Power series
Taylor series
A Taylor series can be viewed in two ways:
- As a way of approximating any analytic function as a polynomial
- A one-to-one correspondence between every analytic function and an order 𝜔 polynomial
(which may be conceived as a vector space).
An order-𝑛 Taylor polynomial is constructed
so that its value
and all its derivatives up to order 𝑛
match that of the original function 𝑓
at a point 𝑓(𝑎).
It is determined to be
𝑇𝑓𝑛,𝑎(𝑥)=𝑛∑𝑚=0𝑓(𝑚)(𝑎)𝑚!(𝑥−𝑎)𝑚
In the case of 𝑎 =0,
it is called a Maclaurin polynomial
𝑇𝑓𝑛,0(𝑥)=𝑛∑𝑚=0𝑓(𝑚)(0)𝑚!𝑥𝑚
As a correspondance,
we have the statement
𝑓(𝑥)=𝑇𝑓∞,0(𝑥)=∞∑𝑚=0𝑓(𝑚)(0)𝑚!𝑥𝑚
Error
The error of an order 𝑛 Taylor polynomial is given by Taylor's theorem,
which converges to zero as 𝑛 →∞ if and only if 𝑓 is an analytic function.
Complex functions
The notion of the Taylor series applies to complex functions as well.
For any function 𝑓 that is analytic for some domain 𝐷 ⊆ℂ where 𝑧0 ∈𝐷,
𝑓(𝑧)=𝑇𝑓∞,𝑧0(𝑧)=∞∑𝑚=0𝑓(𝑚)(𝑧0)𝑚!(𝑧−𝑧0)𝑚
This series converges for any disc centred at 𝑧0 contained by 𝐷.1
In Complex analysis MOC, a similar power series called the Laurent series includes negative powers, that is 𝑚 ranges through all of ℤ.
This generalises the Taylor series to cover some non-analytic functions.
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