Power Series and Analytic Functions

Holomorphic functions ARE power series. That is the rigidity statement: every holomorphic function on a domain equals its Taylor series in a disk around each point, with radius determined by the nearest singularity. Around an isolated singularity, the more general Laurent series adds negative powers of $(z - z\_0)$, and the tail of those negative powers classifies the singularity: zero negative terms means removable, finitely many negative terms means a pole, infinitely many means essential. This is the algebraic engine behind the residue calculus.

Power series and radius of convergence

A power series centred at $z\_0$ has the form

The radius of convergence $R$ is the supremum of $|z - z\_0|$ for which the series converges. The Cauchy-Hadamard formula gives it:

Inside the disk $|z - z\_0| < R$ the series converges absolutely (and uniformly on compact subsets) to a holomorphic function. On the boundary $|z - z\_0| = R$, convergence is delicate, some series converge everywhere on the boundary, others nowhere. Outside the disk the series diverges. The radius $R$ equals the distance from $z\_0$ to the nearest singularity of the limit function.

Taylor series of holomorphic functions

If $f$ is holomorphic on a disk $D$ centred at $z\_0$, then $f$ equals its Taylor series there:

This is genuinely an equality, not merely an asymptotic expansion. The radius of convergence equals the distance from $z\_0$ to the boundary of the maximal disk where $f$ remains holomorphic. For example, $\\frac{1}{1 - z}$ has Taylor series $1 + z + z^2 + \\cdots$ around $z = 0$ with $R = 1$, the nearest singularity is at $z = 1$.

The widget visualizes the disk of analyticity. Try $f(z) = 1/(1 - z)$: the Taylor series at $0$ converges inside $|z| < 1$ and diverges outside. The singularity at $z = 1$ on the boundary blocks any extension.

Laurent series and isolated singularities

If $f$ is holomorphic on an annulus $r < |z - z\_0| < R$ (typical case: $f$ has an isolated singularity at $z\_0$), then $f$ has a Laurent series there:

with both positive and negative powers. The negative-power tail $\\sum\_{n=1}^{\\infty} \\frac{a\_{-n}}{(z - z\_0)^n}$ is the principal part. The coefficient $a\_{-1}$ is the residue (used in the residue theorem).

The principal part classifies the singularity at $z\_0$:

• Removable singularity: all $a\_{-n} = 0$. Defining $f(z\_0) = a\_0$ makes $f$ holomorphic at $z\_0$. Example: $\\sin z / z$ at $z = 0$.
• Pole of order $m$: $a\_{-m} \\neq 0$ but $a\_{-n} = 0$ for $n > m$. The principal part is a polynomial in $1/(z - z\_0)$ of degree $m$. Example: $\\frac{1}{(z - 1)^2}$ has a pole of order 2 at $z = 1$.
• Essential singularity: infinitely many $a\_{-n} \\neq 0$. By Picard's great theorem, near such a singularity $f$ takes every complex value (with at most one exception) infinitely many times. Example: $e^{1/z}$ at $z = 0$.

• Numerical analysis: Pade approximations and continued-fraction extrapolations rely on the Laurent and Taylor structure of meromorphic functions to compute extreme-precision approximations of special functions (gamma, zeta, polylog) far outside their naive Taylor disks.
• Control theory: Transfer functions are rational and live entirely in the complex plane. The pole-zero diagram, the locations of $a\_{-1} \\neq 0$ residues, encodes system stability, transient response, and resonance frequencies.
• Optics: The Gouy phase shift of a focused Gaussian beam is computed from the Laurent expansion of the beam's complex amplitude around the focus, an essential-singularity-like behaviour gives the $\\pi/2$ phase anomaly observed experimentally.
• Statistical physics: The radius of convergence of a partition-function series gives the location of the nearest critical point in the complex temperature plane (the Lee-Yang programme). Phase transitions are diagnosed by where the Taylor series breaks.

Pause and think: What is the radius of convergence of $\\sum n! \\, z^n$? Apply the ratio test: $|a\_{n+1}/a\_n| \\cdot |z| = (n+1)|z| \\to \\infty$ for any $z \\neq 0$. So $R = 0$: the series only converges at $z = 0$. Compare to $\\sum z^n/n!$, which has $R = \\infty$, convergent everywhere, and equals $e^z$.

Try it

• Compute the radius of convergence of $\\sum\_{n \\geq 1} z^n / n^2$. (Hint: $|a\_n|^{1/n} = (1/n^2)^{1/n} \\to 1$, so $R = 1$.)
• Classify the singularity at $z = 0$ of $f(z) = (1 - \\cos z) / z^2$. (Hint: expand $\\cos z = 1 - z^2/2 + z^4/24 - \\cdots$.)
• Find the Laurent expansion of $f(z) = 1/(z(z - 1))$ on $0 < |z| < 1$. (Hint: partial fractions, then geometric series in $z$.)
• True or false: if $f$ has an essential singularity at $z\_0$, there exists a sequence $z\_n \\to z\_0$ with $f(z\_n) \\to \\infty$ and a different sequence $w\_n \\to z\_0$ with $f(w\_n) \\to 0$. (Picard predicts even more: every value is hit infinitely often.)

A trap to watch for

The Laurent expansion of a function depends on which annulus you are in. The function $f(z) = 1/(z(z-1))$ has TWO different Laurent expansions: one on $0 < |z| < 1$ (negative powers from the $1/z$ pole, positive powers from the $1/(1-z)$ geometric series) and a completely different one on $|z| > 1$ (negative powers from both). Beginners conflate these. Always specify the annulus before computing the Laurent series, and remember that the geometric series $\\sum z^n$ converges only for $|z| < 1$; for $|z| > 1$ rewrite $1/(1 - z) = -1/z \\cdot 1/(1 - 1/z) = -\\sum 1/z^{n+1}$.

What you now know

You can apply the Cauchy-Hadamard formula, expand holomorphic functions in Taylor series, work in annuli with Laurent series, and classify singularities by their negative-power tail. The next section is the geometric side of complex analysis: conformal maps, which preserve angles and let us transport Laplace's equation between geometrically very different domains.

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References

• Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, Sections 9.6-9.7.
• Ahlfors, L. V. (1979). Complex Analysis (3rd ed.). McGraw-Hill, ch. 5.
• Stein, E. M., Shakarchi, R. (2003). Complex Analysis. Princeton University Press, ch. 2 and 3.
• Conway, J. B. (1978). Functions of One Complex Variable I (2nd ed.). Springer, ch. 5.
• Brown, J. W., Churchill, R. V. (2014). Complex Variables and Applications (9th ed.). McGraw-Hill, ch. 5.