So the convolution theorem well, actually, before I even go to the convolution theorem, let me define what a convolution is.

So we just showed you that the convolution if I define

We're taking the convolution of sine of t with cosine of t.

And to kind of give you that comfort, let's actually compute a convolution.

So the convolution of sine of t with cosine of t is 1 2t sine of t.

So the convolution of f with g, and this is going to be a function of t, it equals this.

But in this video I just want to make you comfortable with the idea of a convolution, especially in the context of taking Laplace transforms.

And in this video, I'm not going to dive into the intuition of the convolution, because there's a lot of different ways you can look at it.

In this video, I'm going to introduce you to the concept of the convolution, one of the first times a mathematician's actually named something similar to what it's actually doing.

So if I convolute f with g so this means that I'm going to take the convolution of f and g, and this is going to be a function of t.

So, hopefully, you have a little of intuition of well, not intuition, but you at least have a little bit of hands on understanding of how the convolution can be calculated.

It has a lot of different applications, and if you become an engineer really of any kind, you're going to see the convolution in kind of a discreet form and a continuous form, and a bunch of different ways.

And the whole reason why I went through all of this mess and kind of bringing out the neurons that had the trig identities memorized or having to reproof them or whatever else is to just show you that this convolution, it is convoluted and it seems a little bit bizarre, but you really can take the convolutions of actual functions and get an actual answer.

And the definition of a convolution, we're going to do it over a well, there's several definitions you'll see, but the definition we're going to use in this, context there's actually one other definition you'll see in the continuous case, is the integral from 0 to t of f of t minus tau, times g of t let me just write it sorry, it's times g of tau d tau.

So if we make this subsitution, this you'll find on the inside cover of any trigonometry or calculus book, you get the convolution of f and g is equal to I'll just write that f star g I'll just write it with that is equal to the integral from 0 to t of, instead of sine of t minus tau, I'm going to write this thing right there.