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Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.31....

Question

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.31.

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.31.



Answers

A finite Fourier series is given by the sum
$$ f(x) = \sum_{n =1}^{N} a_n \sin nx $$
$$ f(x) = a_1 \sin x + a_2 \sin 2x + \cdots + a_N \sin Nx $$








Show that the $ m $ th coefficient am is given by the formula

$$ a_m = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin mx dx $$

In the question we have. Could it remind the function for which the four year Siri's off the function? F X equals two x ex going from minus by who? By convergence that given function if is continuous on the new US on the limit, minus by who by and it's better with is effects. Half Nash XT equals two one when minus by access from minus by 20 and one when access from zero Dubai that is have their sex would be fertile. One when x is from minus. Bye bye, which is a piece vice continuous on minus by WHO by by Children Toe the food year. Cities off F obtained in problem nine. Gone were just so the function g where G X is a Callisto effects four ex belonging. Toa minus by don't buy since F is 12 new us Andi g Less minus by would be quite less minus I plus less self by negative on which for vapor toe by a blast Bye bye to that is by So therefore the food year cities one word Yes, so that go by very or dick function G defined us gxe Well, it's two x when x it belongs to minus by you by. Thank you

In this video, we're gonna go through the answer to question number 36 from chapter 10.3. Split up into three parts A B and C two starting with part, eh? I've written down or was formula for when CTU is equal to and pie on dhe minus and sorry, but this was equal to end X and people to minus annex. Using these, we can act together, eats the eye an ex, um e it's the minus I and X on dry by two. That's that's gonna give us taking half out of food from the top line. Yeah, we're gonna have because And next they're gonna have plus, I sine annex. But then, of course, we're gonna add another. Caused an X which is gonna make this too. We're gonna minus and I sign an X. So the plus I sine and exit at minus I sign an ex gonna be equals zero. And of course, this is just cause the next Okay, Similarly, we can take eats the eye and x minus. It's the minus I n x on divided by two I on dde going through this, we're going to get from the top line We're gonna have a cause and X plus I sine annex They were gonna subtract cause the next So the cause An exit closet and the mind has an excellent counsel. Yes, and we're gonna We're gonna subtract minus I sign an ex. So what have I signed next? Plus I sine and expect kisses too. I signed an ex divided about to I is just signed an ex. Okay, on to you. Uh hey. So acid air show a slightly different form for the Fourier series. So x, uh, studied various areas has a zero over to, but we could just define See, there ought to be a zero to That's fine. Plus Cem between and is equal to one on infinity off I am cause an ex being sign an ex so a n cause an ex which was just showing to be equal to huh? It's the eye and X minus e to the minus. I and X. It's a who's being which is showing to be equal to eat that, uh, it's the annex minus e minus I and eggs over to I Okay, so just looking, uh, this bit seeing factors off E to the eye and X from the first term, we're gonna have a n over too. From the second story, we're gonna happen. Bien over to I now one over eyes equal to minus eyes. That's the same as minus I be activity. Okay. On coefficients of eats the minus I and X from the first term we're gonna have Excuse me. This should be a close as that gives us a n of a pig. From the second term, we're gonna have minus bien over to I. One of the eye is minus I. That's the same as a plus, huh? Bien? Huh? And then what we can do is just define see end to be this defined. See? Minus said through this, which takes it into the form that was required. Okay, Potsie, I know it's asking us to have to find out what they see. Ends, CNN, cnn C minus on CNN is a n minus I lots of bien or divided by two. So taking up one of the team out the front, uh, a ends are defined by well on the range minus pi pi that if I buy one of the pie integral between minus pi pi of the function f times because And necks the ex minus one over. I s so minus Um ah, yeah. Minus. I lost of bien over to go taking the one of two out, um minus I lost bien working The end could be written in an inter perform on this as well. So putting the inte grants together throwing the integrated together it's gonna be f of x times by cause and x minus I lost of sign X the ex with Okay, So taking the heart out front Yeah, Taking half of the one of a pious to fun could be one of the tea pie It's reminds pie and pie sound FX times this cause an X minus I sign an ex is just my part A is equal to ease the minus I and x the ex not to see my c school spirit subscript minus. And that's equal to a n plus ibn over to so similarly take the one of the two out the front. It's gonna be one of the two times one of the pie times interview between minus pi pi of events times by well, have the indifference together we're gonna cause N X plus I sine an X on just my oldest former formula that's equal to e to the eye. Thanks. The eggs. Okay, so FX can be written as well. It's see, not that's a no over too clothes. That's, um, n equals one to infinity, That is. Ah, CNN. Okay, in the eye and X, who us see? Minus end eats the minus. I an ex. This is just a result from part B. All right, what? But this term, if we set a, uh, another told me about, you know, that don't be variable m to be defined as minus and what then that's just gonna be that could be absorbed into the first summation we could take. And Toby minus infinity to infinity of CNN, he to the eye and X. And, of course, I'm allowed to do this because, see, nor can be resolved into here as well to fill in that gap between the, um, uh, positive interviews on the negative incidents, because, see, No, it is just the same as Senor Hee to the i x times they were. So it's basically this when and is it with Zephyr? Okay, Ah, Nde. All the sea ends can be defined in the same way now. Because if we set uh, yeah, minus end vehicle to M. Then this becomes minus m on your writing. All of your negative on and see ends in the same way as the positives. So they know we ran as one of the tea pie insecurity minus pi pi fx. He's the minus I and next the ex, which is the form of their surprise.

For the glory series of the function. Sign acts over, too. We'll use the result. MacLaurin series off Sighel X is equal to some from zero to infinity. Make you want to party in acts to party to end plus one over Shoot and blast Juan pictorial then replaced X by Axel or two we have McAra. Siri's off the function. Sign off, Axel. Where to? Is he going to some From zero to infinity Makes you want Thio axe to to end. Plus one off shoe to the power. You damn Paswan much. Bye. Shoot and plus one factorial.


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