Basic integration identity- please jog my memory

[Oscur]

Hello all, I'm doing a question for the maths module in my physics degree (I'm a second year undergrad) and there's a question I'm doing on basis functions.

Homework Statement

Verify that functions of the type $$f_{n}(x) = A cos \frac{2\pi n x}{L}$$ where n = 0,1,2... can be used as a basis function for 0<x<L.

There are then three parts:

i. Normalise (I've done this part, I found a normalising factor of $$A=\sqrt{\frac{2}{L}}$$

ii. Check orthogonality (this is the part causing the trouble)

iii. Would it be possible to use $$f_{n}(x)$$ as the basis for all functions in this interval? (this clearly follows from the other two parts, so once I've done ii, I should be able to answer this)

Homework Equations

To check orthogonality, I need to demonstrate that the condition:

$$\int f_{n}(x) f_{m}(x)dx=\delta_{n,m}$$, where the limits of the integral are the same as the interval of interest, is met.

The Attempt at a Solution

This means I need to integrate the product of two cosines with different arguments, and if I try and do this with integration by parts, I end up in an infinite loop with products of cyclic functions. Is there an identity for this?

Thanks in advance!

Matt

 

[tiny-tim]

Hello Matt!

This means I need to integrate the product of two cosines with different arguments … Is there an identity for this?

It's one of the standard trigonometric identities …

cosAcosB = 1/2 [cos(A+B) + cos(A-B)] ​

 

[Oscur]

Much appreciated, thank you!

 

[phyzguy]

Try using a trig identity to write cos(nx)cos(mx) in terms of cos((n-m)x) and cos((n+m)x).