Fourier Series/4 minus x squared over Range of 2
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Theorem
Let $\map f x$ be the real function defined on $\openint 0 2$ as:
- $\map f x = 4 - x^2$
Then its Fourier series can be expressed as:
- $\map f x \sim \ds \frac 8 3 - \frac 4 {\pi^2} \sum_{n \mathop = 1}^\infty \frac {\cos n \pi x} {n^2} + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\sin n \pi x} n$
Proof
By definition of Fourier series:
- $\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \pi x + b_n \sin n \pi x}$
where:
\(\ds a_n\) | \(=\) | \(\ds \int_0^2 \map f x \cos n \pi x \rd x\) | ||||||||||||
\(\ds b_n\) | \(=\) | \(\ds \int_0^2 \map f x \sin n \pi x \rd x\) |
for all $n \in \Z_{>0}$.
Thus:
\(\ds a_0\) | \(=\) | \(\ds \int_0^2 \map f x \rd x\) | Cosine of Zero is One | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^2 \paren {4 - x^2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {4 x - \frac {x^3} 3} 0 2\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {4 \times 2 - \frac {2^3} 3} - \paren {0 - 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 - \frac 8 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {16} 3\) |
$\Box$
For $n > 0$:
\(\ds a_n\) | \(=\) | \(\ds \int_0^2 \map f x \cos n \pi x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^2 \paren {4 - x^2} \cos n \pi x \rd x\) | Definition of $f$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \int_0^2 \cos n \pi x \rd x - \int_0^2 x^2 \cos n \pi x \rd x\) | Linear Combination of Definite Integrals |
Splitting this up into bits:
\(\ds \) | \(\) | \(\ds 4 \int_0^2 \cos n \pi x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \intlimits {\frac {\sin n \pi x} {n \pi} } 0 2\) | Primitive of $\cos n \pi x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\frac {\sin 2 n \pi} {n \pi} } - 4 \paren {\frac {\sin 0} {n \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Sine of Multiple of Pi |
\(\ds \) | \(\) | \(\ds \int_0^2 x^2 \cos n \pi x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\frac {2 x \cos n \pi x} {\paren {n \pi}^2} + \paren {\frac {x^2} {n \pi} - \frac 2 {\paren {n \pi}^3} } \sin n \pi x} 0 2\) | Primitive of $x^2 \cos n \pi x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {4 \cos 2 n \pi} {n^2 \pi^2} + \paren {\frac 4 {n \pi} - \frac 2 {\paren {n \pi}^3} } \sin 2 n \pi} - \paren {\frac {0 \cos 0} {n^2 \pi^2} + \paren {\frac 0 {n \pi} - \frac 2 {\paren {n \pi}^3} } \sin 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 \cos 2 n \pi} {n^2 \pi^2}\) | Sine of Multiple of Pi and removal of vanishing terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 4 {n^2 \pi^2}\) | Cosine of Multiple of Pi |
Reassembling $a_n$ from the remaining non-vanishing terms:
\(\ds a_n\) | \(=\) | \(\ds 0 - \frac 4 {n^2 \pi^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 4 {n^2 \pi^2}\) |
$\Box$
Now for the $\sin n \pi x$ terms:
\(\ds b_n\) | \(=\) | \(\ds \int_0^2 \map f x \sin n \pi x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^2 \paren {4 - x^2} \sin n \pi x \rd x\) | Definition of $f$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \int_0^2 \sin n \pi x \rd x - \int_0^2 x^2 \sin n \pi x \rd x\) | Linear Combination of Definite Integrals |
Splitting this up into bits:
\(\ds \) | \(\) | \(\ds 4 \int_0^2 \sin n \pi x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \intlimits {\frac {-\cos n \pi x} {n \pi} } 0 2\) | Primitive of $\sin n \pi x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\frac {-\cos 2 n \pi} {n \pi} } - 4 \paren {\frac {-\cos 0} {n \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\frac {-1} {n \pi} } - 4 \paren {\frac {-1} {n \pi} }\) | Cosine of Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
\(\ds \) | \(\) | \(\ds \int_0^2 x^2 \sin n x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\frac {2 x \sin n \pi x} {\paren {n \pi}^2} + \paren {\frac 2 {\paren {n \pi}^3} - \frac {x^2} {n \pi} } \cos n \pi x} 0 2\) | Primitive of $x^2 \sin n \pi x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {4 \sin 2 n \pi} {n^2 \pi^2} + \paren {\frac 2 {\paren {n \pi}^3} - \frac 4 {n \pi} } \cos 2 n \pi} - \paren {\frac {0 \sin 0} {n^2 \pi^2} + \paren {\frac 2 {\paren {n \pi}^3} - \frac 0 {n \pi} } \cos 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac 2 {\paren {n \pi}^3} - \frac 4 {n \pi} } \cos 2 n \pi - \frac 2 {\paren {n \pi}^3} \cos 0\) | Sine of Multiple of Pi and removal of vanishing terms | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 4 {n \pi}\) | Cosine of Multiple of Pi and simplifying |
Reassembling $b_n$ from the remaining non-vanishing terms:
\(\ds b_n\) | \(=\) | \(\ds 0 - \paren {-\frac 4 {n \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 4 {n \pi}\) |
$\Box$
Finally:
\(\ds \map f x\) | \(\sim\) | \(\ds \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {\frac {16} 3} + \sum_{n \mathop = 1}^\infty \paren {-\frac 4 {n^2 \pi^2} \cos n \pi x + \frac 4 {n \pi} \sin n \pi x}\) | substituting for $a_0$, $a_n$ and $b_n$ from above | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 8 3 - \frac 4 {\pi^2} \sum_{n \mathop = 1}^\infty \frac {\cos n \pi x} {n^2} + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\sin n \pi x} n\) | rearranging |
$\blacksquare$
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 3$. Other Types of Whole-Range Series: Example $2$