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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · taylor series of 1/(1-x)

Calculus · Taylor-Series

Taylor series of 1/(1 - x)

\frac{1}{1 - x}

Àwọn ìṣàmúlò-ètò

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.

Fi àwọn àgbèwọlé hàn

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

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Bii o ṣe le: 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.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Complex Analysis

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