Taylor Series and Analyticity of Holomorphic Functions

Taylor Series and Analyticity

Motivation: From Integral to Power Series

The Cauchy integral formula allows the computation of derivatives of all orders. From this, it immediately follows that a holomorphic function can be expanded as a convergent Taylor series—and this is an equality, not just an approximation. Holomorphy and analyticity in the complex case are the same concept, unlike in real analysis.

The Taylor Series of a Holomorphic Function

Theorem: If $f$ is holomorphic in the disk $|z - z_0| < R$, then for all $z$ in this disk:

$ f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n, \quad a_n = \frac{f^{(n)}(z_0)}{n!} = \frac{1}{2\pi i} \oint \frac{f(\zeta)}{(\zeta-z_0)^{n+1}} d\zeta. $

The series converges uniformly on each compact subset inside the disk.

Radius of convergence $R = \mathrm{dist}(z_0, \partial D)$ — the distance to the nearest singular point. There is no “random” breakdown of convergence: the series converges precisely up to the nearest singularity.

Important Expansions

Geometric series: $1/(1−z) = \sum_{n=0}^{\infty} z^n$ for $|z| < 1$.

Exponential: $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$ for all $z \in \mathbb{C}$.

Logarithm: $\ln(1+z) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{z^n}{n}$ for $|z| < 1$.

sin z: $\sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!}$ for all $z \in \mathbb{C}$.

Holomorphy $\iff$ Analyticity

In real analysis, $C^\infty \neq$ analyticity: $f(x) = e^{-1/x^2}$ (for $x \neq 0$), $f(0) = 0$ — infinitely differentiable, but the Taylor series at zero equals $0 \neq f$.

In complex analysis: holomorphy $\iff$ analyticity (equality to the power series). The reason: the Cauchy integral formula makes expansion into a series automatic.

The Principle of Analytic Continuation

If $f$ and $g$ are holomorphic in the domain $D$ and coincide on a sequence of points with a limit point inside $D$, then $f \equiv g$ in $D$.

Corollary: a holomorphic function is determined by its values on any “piece” of the domain—for example, on a small segment of the real axis.

The Mean Value Theorem

$ f(z_0) = \frac{1}{2\pi} \int_{0}^{2\pi} f(z_0 + re^{i\theta}) d\theta. $

The value at the center equals the mean value over the circle. This theorem implies the maximum principle.

Numerical Example

Problem: Find the radius of convergence of the Taylor series $f(z) = 1/(1+z^2)$ at the point $z_0 = 0$ and write the first four terms.

Step 1. Zeros of the denominator: $1 + z^2 = 0 \rightarrow z = \pm i$. Distance from $0$ to the nearest singular point: $R = |\pm i| = 1$. The series converges in the disk $|z| < 1$.

Step 2. Expand via geometric series: $ \frac{1}{1+z^2} = \frac{1}{1 - (-z^2)} = \sum_{n=0}^{\infty} (-z^2)^n = \sum_{n=0}^{\infty} (-1)^n z^{2n}. $ $f(z) = 1 - z^2 + z^4 - z^6 + \ldots$

Step 3. Check via derivatives: $f'(z) = -2z/(1+z^2)^2 \rightarrow f'(0) = 0$. The coefficient of $z$: $a_1 = 0$ ✓. $f''(z)|_{z=0} = -2 \rightarrow a_2 = -2/2! = -1$ ✓.

Step 4. On the real axis: $1/(1+x^2) = 1 - x^2 + x^4 - \ldots$ — the well-known series for $\arctan'(x)$. Integrating: $\arctan x = x - x^3/3 + x^5/5 - \ldots$ (Leibniz series, $|x| < 1$).

Practical Application

Digital signal processing: expansion of transfer functions into power series in $z$ ($z$-transform) underlies the design of digital filters. The radius of convergence determines the stability region of the filter.

Additional Aspects

The radius of convergence $R$ is associated with the distance to the nearest singular point: $R =$ distance from the center of expansion to the nearest pole/branch point. The Cauchy–Hadamard formula $1/R = \limsup |a_n|^{1/n}$ provides a way to estimate $R$ via the coefficients. In practice, truncation of the Taylor series is used as a numerical method (for example, expansion of $\exp$, $\sin$, $\cos$ in libraries with fixed precision), as well as a theoretical tool: the series coefficients carry information about the growth of the function (Hadamard's factorization theorems). In combinatorics, generating functions are essentially Taylor series, and the theorem on expansion gives the asymptotics of a numerical sequence through analysis of the nearest singularity (Darbo's method, saddle point method).

Connection with Other Areas of Mathematics

In the theory of differential equations, analytic solutions are described through Taylor series via the Frobenius method: Legendre, Hermite, and Bessel obtained orthogonal systems of functions as solutions to linear second order equations with regular singularities. The Cauchy–Kovalevskaya theorem states the existence and uniqueness of a local analytic solution for analytic coefficients and initial data; the proof relies on expanding the unknown function into a power series and solving the recurrence coefficientwise.

In functional analysis, power series define analytic functional calculi: via the spectral theorem for self-adjoint operators, $f(A)$ is defined for an analytic function $f$, using a series expansion around the spectrum of $A$. The Dunford–Schwartz approach to functional calculus for bounded operators on Banach spaces is based on the Cauchy integral formula and subsequent expansion into series.

In algebra, formal power series form local rings appearing in algebraic geometry: the local ring of an analytic function at a point is isomorphic to the ring of formal series over the maximal ideal. In the theory of modules and deformations (Serre, Grothendieck), such local structures describe the infinitesimal properties of manifolds.

In probability, analytic properties of generating functions of moments and probability distributions are used for deriving limit theorems. Cramér's work in 1938 on large deviations relies on the analyticity of the logarithm of the moment generating function. Analytic continuation of characteristic functions is also connected with exponential series.

Numerical methods, starting from Newton and Lagrange, use truncated Taylor series in Runge–Kutta schemes, linearization methods, and construction of multipoint formulas. In approximation theory, Chebyshev and Weierstrass linked polynomial approximations with local Taylor series, although Weierstrass's global uniform approximation result is based on more general polynomials, not just partial sums of series.

Historical Background and Development of the Idea

The first expansions into power series appeared in Newton’s manuscripts at the end of the 17th century during the analysis of binomial expressions. Brook Taylor in the 1715 treatise Methodus incrementorum directa et inversa systematized the formula for expanding a function near a point, however, strict justification of convergence was absent. Lagrange in 1797 in Théorie des fonctions analytiques considered functions as generated by their series, effectively taking analyticity as the definition. The shift toward rigorous theory is associated with Cauchy: his Cours d’analyse (1821) introduced the concept of convergence and justified the use of power series via integral formulas. Development of complex analysis by Weierstrass, Riemann, and Laurent in the mid-19th century led to a precise description of the radius of convergence through the nearest singularities, as well as to representing holomorphic functions as sums of power series in each domain. At the end of the 19th and beginning of the 20th centuries, Poincaré and Bourchs extended the application of power series to asymptotic expansions and divergent series.