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MATH-M 344 Chapter Notes - Chapter 10: Fourier Series, Even And Odd Functions

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19 Apr 2017

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Benjamin Melinand

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M344 section 10. 4 notes- even and odd functions. For function (cid:1858): , defined for all (cid:1876): even if (cid:1858)(cid:4666) (cid:1876)(cid:4667)=(cid:1858)(cid:4666)(cid:1876)(cid:4667) for all (cid:1876) . |(cid:1876)| cos(cid:1866)(cid:1876: odd if (cid:1858)(cid:4666) (cid:1876)(cid:4667)= (cid:1858)(cid:4666)(cid:1876)(cid:4667) for all (cid:1876) . Let (cid:1858) be even and (cid:1859) be odd, and let (cid:1876) (cid:4666)(cid:1858)(cid:1859)(cid:4667)(cid:4666) (cid:1876)(cid:4667)=(cid:1858)(cid:4666) (cid:1876)(cid:4667)(cid:1859)(cid:4666) (cid:1876)(cid:4667) Quotients: even/even = even, odd/odd = even, odd/even = odd, even/odd = odd. If (cid:1858) even, (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1856)(cid:1876)=(cid:884) (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1856)(cid:1876), where (cid:3410)(cid:882) (cid:2868) Symmetry with (cid:1877) axis adds equal areas. Symmetry with (cid:1876) axis cancels out equal areas. If (cid:1858) is (cid:884)-periodic and even, then (cid:1853)(cid:3040)=(cid:2869) (cid:1858)(cid:4666)(cid:1876)(cid:4667)cos(cid:4672)(cid:3040) (cid:1876)(cid:4673) (cid:1854)(cid:3040)=(cid:2869) (cid:1858)(cid:4666)(cid:1876)(cid:4667)sin(cid:4672)(cid:3040) (cid:1876)(cid:4673) (cid:1856)(cid:1876)=(cid:882) for (cid:1865)=(cid:883),(cid:884), . Integral representing (cid:1853)(cid:3040) deals with even: even = even function. Integral representing (cid:1854)(cid:3040) deals with even: odd = odd function. If (cid:1858) and (cid:1858) piecewise continuous ((cid:1858) is equal to its fourier series), then we talk about the cosine series (cid:1858)(cid:4666)(cid:1876)(cid:4667)=0(cid:2870)+ (cid:1853)(cid:3040)cos(cid:4672)(cid:3040) (cid:1876)(cid:4673)

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Math 151 Sections 34, 35, 36 Workshop 11 ted to arn This workshop deals with an addition formula in section 5.1. This formula is rel oft told story about the great mathematician Gauss. Here is one version of the story When Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest batch of math homeworks. This is why he decided to finish the grading during his math period. To keep the students busy while he graded, he gave them the following task at th of the period: Use your slate (the small personal blackboard that every student had start in class) to add up the numbers 1,2,3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointed to his slate Here is the answer." Then Gauss did nothing for an hour, while the other students labored with the additions and the teacher graded. At the end of the perioc Gauss showed the correct answer 5050 on his slate. Only Gauss got the correct answer. Then Gauss told everybody that the sum 12 +3.+ 100 is (1+100)+(2+99) + (3498)+ Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be an even number Problem 2. Let N be an even positive integer. Use the method of Problem 1 to prove the formula 1 +2 +3+.. . +N(N. Include the reason why it is important that N +(50+51) = 101 + 101 +101 + +101-101.50-5050 an even number. Problem 3. The formula is still valid when N is an odd integer. Prove this, using a variation on the argument given by Gauss The story continues: Gauss then gave the class a lecture on how his method could be used to sum an arithmeric series (a +1.b) (a +2.b) (a +3.b)+. (a +N b), where a and b are constants Problem 4. Find a formula for the sum (a+1.b)+ (a+2.b) + (a +3.b)+ +(a+ N.b) Prove this formula the way Gauss would have done it. You have to do the case when N is even and the case when N is odd Now let us consider a fictional alternate history where Gauss had not thought of this very clever trick. We can imagine the ten year old Gauss going home and doing the sequential computations 1+2 1+2+3 1+2+3+4 1+2+3+4+5 and so forth, just to see if he could find a pattern in the sequence of answers. He would have discovered somethin like this: once he had the sum 1+2+3+...+ N, he could simply add N +1 to that sum and that would give him the sum 1+2+3+ + (N +1) very quickly Now ass N(N+1 ume that, by trial and error, Gauss discovered that the formula 1+2+3+ is true for several values of N. How could he prove that the formula is r for ALL N? The insight described in the previous paragraph might have led Gauss to other words, he only needed to show: Formula for 1+2+3+ + N plus N +1 equals formula for 1+2+3++N+1). He would have discovered the principle of mathema -.. + N eally true 1)+1) think the following: All he had to do was prove mm) + (N + 1) NN+1 (N+1)(( In ered the principle of mathematical induction Problem 5. Prove N(N+1) (N + 1 of the formula for 1+2+3+.+N. (Nt1MNHHI) Show how this leads to a uCtion a proof