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Basic integration identity- please jog my memory!

  • Thread starter Oscur
  • Start date
  • #1
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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 [tex]f_{n}(x) = A cos \frac{2\pi n x}{L}[/tex] 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 [tex]A=\sqrt{\frac{2}{L}}[/tex]

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

iii. Would it be possible to use [tex]f_{n}(x)[/tex] 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:

[tex]\int f_{n}(x) f_{m}(x)dx=\delta_{n,m}[/tex], 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

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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Hello Matt! :smile:
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)] :wink:
 
  • #3
31
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Much appreciated, thank you!
 
  • #4
phyzguy
Science Advisor
4,493
1,443
Try using a trig identity to write cos(nx)cos(mx) in terms of cos((n-m)x) and cos((n+m)x).
 

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