Series & Singularities

Taylor series, Laurent series, and the classification of singularities

Series & Singularities

Zoom in on an analytic function and it becomes a power series. Zoom in on a point where it breaks and the series grows negative powers, which tell you what kind of break it is.

1. Taylor Series Animator

Choose a function and raise N to add terms, toggling between the partial sum and the actual function. The match holds inside the radius of convergence, which ends at the nearest singularity.

Exponential
e^z = Σ z^n / n!
Radius of Convergence: ∞
Converges everywhere (R = ∞). An entire function.
Sₙ(z) = Σ(k=0 to 5) aₖ·zᵏ

Partial Sum (N = 5)

Taylor Coefficients (first 6 terms):

a0 = 1.0000
a1 = 1.0000
a2 = 0.5000
a3 = 0.1667
a4 = 0.0417
a5 = 0.0083

📊 Understanding Taylor Series:

  • Partial sums S_N = a₀ + a₁z + a₂z² + ... + aₙzᴺ approximate the actual function
  • As N increases, the approximation gets better (within the radius of convergence)
  • For entire functions (R = ∞), series converges everywhere
  • For functions with poles, series stops at the nearest singularity
  • Toggle between partial sum and actual function to see convergence quality
Go deeper:Taylor Series

One of the most powerful features of complex analysis is that every analytic function can be represented as a power series (Taylor series). Near singularities, we need Laurent series which include negative powers.

Understanding singularities - points where functions "break" - is crucial for the Residue Theorem, the crown jewel of complex analysis that allows us to evaluate difficult real integrals using complex methods.

Every analytic function can be represented as a Taylor series around any point z₀ where it's analytic:

f(z) = Σn=0∞ aₙ(z-z₀)ⁿ    where    aₙ = f(n)(z₀) / n!

Watch as partial sums converge to the actual function. The series converges within a circle (the radius of convergence) until it hits the nearest singularity.

2. Laurent Series Explorer

Choose a function with a singularity and read its series split in two: negative powers (the principal part) and ordinary ones. The a₋₁ coefficient is the residue.

Reciprocal: f(z) = 1/z
Singularity Type: Simple pole at z = 0+0i
Simplest pole. Residue = 1.

Domain Coloring Visualization:

White regions indicate singularities (poles). Essential singularities show wild color oscillations.

Laurent Series Decomposition:

Principal Part (Negative Powers):
1/z
This part captures the singular behavior
Regular Part (Non-negative Powers):
0
This is just a Taylor series (analytic part)

Residue (a₋₁ coefficient):

Res(f, 0+0i) =
1.00 + 0.00i
The residue is the coefficient of the 1/z term in the Laurent expansion
Simple Pole
Laurent series: a₋₁/z + a₀ + a₁z + ...
Only one negative power.
Pole of Order n
Laurent series: a₋ₙ/zⁿ + ... + a₋₁/z + a₀ + ...
Finitely many negative powers.
Essential Singularity
Laurent series: ... + a₋₃/z³ + a₋₂/z² + a₋₁/z + ...
Infinitely many negative powers!

⚡ Laurent Series Key Points:

  • Laurent series = Taylor series + principal part (negative powers)
  • Valid in an annulus around the singularity (between two circles)
  • The residue (a₋₁) is the key to contour integrals: ∮ f(z) dz = 2πi · Res(f, z₀)
  • Essential singularities have infinitely many negative terms - extremely wild behavior!
  • For poles of order n, residue can be computed using: Res = lim (1/(n-1)!) d^(n-1)/dz^(n-1) [(z-z₀)ⁿ f(z)]
Go deeper:Laurent Series

Near singularities, we need Laurent series which include negative powers:

f(z) = Σn=-∞∞ aₙ(z-z₀)ⁿ = ... + a₋₂/(z-z₀)² + a₋₁/(z-z₀) + a₀ + a₁(z-z₀) + a₂(z-z₀)² + ...
The principal part (negative powers) captures the singular behavior

The coefficient a₋₁ is called the residue - it's the key to evaluating contour integrals via the Residue Theorem.

