Scaled Sine Functions of Integer Multiples form Orthonormal Set

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Theorem

For all $n \in \Z_{>0}$, let $\map {\phi_n} x$ be the real function defined on the interval $\openint 0 \lambda$ as:

$\map {\phi_n} x = \sqrt {\dfrac 2 \lambda} \sin \dfrac {n \pi x} \lambda$

Let $S$ be the set:

$S = \set {\phi_n: n \in \Z_{>0} }$

Then $S$ is an orthonormal set.


Proof

Consider the definite integral:

$I_{m n} = \ds \int_0^\lambda \map {\phi_m} x \map {\phi_n} x \rd x$


From Sine Function is Odd, each of $\map {\phi_n} x$ is an odd function.

From Odd Function Times Odd Function is Even, $\map {\phi_m} x \map {\phi_n} x$ is even.

That is:

$\paren {\sqrt {\dfrac 2 \lambda} \sin \dfrac {m \pi x} \lambda} \paren {\sqrt {\dfrac 2 \lambda} \sin \dfrac {n \pi x} \lambda}$

is an even function.


Let $u = \dfrac {\pi x} \lambda$.


We have:

\(\ds \dfrac {\d u} {\d x}\) \(=\) \(\ds \dfrac \pi \lambda\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d x} {\d u}\) \(=\) \(\ds \dfrac \lambda \pi\)


\(\ds x\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds u\) \(=\) \(\ds \dfrac {\pi \times 0} \lambda\)
\(\ds \) \(=\) \(\ds 0\)


\(\ds x\) \(=\) \(\ds \lambda\)
\(\ds \leadsto \ \ \) \(\ds u\) \(=\) \(\ds \dfrac {\pi \times \lambda} \lambda\)
\(\ds \) \(=\) \(\ds \pi\)

So:

\(\ds I_{m n}\) \(=\) \(\ds \int_0^\lambda \map {\phi_m} x \map {\phi_n} x \rd x\)
\(\ds \) \(=\) \(\ds \int_0^\lambda \paren {\sqrt {\dfrac 2 \lambda} \sin \frac {m \pi x} \lambda} \paren {\sqrt {\dfrac 2 \lambda} \sin \frac {n \pi x} \lambda} \rd x\)
\(\ds \) \(=\) \(\ds \frac 2 \lambda \int_0^\lambda \sin \frac {m \pi x} \lambda \sin \frac {n \pi x} \lambda \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \frac 2 \lambda \int_0^\pi \frac \lambda \pi \sin m u \sin n u \rd u\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 2 \lambda \frac \lambda \pi \int_0^\pi \sin m u \sin n u \rd u\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \frac 2 \pi \int_0^\pi \sin m u \sin n u \rd u\)
\(\ds \) \(=\) \(\ds \frac 1 \pi \int_{-\pi}^\pi \sin m u \sin n u \rd u\) Definite Integral of Even Function
\(\ds \) \(=\) \(\ds \frac 1 \pi \pi \delta_{m n}\) Integral over $2 \pi$ of $\sin m u \sin n u$
\(\ds \) \(=\) \(\ds \delta_{m n}\)

Hence the result by definition of orthonormal set.

$\blacksquare$


Sources

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