
Fourier Series 9

Lecture1.1

Lecture1.2

Lecture1.3

Lecture1.4

Lecture1.5

Lecture1.6

Lecture1.7

Lecture1.8

Lecture1.9

Fourier Series Part 4
So how do we decide whether or not a function is even or odd? So that we can decide which terms are going to go away in the fourier analysis So here are the conditions? If you remember from Algebra here are the conditions [for] even and odd or for odd and even the way that I’ve written them f? of negative x has to be equal [to] negative negative f of x to be odd, and that’s for all x values so Look at this this one [here] so pick an x value. Let’s say 0.5, so if this whole distance is is 1 from [here] to here is 1 Then if we’re picking an x is an x of 0.5 x is equal to point 5 the magnitude of the number is 0.5 so f of negative 0.5 means you come right here and then you drop down to the function and so that’s a negative function value and Then you take the Mac that number that number is 0.5. And you come out here now We need to do f of x you come out here 0.5 You come up you have a positive function value But then you have to [multiply] it times as negative so you get a negative [numbers] So you have a negative you have this negative number is equal to this negative number on this other side So even though the function values is positive when you multiply that times that negative it comes down back down here And so those are equal So this is an odd function right and you can show that’s true for all x in this function that I’ve drawn there Okay for even and I’ve erased some of my [f’s] here for an even function f of [negative] x is equal to f of x Here’s an example to put this aside for a moment x squared So f of negative 2 negative 2 Squared negative 2 times negative 2 is positive [for] [f] Of 2 2 squared is 4 So f of [negative] 2 is 4 and f of 2 gives 4 so? That that is an even [function] so let’s again pick a number pick negative [point] [5] Just like before so [if] we come out So well no not negative 0.5 pick a pick a positive number. That’s the key to the thing [do] not get confused so 0.5 So we come out here and but if We’re picking the magnitude of x to be 0.5. And it’s negative of that number, so we’re coming over negative 0.5, right and so that gives a positive function value and Then the magnitude of the number x is 0.5, so we come over 0.5, and then we come up to the function It’s a positive function value, and they’re the same Being let’s head it as a curse Okay, so we see that this satisfies the even condition for all x so this one here is odd this one [is] even Now again, I only picked. I only did this [for] two x values you need to do it for more, but We can see you could go through and do this process for other x values and you would see that that was true for all The other x values okay, what about this one? This is the graph y is equal to e to the minus x Now real quick. Let me back up if this is odd So if you want to use which which 48 terms are you’re going to use well odd functions you? You model, odd functions with odd? functions, so you would use these terms and You would not have any of these terms you also would not have this by the way So these terms this term in this term these these terms because the [sum] are used to model eat to model even functions and This term right [here] all these terms because it’s a sum these sinusoidal terms are used to model the odd functions Okay What about e to the minus x so for this one it would be this you would use the sine terms For this even one you would use this and this Okay, what about this e to the minus x? This one is neither odd nor even even it’s neither. So this one’s neither because look if you pick it a value 0.5 Come out 0.5 Or you know f of negative 0.5 You always you want to do this part first Because this is the condition always to start with and then you look at what? you look at these other values after you look at this first, so come out negative 0.5, and There’s the function value then come [over] positive 0.5, and there’s a function value the Magnitudes of those function values that are definitely not the same [doesn’t] [matter] what the signs They are the net designs are the magnitudes are the same so they [definitely] cannot be the same as this So this thing is neither odd nor even so this is neither So this is just one example Where you would need to use the whole enchilada? Good luck if you get that on an exam, [and] it’s not likely as we discussed previously It’s more likely that you’ll get an odd or an even function whether it be some stuff [cancelling] to make it easy on you, okay? Little Blonde [yup] what about a function that is both odd and even can you think of one that’s [a] difficult one that’s a tough one, [so] what about this? I’m going to [write] it down analytically. What about f of x Is equal to [zero] for all x? Okay, because look let’s use our test So if ever Fx is equal to zero that means it’s the line. It’s the line that runs along the [x] [axis] This is the function. It’s a line. It’s a constant value zero So if you pick negative 0.5 X right then this is [zero] f of 0.5 this is zero zero equals zero [so] it’s odd satisfies our condition for all x What about this one f of negative 0.5? 0 f 0.5 0 satisfies that condition this one’s both and I would challenge you [to] find one that is not both and If you get it, you can post it on the website Post it all in the comments page, so this one is both To be honest, I don’t think you should waste your time I don’t think I don’t think there’s another one [that] satisfies both I think that’s the only one that will that is both odd and even is that one, so [okay], that is it for Even and odd functions and deciding which for announce you know which [forty] terms to use I’m gonna step up the side and give you clear screenshot