# Taylor and Fourier analysis

## Power series

### Power series

of the form:

$$\sum_{n=0}a_n(x-c)^n$$

#### Smoothness of power series

Power series are all smooth. That is, they are infinitely differentiable.

## Taylor series

### Taylor series

$$f(x)$$ can be estimated at point $$c$$ by identifying its repeated differentials at point $$c$$.

The coefficients of an infinate number of polynomials at point $$c$$ allow this.

$$f(x)=\sum_{i=0}^{\infty }a_i(x-c)^i$$

$$f'(x)=\sum_{i=1}^{\infty }a_i(x-c)^{i-1}i$$

$$f''(x)=\sum_{i=2}^{\infty }a_i(x-c)^{i-2}i(i-1)$$

$$f^j(x)=\sum_{i=j}^{\infty }a_i(x-c)^{i-j}\dfrac{i!}{(i-j)!}$$

For $$x=c$$ only the first term in the series is non-zero.

$$f^j(c)=\sum_{i=j}^{\infty }a_i(c-c)^{i-j}\dfrac{i!}{(i-j)!}$$

$$f^j(c)=a_ij!$$

So:

$$a_j=\dfrac{f^j(c)}{j!}$$

So:

$$f(x)=\sum_{i=0}^\infty (x-c)^i \dfrac{f^i(c)}{i!}$$

### Convergence

If $$x=c$$ then the power series will be equal to $$a_0$$.

For other values the power series may not converge.

$$\dfrac{1}{R}={\lim \sup}_{n_\rightarrow \infty} (|a_n|^{\dfrac{1}{n}})$$

### Maclaurin series

A Taylor series around $$c=0$$.

$$f(x)=\sum_{i=0}^\infty (x-c)^i \dfrac{f^i(c)}{i!}$$

$$f(x)=\sum_{i=0}^\infty (x)^i \dfrac{f^i(0)}{i!}$$

For example, for:

$$f(x)=(1-x)^{-1}$$

$$f^i(0)=i!$$

So, around $$x=0$$:

$$f(x)=\sum_{i=0}^\infty (x)^i$$

### Fourier transforms

#### Taylor series of matrices

We can also use Taylor series to evaluate functions of matrices.

Consider $$e^M$$

We can evaluate this as:

$$e^M=\sum_{k=0}^\infty \dfrac{1}{k!}M^k$$

### Analytic functions

(root test, direct comparison test, rate of convergence, radius of convergence)

## Fourier analysis

### Representing wave functions

Wave function are of the form:

$$\cos(ax + b)$$

$$\sin(ax + b)$$

We can use the following identities:

• $$\cos(x)=\sin(x+\dfrac{\tau }{8})$$

• $$\sin(-x)=-\sin(x)$$

• $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$$

So we can write any function as:

### Fourier series

#### Fourier series

Motivation: we have a function we want to display as another sort of function.

More specifically, a function can be shown as a combination of sinusoidal waves.

To frame this let’s imagine a sound wave, with values $$f(t)$$ for all time values $$t$$. We can imagine this as a summation of sinusoidal functions. That is:

$$f(t)=\sum_{n=0}^{\inf } a_ncos(nw_0t)$$

We want to get another function $$F(\xi )$$ for all frequencies $$\xi$$.

#### Combinations of wave functions

We can add sinusoidal waves to get new waves.

For example

$$s_N(x)=2\sin(x+3)+\sin(-4x)+\dfrac{1}{2}\cos(x)$$

#### As a summation of series

We can simplify arbitrary series using the following identities:

$$\cos(x)=\sin(x+\dfrac{\tau }{8})$$

$$\sin(-x)=-\sin(x)$$

So we have:

$$s(x)=2\sin(x+3)-\sin(4x)+\dfrac{1}{2}\sin(x+\dfrac{\tau }{8})$$

We can put this into the following format:

$$s(x)=\sum^m_{i=1}a_i\sin(b_ix+c_i)$$

Where:

$$a=[2,-1,\dfrac{1}{2}]$$

$$b=[1,4,1]$$

$$c=[3,0,\dfrac{\tau}{8}]$$

#### Ordering by $$b$$

We can move terms around to get:

$$s(x)=\sum^m_{i=1}a_i\sin(b_ix+c_i)$$

Where:

$$a=[2,\dfrac{1}{2},-1]$$

$$b=[1,1,4]$$

$$c=[3,\dfrac{\tau}{8},0]$$

#### Adding waves with same frequency

We know that:

$$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$$

So:

$$\sin(b_ix+c_i)=\sin(b_ix)\cos(c_i)+\sin(c_i)\cos(b_ix)$$

If $$2$$ terms have the same value for $$b_i$$, then:

$$a_i\sin(b_ix+c_i)+a_j\sin(b_jx+c_j)=a_i\sin(b_ix+c_i)+a_j\sin(b_ix+c_j)$$

$$a_i\sin(b_ix+c_i)+a_j\sin(b_jx+c_j)=a_i\sin(b_ix)\cos(c_i)+a_i\sin(c_i)\cos(b_ix)+a_j\sin(b_ix)\cos(c_j)+a_j\sin(c_j)\cos(b_ix)$$

So we now get for:

$$s(x)=\sum^m_{i=1}a_i\sin(b_ix+c_i)$$

$$a=[,-1]$$

$$b=[,4]$$

$$c=[,0]$$

### Fourier transforms

#### Fourier transform

$$\hat f(\Xi )=\int_{-\infty}^{\infty }f(x)e^{-2\pi ix\Xi }dx$$

#### Inverse Fourier transform

$$f(x)=\int_{-\infty}^{\infty }\hat f(\Xi )e^{2\pi ix\Xi }d\Xi$$