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:
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 \(\xi\).
We can add sinusoidal waves to get new waves.
For example
\(s_N(x)=2\sin(x+3)+\sin(-4x)+\dfrac{1}{2}\cos(x)\)
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:
\(a=[2,-1,\dfrac{1}{2}]\)
\(b=[1,4,1]\)
\(c=[3,0,\dfrac{\tau}{8}]\)
We can move terms around to get:
\(s(x)=\sum^m_{i=1}a_i\sin(b_ix+c_i)\)
Where:
\(a=[2,\dfrac{1}{2},-1]\)
\(b=[1,1,4]\)
\(c=[3,\dfrac{\tau}{8},0]\)
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:
\(s(x)=\sum^m_{i=1}a_i\sin(b_ix+c_i)\)
\(a=[,-1]\)
\(b=[,4]\)
\(c=[,0]\)
\(\hat f(\Xi )=\int_{-\infty}^{\infty }f(x)e^{-2\pi ix\Xi }dx\)
\(f(x)=\int_{-\infty}^{\infty }\hat f(\Xi )e^{2\pi ix\Xi }d\Xi\)