About this video
This video lays the groundwork for Fourier series by proving two definite integral facts over the interval zero to two pi. The integral of sin(mt) and the integral of cos(mt) are both zero for any nonzero integer m. The proofs use antiderivatives and the fact that the trig functions repeat every two pi, and these results will be used to find the Fourier coefficients in later videos.
You'll be able to
- state why a periodic function can be written as a sum of weighted sines and cosines.
- evaluate the integral of sin(mt) and cos(mt) over zero to two pi for nonzero integer m.
- explain why the interval length of the period matters when integrating trig functions.
Key words
- Periodic function
- A function that repeats its values after a fixed period, here two pi.
- Fourier coefficients
- The weights of the sine and cosine terms that reproduce a periodic function.
- Definite integral
- The integral of a function between two limits, here zero and two pi.
- Nonzero integer m
- A whole number other than zero, used as the frequency multiplier in sin(mt) and cos(mt).
- Antiderivative
- A function whose derivative is the given function, used to evaluate definite integrals.
Transcript
- [Voiceover] In the last video we introduced the idea that we could represent any arbitrary, periodic function by a series of weighted cosines and sines. And what I'm gonna start doing in this video, is starting to establish our mathematical foundation. So it'll be pretty straightforward for us to find these coefficients that give us that function.
So the first thing I wanna do, the first thing I'm gonna do is establish some truths using, or some truths with the definite integrals. I'm gonna focus over the intervals zero to pi over this video and the next few videos because the function we're approximating has a period of two pi, completes one cycle from zero to two pi, we could've done it over other intervals of length two pi, and if this period was other than two pi, we would've done it over intervals of that period, but I'm focusing on two pi because it makes the math a little bit cleaner, a little bit simpler, and then we can generalize in the future.
So let's just establish some things about definite integrals of trig functions. So the first I wanna establish, I wanna establish that the definite integral from zero to two pi of sine of mx, dx, actually let me stay in t, since our original function is in terms of t.
Sine of mt, dt, I wanna establish that that is equal to zero for any non-zero integer m. For non-zero, non-zero integer, integer m, and I also wanna establish that the integral from zero to two pi of cosine of mt, dt, is equal to zero for any non-zero integer m.
And you might already take this for granted or you feel good about it, or you've already proven it to yourself, and if so, you could actually skip this video. But let's work through it, 'cause it's actually a good review of some integral calculus here. So let's first do this top one, so let me just re-write the integral.
So we're gonna take the integral from zero to two pi of sine mt, dt, now we know we won't take the antiderivative of sine of mt, so we know that the derivative with respect to t of cosine mt, is equal to, what is this, this is going to be equal to m. The derivative of mt with respect to t times the derivative of cosine mt with respect to mt.
So times negative sine of mt, or we could write this is gonna be equal to negative m sine of mt, and I could put a parentheses there if I like. And so I almost have negative m sine of mt, I just wanna have a negative m here. So what if I put a negative m there, but I can't just do that, that would change the value of the expression, but I could also multiply by negative one over m.
Now these two would, if we take the product they're gonna cancel out, we're gonna get our original expression. But this is useful because now we can say this is equal to negative one over m and now the antiderivative of this business right over here, we know is cosine mt.
So it's gonna be cosine mt, cosine mt, evaluated at two pi and zero, two pi and zero and so this is going to be equal to, this is equal to negative one over m times cosine of m times two pi, right this way, cosine, cosine of m times two pi minus cosine of, well it's gonna be m times zero, which is just gonna be, we can just write that as zero.
And so let's see, cosine of any multiple of two pi, well that's just gonna be one, and cosine of zero, well that's also one. So you have one minus one is zero, times negative one over m, well this is all going to evaluate to the result we wanted, this is all going to evaluate to zero.
So we have just proven that first statement, so now let's prove the second one. It's gonna be a very similar argument, so let's re-write it. We're gonna get the integral from zero to two pi of cosine of mt, dt, and now let me engineer this a little bit, we know that the derivative of sine of mt is m cosine mt, so let me multiply and divide by m, and we multiply by an m and divide by an m, not changing the actual value, and so this is going to be equal to one over m, and then the antiderivative of, let me find a nice color here, the antiderivative of that right over there, that should say, the antiderivative of this right over here is sine of mt, so sine of mt.
Notice the derivative of sine of mt is m cosine mt, and we're going to evaluate that at from zero to two pi, so this is going to be equal to, we still have our one over m out front, one over m, and so this is going to be sine of m times two pi, or we could say two pi m, two pi m, minus sine of m times zero, so sine, we can just write that as sine of zero, and what's the sine of any multiple of two pi.
Remember m is a non-zero integer, so any multiple that's gonna be a multiple of two pi here, well that's just going to be zero, and sine of zero is just going to be zero, so this whole thing, this whole thing is just going to be zero, and so we have established our second statement there.
So this is going to be a nice base to build from, and now we're going to do slightly more complex integrals in the next few videos, so there's gonna be hopefully pretty straightforward to find our Fourier coefficients with a little bit of calculus and algebraic manipulation.
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