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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.

ຕົວຢ່າງທີ່ໄດ້ເຮັດ: `taylor series of 1/(1-x)`

Calculus · Taylor-Series

Taylor series of 1/(1 - x)

\frac{1}{1 - x}

ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ

f(x) = \frac{1}{1 - x}
Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.
f^{(0)}(0) = \frac{1}{1 - x}\big|_{x=0} = 1
Derivative 0 at the centre.
f^{(1)}(0) = \frac{1}{\left(1 - x\right)^{2}}\big|_{x=0} = 1
Derivative 1 at the centre.
f^{(2)}(0) = - \frac{2}{\left(x - 1\right)^{3}}\big|_{x=0} = 2
Derivative 2 at the centre.
f^{(3)}(0) = \frac{6}{\left(x - 1\right)^{4}}\big|_{x=0} = 6
Derivative 3 at the centre.
f^{(4)}(0) = - \frac{24}{\left(x - 1\right)^{5}}\big|_{x=0} = 24
Derivative 4 at the centre.
f^{(5)}(0) = \frac{120}{\left(x - 1\right)^{6}}\big|_{x=0} = 120
Derivative 5 at the centre.
x^{5} + x^{4} + x^{3} + x^{2} + x + 1
Assemble the terms up to degree 5.

ເປີດເຜີຍຄຳຕອບ

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

Symbols used here

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

O(n^2),\ \Theta,\ \Omega

big-O notation
Grows no faster than n² (up to a constant), for large n.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\theta

theta
The usual name for an angle.

\frac{\partial f}{\partial x}

partial derivative
Derivative with respect to x, holding the other variables fixed.

\iint,\ \oint

double / contour integral
Integral over a region of the plane; integral around a closed curve.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\zeta(s)

Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.

\operatorname{Res}_{z=a} f

residue
Coefficient of 1/(z − a) at a singularity; drives contour integrals.

f: \mathbb{C} \to \mathbb{C},\ \bar{z}

complex function, conjugate
A map of the complex plane; x − iy.

How to: Power series and analytic continuation

Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.
Derivative 0 at the centre.
Derivative 1 at the centre.
Derivative 2 at the centre.
Derivative 3 at the centre.
Derivative 4 at the centre.
Derivative 5 at the centre.
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.

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ເພີ່ມເຕີມໃນ Complex Analysis

The complex plane and Euler's formula Holomorphic functions and the Cauchy–Riemann equations Contour integrals and the residue theorem

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