Series & Singularities

Taylor series, Laurent series, and the classification of singularities

Series & Singularities

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.

1. Taylor Series Animator

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.

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

2. Laurent Series Explorer

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.

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)]

3. Singularity Classifier

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
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

4. 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.

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!

Key Takeaways

1. Taylor Series: Every analytic function equals its Taylor series within the radius of convergence. This is much stronger than in real analysis!

2. Laurent Series: Near singularities, we need negative powers. The principal part captures the singular behavior.

3. Singularity Types: Removable (harmless), poles (controlled blow-up), essential (chaotic). Classification determines residue calculation.

4. Residues: The a₋₁ coefficient in Laurent series. This single number determines contour integrals via the Residue Theorem!

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