When you involve sines and cosines, but not as To him in the study of differential equations,īecause a lot of differential equations are easy to solve Weighted sines and cosines, this was explored originally by Fourier, and they're called Fourier Series. Series where you represent something by essentially Well, okay, this seems like a fun little mathematical exercise, but why do folks even do this? Well this was firstĮxplored, and they're named, series like this, infinite
This has a frequency of one over two pi, this has twice the frequency, this has a frequency of one over pi. So let's add a sub two, soĪnother waiting coefficient, times cosine of two t. Is we're gonna add sinusoids of frequencies that are This would look likeĪ very clean sinusoid, not like a square wave. To describe this function, because we know what this would look like. Of cosine of t in it, than it has of sine of t in it. Than b one, well it says, okay, this has a lot more
It would make sense that it would involve some functions Has a period of two pi, and I just set up this one so it does have a period of two pi, well Why am I starting withĬosine of t and sine of t? Well, if our original function Now, why am I starting with cosine of t? And I could also add a sine of t, so plus b sub one, of sine of t. Then, let's start adding some periodic functions here. Going to be based on the average value of theįunction over one period. That'll shift it up or down, and as we'll see, that's It's going to be sum, let's say baseline constant, Of sines and cosines? So can we write it, so So can we take our f of t, and write it as the sum Little bit more clearly, can we take our f of t, Sum of sines and cosines of different periods orĭifferent frequencies? So to write that out a And what we're gonnaĮxplore in this video, is can we take a periodicįunction like this and represent it as an infinite So we could write itsįrequency is equal to one over two pi cycles per second,Ĭycles per second, it could also be described as hertz. We could say two pi, two pi seconds per cycle,
Period is equal to two pi, if we wanna put the units It is a periodic function, that it completes oneĬycle every two pi seconds. Is often described as a square wave, and we see that The graph of y is equal to f of t here, our horizontalĪxis is in terms of time, in terms of seconds.
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