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Problem. Derive the formula


ole.gif


where


ole1.gif



and f(x) is a sectionally continuous function defined on an interval c < x < c + 2L



Derivation. In deriving the above formulas we will use the following definite integrals where m and n are positive integers. The integrals are valid for all values of c.


ole2.gif


Step 1. We first consider the case of L = π i.e. the case where f(x) is a sectionally continuous function defined on an interval c < x < c + 2π. In this case 1) and 2) become


ole3.gif


where


ole4.gif


We will make a starting assumption that f(x) can be represented by series 10). We will not attempt to prove that here. We now give the derivation of the coefficients an and bn of 11).


First, we shall assume that we can legitimately integrate 10) term by term (integrating a series term by term is not always legitimate). Integrating from x = c to x = c + 2π we get


ole5.gif


The integral on the left can always be evaluated, since f(x) is a known function. If necessary some method of approximate integration such as the trapezoid rule can be used. The first term on the right is


             ole6.gif


By 3) all integrals with a cosine vanish and by 4) all integrals with a sine vanish. Thus 12) reduces to


             ole7.gif


or


ole8.gif

 


To find an (n = 1, 2, 3, .....), we multiply each side of 10) by cos nx and then integrate from c to c + 2π to obtain



ole9.gif


The integral on the left can always be evaluated. By 3) and 4) all the integrals on the right that contain only cosine terms are zero except the one containing cos2 nx which, by 6), is equal to π. By 7) all the integrals containing sine terms are zero. Thus


             ole10.gif


or


ole11.gif


We now determine bn by essentially the same procedure. We multiply each side of 10) by sin nx and then integrate from c to c + 2π to obtain


we multiply each side of 10) by cos nx and then integrate from c to c + 2π to obtain            


ole12.gif


As before all the integrals on the right vanish except for one, giving


             ole13.gif


or


ole14.gif



Formulas 13), 14) and 15) are known as the Euler or Euler-Fourier formulas.



Step 2. In Step 1 we have proven that if the given function has a period of 2π, then its Fourier series representation is given by 10) and 11). We now deal with the following question: If the function f(x) that we are given has a period of 2L instead of 2π, what is its Fourier series representation? We will do a change of variable which effects a change of scale. We need the formula for the proper change of variable.


ole15.gif

Let us first observe that the period p of a general function y = sin ax is given by p = 2π/a. See Fig. 1. The period corresponds to the range of values of x corresponding to a complete cycle of the function. Thus it is found by solving the equation ax = 2π for x i.e. p = 2π/a. (for example, the range of values of x corresponding to a complete cycle of the function y = sin 3.2x is found from the equation 3.2x = 2π so the period is x = 2π /3.2).


Thus the period of the functions sin ax and cos ax in a Fourier series is 2π/a. 


If the function f(x) that we wish to represent by a Fourier series has a period of 2L, then the a1 and b1 terms in its series representation must have the form a1 cos ax and b1 sin ax with a period of 2L = 2π/a. Thus a = π / L and the a1 and b1 terms are


             ole16.gif


We now ask ourselves the question: What change of variable that will change the given function f(x) into one with a period 2π? The answer: The change of variable that will accomplish this is t = πx/L, or x = Lt/π. If we make the substitution x = Lt/π in the given function y = f(x) we obtain a function


             ole17.gif


which has a period of 2π.


As a function of t of period 2π, F(t) is represented by the Fourier series


ole18.gif


where


ole19.gif


If, now, we make the inverse substitutions


             ole20.gif


in 16) and 17), they become


ole21.gif


where


ole22.gif


 


References.

  Wylie. Advanced Engineering Mathematics. Chap. 7


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