maths.free › Complex Analysis › Power series and analytic continuation

Power series and analytic continuation

Radius of convergence, the identity theorem, and extending ζ(s) beyond its series.

A holomorphic function equals its Taylor series on the largest disc avoiding singularities. Two holomorphic functions agreeing on a small set agree everywhere they are both defined, so a function can be extended uniquely, which is how ζ(s) exists at s = −1 (and equals −1/12). Picture it: the geometric series 1/(1 − x) converging inside |x| < 1, though the function lives everywhere except x = 1. Think it: this rigidity is the engine of the Riemann Hypothesis page.

Worked example · taylor series of 1/(1-x)

Calculus · Taylor-Series

Taylor series of 1/(1 - x)

\frac{1}{1 - x}

Step by step

  1. f(x) = \frac{1}{1 - x}

    Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

  2. f^{(0)}(0) = \frac{1}{1 - x}\big|_{x=0} = 1

    Derivative 0 at the centre.

  3. f^{(1)}(0) = \frac{1}{\left(1 - x\right)^{2}}\big|_{x=0} = 1

    Derivative 1 at the centre.

  4. f^{(2)}(0) = - \frac{2}{\left(x - 1\right)^{3}}\big|_{x=0} = 2

    Derivative 2 at the centre.

  5. f^{(3)}(0) = \frac{6}{\left(x - 1\right)^{4}}\big|_{x=0} = 6

    Derivative 3 at the centre.

  6. f^{(4)}(0) = - \frac{24}{\left(x - 1\right)^{5}}\big|_{x=0} = 24

    Derivative 4 at the centre.

  7. f^{(5)}(0) = \frac{120}{\left(x - 1\right)^{6}}\big|_{x=0} = 120

    Derivative 5 at the centre.

  8. x^{5} + x^{4} + x^{3} + x^{2} + x + 1

    Assemble the terms up to degree 5.

Reveal the answer
x^{5} + x^{4} + x^{3} + x^{2} + x + 1 + O((x-0)^{6})

Now you Pick a problem, or type or draw your own. Every step, a picture, the answer hidden until you ask.

Keep your own working

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Sign Up Login

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

How to: Power series and analytic continuation

  1. Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.
  2. Derivative 0 at the centre.
  3. Derivative 1 at the centre.
  4. Derivative 2 at the centre.
  5. Derivative 3 at the centre.
  6. Derivative 4 at the centre.
  7. Derivative 5 at the centre.
  8. Assemble the terms up to degree 5.

Questions people ask

Why is complex differentiability so much stronger than real?

The limit must be the same from every direction in the plane, not just two. That forces the Cauchy-Riemann equations, which in turn force infinitely many derivatives and a convergent Taylor series.

What is a residue?

The coefficient of 1/(z − a) in the Laurent series at a singularity a. The residue theorem says a contour integral equals 2πi times the sum of the residues inside, which evaluates many real integrals in one line.

More in Complex Analysis

The complex plane and Euler's formulaHolomorphic functions and the Cauchy-Riemann equationsContour integrals and the residue theorem