Wright State University Lake Campus/2020-1/Intro to imaginary numbers

Students have trouble appreciating the square root of minus 1, or
 * $$\sqrt{-1}=i$$

For this introduction, all you need is the following ideas:

Derivative of a polyomial

 * $$ \frac{d}{dt}Ct^n=nCt^{n-1}$$

Newton's law for an ideal spring

 * $$F = -kx = ma = m\frac{dv}{dt} =m\frac{d^2x}{dt^2}$$

The first step in understanding the solution to this is to imagine that k=m=1:
 * $$\frac{d^2x}{dt^2}=-x$$

What function of time is the negative of it's own derivative?

The following infinite series famously equals it's own second derivative:


 * $$f(t)=1 + t + \frac{1}{2\cdot 1}t^2 + \frac{1}{2\cdot 1}t^2 + \frac{1}{3\cdot 2\cdot 1}t^3 + \frac{1}{4\cdot 3\cdot 2\cdot 1}t^4 + ...$$

See
 * Exponential function
 * commons:File:Serie di Taylor della funzione seno.gif
 * Taylor_series