3. Singularity Classifier

Choose a function and see its singularity classified as removable, a pole, or essential. Compare the calm colors around sin(z)/z with the storm around e^(1/z).

f(z) = sin(z)/z at z = 0+0i
lim(z→0) sin(z)/z = 1. We can "remove" the singularity by defining f(0) = 1.

Singularity Classification:

✓
Removable
Limit exists
○
Pole
Finite neg. powers
○
Essential
Infinite neg. powers

Limit as z → 0+0i:

lim f(z) = 1.0000 + 0.0000i
Since the limit exists and is finite, we can define f at the singularity to make the function analytic everywhere!

Domain Coloring Visualization:

Removable singularities appear as smooth color transitions.

🔬 How to Classify Singularities:

Removable:
  • lim(z→z₀) f(z) exists and is finite
  • Laurent series has no negative powers
  • Can be "fixed" by defining f(z₀) = lim value
Pole of order n:
  • lim(z→z₀) |f(z)| = ∞
  • Laurent series: aₙ/zⁿ + ... + a₋₁/z + (regular part)
  • Test: (z-z₀)ⁿ·f(z) has removable singularity at z₀
Essential:
  • lim(z→z₀) f(z) does not exist (not even ∞)
  • Laurent series has infinitely many negative powers
  • Picard's theorem: f gets arbitrarily close to every value near z₀
Go deeper:Three Types of Singularities

Singularities come in three types, each with distinct behavior:

Removable Singularity
Laurent series has no negative powers. Can be "fixed" by defining f(z₀) appropriately.
Example: sin(z)/z at z=0
Pole (Order n)
Laurent series has finitely many negative powers. The highest is the pole order.
Example: 1/z² has pole of order 2
Essential Singularity
Laurent series has infinitely many negative powers. Wildly oscillating behavior.
Example: e^(1/z) at z=0

4. Analytic Continuation

Switch between the series and its continuation. The series 1 + z + z² + ... only converges for |z| < 1, but 1/(1-z) lives everywhere except z = 1.

Geometric Series: 1/(1-z)
Series: 1 + z + z² + z³ + ...
The geometric series converges only for |z| < 1, but the function 1/(1-z) is defined everywhere except z = 1. Analytic continuation extends the series beyond its circle of convergence.
Original Domain (Series)
|z| < 1
Where the power series converges
Extended Domain (Function)
ℂ \ {1}
Where the analytic continuation is defined
Partial sum with 10 terms

Series Approximation (N = 10)

Series diverges outside its circle of convergence (artifacts/discontinuities visible)

🔄 Analytic Continuation Explained:

  • A power series has a radius of convergence R - it only converges for |z-z₀| < R
  • But the function it represents may be defined on a much larger domain
  • Analytic continuation is the process of extending a function beyond its series' convergence radius
  • For 1/(1-z): series converges in |z| < 1, but function is defined on all of ℂ except z=1
  • This leads to multi-valued functions (like log, √) which need branch cuts to be single-valued
  • Analytic continuation is unique: if two analytic functions agree on any open set, they agree everywhere they're both defined!
Go deeper:Analytic Continuation

Sometimes a function defined on a small domain can be extended to a larger domain while preserving analyticity:

For example, the geometric series 1 + z + z² + z³ + ... = 1/(1-z) converges only for |z| < 1, but the function 1/(1-z) is defined everywhere except z = 1.

This process of extending functions is called analytic continuation and reveals deep connections between seemingly different functions.

Analytic Continuation: Functions can often be extended beyond their original domain. This leads to concepts like Riemann surfaces and multi-valued functions.

Key Takeaways

  • Inside its radius of convergence, an analytic function equals its Taylor series
  • Laurent series add negative powers to describe a function near a singularity
  • Singularities are removable, poles, or essential
  • The a₋₁ coefficient, the residue, drives the Residue Theorem

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