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Basic integration identity- please jog my memory!

  1. Mar 15, 2010 #1
    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.

    1. The problem statement, all variables and given/known data

    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)

    2. Relevant 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.

    3. 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!!

    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 15, 2010 #2


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    Homework Helper

    Hello Matt! :smile:
    It's one of the standard trigonometric identities …

    cosAcosB = 1/2 [cos(A+B) + cos(A-B)] :wink:
  4. Mar 15, 2010 #3
    Much appreciated, thank you!
  5. Mar 15, 2010 #4


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    Science Advisor

    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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