00:06In this video we're going to talk about the trigonometric Fourier series.
00:10Have you ever wondered how periodic functions can be decomposed into a
00:15A combination of sines and cosines? Well, the Fourier series is the answer to that question.
00:22A Fourier series is the representation of an integrable periodic function with a
00:26series of constants to zero at n, bn, multiplied by convergent trigonometric functions
00:31of sine and cosine, respectively. We have coefficients a0, an, bn,
00:37It is calculated using these piecewise integrable functions, respectively. Even and
00:43odd numbers. Imagine you have a vertical mirror that divides a graph into two equal parts. If the
00:50Fold the graph in that mirror image, the two parts match perfectly, then you have a
00:55even function. Formal definition, a function fx is even, if for any value of x in its domain,
01:02It is true that the function of negative x is equal to the function of x. This means that if you replace x
01:10By subtracting x from the function, you get the same result. For example, the function x equals x.
01:17square is even, because the function of negative x equals x minus squared, which is equal to x squared.
01:23is equal to the function of x. Graphically, even functions have symmetry with respect to the y-axis.
01:30For the odd function, now imagine you have a point on the graph and you rotate it 180 degrees
01:35around the origin 0,0. If the point ends at another point on the graph, then you have a function
01:43odd. A function f(x) is odd if for any value of x in its domain, the following holds true, function of
01:50negative x is equal to negative x. This means that if you replace x with negative x in the
01:56for this function, you get the opposite of the original value. For this, the function of x equals the cube of x.
02:02It is odd because when replacing, negative x cube, is equal to negative x cube and is equal to negative the
02:08function.
02:09When working with Fourier series, even and odd functions take on a fundamental role,
02:14since they allow us to simplify the calculations and obtain more compact expressions for the series.
02:20Even functions. Definition: A function f(x) is said to be even if, when x is replaced by negative x,
02:26The function does not change. Fourier series of an even function: if fx is even, its series only contains
02:32cosine terms. This function of x is equal to sub 0 over 2, plus, sigma from n values
02:39to 1
02:39infinity, of the coefficient a n, by cosine of n w x. A sub 0 and a n, are the coefficients
02:46and w,
02:46double, is the fundamental frequency. A function fx is said to be odd if when x is replaced by minus
02:53x, the function changes sign, that is, a function of negative x, is equal to negative function of x.
03:01Fourier series of an odd function, if fx is odd, its Fourier series only contains
03:06sine terms, this means that in the general formula if it is sine, its result is odd.
03:12We have bn as the coefficients, and w as the frequency. Even functions.
03:19The graphs of even functions, the most common with respect to the vertical y-axis, are,
03:24Square wave, triangular wave, sinusoidal wave on the y-axis, sinusoidal wave on the axes. Odd functions.
03:30The graphs of odd functions, the most common, a function is said to be odd if its
03:36The graph is symmetric with respect to the origin. We have the following exercise: find the
03:42Trigonometric Fourier series of the sawtooth wave. For this, we establish the intervals
03:48of the function from b goes from zero less than w t less than b, and from zero goes from
03:53minus pi and,
03:54The function is b over pi times w t. The period is 2 pi, now we calculate to zero, this
04:00is 2 over t times
04:01the definite integral from t means to minus t means, a function of t times the differential.
04:07Replacing values with zero equals 1 over pi times the definite integral from pi and minus pi,
04:11by the function b over pi w t by the differential of t, and being a constant we have b
04:17about pi
04:17integrating the square by the variable w and the differential. We have a sub zero equals zero,
04:24and a sub n, is zero since the function by simple inspection is odd, consequently
04:29It only has zero terms. Now we calculate b n, according to the definite integral, of integrand
04:35function of t, by sine of n w t, by the differential of t. Simplifying its limits we have pi,
04:42and 1 over pi. Thus we have a n zero, a zero equals zero, we calculate b n according to the formula,
04:49and integrating we have b over pi squared, by the definite integral from pi and minus pi,
04:54of the integrand w t times sine of n w t times the differential of t, here we apply the integral
05:00by parts, making the differential of u equal to w t, and integrating the sine function of b
05:05of n w
05:06t times the differential of w t, is equal to b minus 1 over n times cosine of n w
05:11t times the differential of
05:12w t. And developing the integration by parts, it is equal to b over pi squared, by the multiplication
05:19of u times b, minus the integral of the function v, establishing limits from pi to minus pi.
05:25And integrating we have less w t over n, times cosine of n w t plus, 1 over n squared times
05:31sine of n w t,
05:32This is with limits from pi and negative pi. Here we have that for every value of sine of pi and negative pi
05:37is equal to
05:38zero, also cosine of plus pi and minus pi is equal to minus 1. Replacing the limits from pi and
05:44less
05:44pi, we have cosine values, simplifying we have the answer minus 2 over n pi times
05:50cosine of n pi, we already have the value of the coefficient of b n, but cosine of n pi is equal to
05:55at minus 1
05:55raised to the power of n, and multiplying by minus 1 in parentheses, we have 2b over n pi times
06:01negative 1 raised to the power of n plus 1. And we have the general Fourier formula, replacing the
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