Show that the functions sin x, sin 2x, sin 3x, are orthogonal

In summary, to show that the functions sin x, sin 2x, sin 3x, ... are orthogonal on the interval (0,pi) with respect to p(x) = 1, one can use the identity \sin{s} \sin{t} = \frac{ \cos{s-t} - \cos{s+t} }{2} and write \sin{(n x)} = \frac{e^{i n x} - e^{-inx} }{2i} to expand the sines as complex exponentials and prove the orthogonality.
  • #1
stunner5000pt
1,461
2
Show that the functions sin x, sin 2x, sin 3x, ... are orthogonal on the interval (0,pi) with respect to p(x) = 1 (where p is supposed to be rho)

i know i have to use this
[tex] \int_{0}^{\pi} \phi (x) \ psi (x) \rho (x) dx = 0 [/tex] and i have no trouble doing it for n = 1, and n=2
but how wouldi go about proving it for hte general case that is

[tex] \int_{0}^{pi} \sin{nx} \sin{(n+1)x} dx [/tex] for n in positive integers

would i use this identity :
[tex] \sin{s) \sin{t} = \frac{ \cos{s-t} - \cos{s+t} }{2} [/tex]
i proved it using this identity... but isn't this identity a bit too obscure? Isnt there a less 'weird' way of doing this?
 
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  • #2
One very easy way of proceeding is to write
[tex]
\sin{(n x)} = \frac{e^{i n x} - e^{-inx} }{2i},
[/tex]
expand both the sines under the integral as complex exponentials and the result falls out immediately.
 

FAQ: Show that the functions sin x, sin 2x, sin 3x, are orthogonal

What does it mean for functions to be orthogonal?

When two functions are orthogonal, it means that their inner product (integral of their product over a given interval) is equal to zero. This indicates that the functions are perpendicular to each other in a mathematical sense.

How do you show that the functions sin x, sin 2x, and sin 3x are orthogonal?

To show that these functions are orthogonal, we need to take the inner product of any two of them and show that it equals zero. This can be done by using trigonometric identities and integrating over a specific interval.

Why is showing orthogonality important in mathematics?

Orthogonality is important because it allows us to define a basis for a vector space. In the case of functions, it allows us to use orthogonal functions as a basis to represent other functions, making calculations and analysis easier.

Can you generalize this concept to other sets of functions?

Yes, the concept of orthogonality can be extended to any set of functions. As long as their inner product is equal to zero, they can be considered orthogonal. This is commonly used in fields such as Fourier analysis and linear algebra.

What other properties do orthogonal functions have?

Besides having an inner product of zero, orthogonal functions also have the property of being linearly independent. This means that no one function in the set can be written as a linear combination of the others. They are also useful in solving differential equations and representing periodic functions.

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