Heaviside Step Function

del

One of the more interesting piecewise functions would be the Heaviside step function, which is useful in function analysis and generating step functions. The fourier series of the function is a lot more complex than you would expect, incorporating the delta function. It was first used for telegraph communications, but it's a lot more useful and pops up in a lot of fields involving a discrete signal change, like in biochemistry.

Ally

Hi!

I've been looking into the heaviside step function a bit.

Interestingly, people can't seem to agree on a single definition for the unit step function in general, I've seen several definitions pop up including $f(0)=\pm 1$, $f(0)=\frac{1}{2}$ or $f(0)$ being completely undefined.

I believe I've seen the fourier series of it before! At least, more so for $\operatorname{sgn}(x)$ - but some definitions of the step function are simply transformations of $\operatorname{sgn}(x)$ interestingly enough.

del

one of the more interesting ideas i came across while discussing heaviside functions with some friends is how a summation of heaviside functions is a reimann summation over the function that the heaviside function summation uniformly converges to. It's fairly obvious once you remember the basic definition of integration being a sum of rectangles, but still.

del

One of the other places the step function pops up is in solving piecewise ODEs using the Laplace transform. It's a great way to solve ODEs algebraically rather than analytically.