane, generalizing interval concepts from real analysis. The instructor reviews the nine types of intervals in real numbers (four bounded and five unbounded) before generalizing to the complex plane. Key geometric definitions include: open disk D(z₀, R) = {z : |z - z₀| < R}, closed disk D̄(z₀, R) = {z : |z - z₀| ≤ R}, and circle ∂D(z₀, R) = {z : |z - z₀| = R}. The core definitions of interior points (points with neighborhoods entirely in A) and open sets (sets where every point is interior) are established. The instructor proves rigorously that open disks are open sets using the triangle inequality, demonstrating why these sets are called 'open'.