Question

Let f(x) be a function of period $2pi$ such that $f(x) = x$ over the interval $-pi < x < pi$ a) Sketch a graph of f(x) in the interval of $-3pi < x < 3pi$ b) Show that the Fourier series for f(x) in the interval $-pi < x < pi$. Show complete solution for $a_0$, $a_n$ and $b_n$ c) By giving an appropriate value to x show that $frac{pi}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + frac{1}{9} - ...$

          Let f(x) be a function of period $2pi$ such that
$f(x) = x$ over the interval $-pi < x < pi$
a) Sketch a graph of f(x) in the interval of $-3pi < x < 3pi$
b) Show that the Fourier series for f(x) in the
interval $-pi < x < pi$.
Show complete solution for $a_0$, $a_n$ and $b_n$
c) By giving an appropriate value to x show that
$frac{pi}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + frac{1}{9} - ...$

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$Let f(x) be a function of period 2pi such that f(x) = x over the interval -pi < x < pi a) Sketch a graph of f(x) in the interval of -3pi < x < 3pi b) Show that the Fourier series for f(x) in the interval -pi < x < pi. Show complete solution for a0, an and bn c) By giving an appropriate value to x show that fracpi4 = 1 - frac13 + frac15 - frac17 + frac19 - ...$

Added by Matthew W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sam Stansfield

01/01/2025

Step 1

The Fourier series for a function typically involves an infinite series of sines and cosines, not a simple sum of numbers like "41/3 + 57 + 9." Therefore, I'll correct the approach to show how to sketch the graph as requested in part a and how to start the process Show more…

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Let f(x) be a function of period 2π such that f(x) = x over the interval -π < x < π. a) Sketch a graph of f(x) in the interval -3π < x < 3π. b) Show that the Fourier series for f(x) in the interval -π < x < π. Show complete solution for a0, an and bn c) By giving an appropriate value to x show that π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

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Transcript

00:01 So we're given this function, drew a picture of it, and then we want to calculate the fourier series.

00:07 Of course, the fourier series looks like this.

00:17 So now the fourier series is written like this.

00:21 We have handy -dandy formulas for calculating what the a's and bs are, and they want us to do that.

00:28 So here we go.

00:32 And that is an odd function that's integrated on a symmetric interval.

00:37 So it's automatically going to be zero.

00:40 We're actually going to use that same statement a lot.

00:44 So the a sub -ns look like this.

01:15 All right.

01:17 And again, it's an odd function on a symmetric interval.

01:32 So that is automatically zero.

01:37 But the b -sab -ns are not zero.

01:43 So b -sab -n is...

02:29 So i'm going to make a substitution u is equal to n -x.

02:45 All right so now to evaluate this um the sign is zero at both limits so we got all right so what's left here is odd when i substitute in the limits what i'm going to do is get two equal things n pi the cosine we got two of them because each limit gives us the same result okay and the cosine of n pi is minus 1 to the n.

04:21 So there's our b sub -ns.

04:25 So we got, so now we want to get this infinite series for pi over 4...

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