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We will briefly discuss zeros and poles of meromorphic functions here. We assume the Laurent series exist in the vicinity of a point z₀:

Clearly if we want f(z₀) is zero we require a_k is zero for k smaller and equals to zero. Consequently we define that z₀ is a zero of order n if:

We can observe some useful properties from (2). Firstly, an n order zero implies that up to the (n-1)^th derivative of f(z) are also zero at z₀ and vice versa. We can use this property to determine the order of zeros of a function, in the case that they are not so obvious. Secondly, The zeros of f(z) are the poles of 1/f(z) for obvious reason, provided that f(z) is not identically zero. We would like to find the properties of the Laurent series of  1/f(z):

The value of m can be determined by multiply the denominator of right hand side : We obtain a series that constantly equals to one. This requires m=n, and we see the coefficients b_k are fixed by the values of a_k .

In summary we found that

In particular, if f(z) is analytic and non-zero at z₀ , we now from (2) know that n=0 and thus from (4) 1/f(z) is also analytic and non-zero at z_0.

Read Zeros and Poles in full

1. Weierstrass Approximation Theorem

We want to show that The set of polynomials is dense in the space of continuous functions. We shall firstly define the Bernstein Polynomials as

where f is a continuous function with domain [0,1] and we can thus see that it is uniformly continuous.This means that in δ-ε language :

here we manually set an interval in the domain. However, we are interested in the general relationship between two outputs without limitation ( other than the distance between two input must be in the range [0,1] ). So we want to now what happens when the |x -ξ| is larger than δ.

We recall the definition of norms in linear algebra. For finite vector space, the infinite norm is defined as:

It is easy to understand that this gives the largest entry (in terms of magnitude) of a vector. Note that this is true even if there exists more than one maximum entry. We expand this property into function space and define M to be the infinite norm of f(x). Then for |x -ξ| is larger than δ we write:

Then combine (3) and (4) we obtain

We now want to use this relation to show that Bernstein polynomials can be used to approximate f(x). We notice that the Bernstein polynomial of f(ξ) is just f(ξ), where binomial expansion has been used to obtain this result. Then, using the fact that Bernstein polynomial is linear for f(x),

In the second line above we putted the first term of the function into the expression of the polynomial and it came out will some calculation. Obviously, we need to set x=ξ to proceed and this yields

What does this means? Remember that we have the freedom to make n as large as we want. This means that we can make the difference between Bernstein polynomials and our function as arbitrarily small as we want by approaching n to infinity.  i.e., Bernstein  polynomials converges to our f(x).

[Literature: Matt Young, MATH 328 Notes, Queen’s University at Kingston, 2006]

2. Taylor’s Theorem

Now we are ready to prove Taylor’s Theorem. Knowing polynomials span the space of continuous function, we now assume that the function is also infinitely differentiable. In this special case, instead of using Bernstein polynomials, we want to use the basis consists of 1,x,…,xⁿ. Matching the value of our series and our function at a particular point yields the ordinary form of single variable Taylor series.

3. Vector fields Taylor Expansion

To be continued…

 

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