# Prove these functions are orthogonal

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• Oct 20th 2009, 05:48 PM
coldfire
Prove these functions are orthogonal
Hi, can anyone help me solve this:

2 functions f and g are orthogonal on [a,b] if:

$\displaystyle \int_a^b f(x)g(x)dx = 0$

Prove that $\displaystyle f_{m}(x) =$sin$\displaystyle mx$ and $\displaystyle f_{n}(x) =$sin$\displaystyle nx$ are orthogonal on the interval $\displaystyle [-\pi,\pi]$ if m and n are integers such that $\displaystyle m^2$ ≠ $\displaystyle n^2$

Thanks.
• Oct 20th 2009, 05:58 PM
mr fantastic
Quote:

Originally Posted by coldfire
Hi, can anyone help me solve this:

2 functions f and g are orthogonal on [a,b] if:

$\displaystyle \int_a^b f(x)g(x)dx = 0$

Prove that $\displaystyle f_{m}(x) =$sin$\displaystyle mx$ and $\displaystyle f_{n}(x) =$sin$\displaystyle nx$ are orthogonal on the interval $\displaystyle [-\pi,\pi]$ if m and n are integers such that $\displaystyle m^2$ ≠ $\displaystyle n^2$

Thanks.

Please post what you've done and state where you get stuck.

Note that $\displaystyle \sin (nx) \sin (mx) = \frac{1}{2} \left(\cos ([n - m]x) - \cos ([n + m]x) \right)$.
• Oct 20th 2009, 05:59 PM
TheEmptySet
Quote:

Originally Posted by coldfire
Hi, can anyone help me solve this:

2 functions f and g are orthogonal on [a,b] if:

$\displaystyle \int_a^b f(x)g(x)dx = 0$

Prove that $\displaystyle f_{m}(x) =$sin$\displaystyle mx$ and $\displaystyle f_{n}(x) =$sin$\displaystyle nx$ are orthogonal on the interval $\displaystyle [-\pi,\pi]$ if m and n are integers such that $\displaystyle m^2$ ≠ $\displaystyle n^2$

Thanks.

Using the identitiy that

$\displaystyle \sin(u) \sin(v)=\frac{1}{2}(\cos(u-v)-\cos(u+v))$

So in your case you get

$\displaystyle \sin(mx) \sin(nx)=\frac{1}{2}(\cos((m-n)x)-\cos((m+n)x))$

It should all be down hill from here.

EDIT: haha too slow :)