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How would you integrate this function?

  1. Apr 11, 2017 #1
    1. The problem statement, all variables and given/known data
    I need to find the coefficients for a fourier expansion. Here is the integral I need to solve: (1/π) ∫sin(x/2)sin(nx)dx where the limits are from -π to π. The original function for the expansion is f(x) = sin(x/2)

    2. Relevant equations
    None

    3. The attempt at a solution
    Should I do this in terms of complex numbers? I think that should be my approach but I am not sure. If this is the right approach can you help me get started with setting up the integral?
     
  2. jcsd
  3. Apr 11, 2017 #2

    Mark44

    Staff: Mentor

    I don't think so. The usual approach is integration by parts, twice. Here's a link to a similar example, https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintsoldirectory/TrigIntSol3.html#SOLUTION 24, problem #24
     
  4. Apr 11, 2017 #3

    Dr Transport

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    use the identity

    $$\frac{1}{2\pi} \int_{-\pi} ^{\pi} \sin(nx)\sin(mx) dx = \delta_{nm}$$

    and the integral pops right out for you . (i might have the normalization wrong.... off the top of my head....)
     
  5. Apr 11, 2017 #4
    Woah, so you're saying that using the kronecker delta I can just say it is δn (1/2)? Can you walk me through those steps? Does it have something to do with the dirac delta function/identities? If the kronecker truly makes it simpler then I'd rather use that lol.
     
  6. Apr 11, 2017 #5

    Dr Transport

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    it is a definition....if you write the sine functions in terms of complex exponential's and do the integrals, it comes out very quickly....
     
  7. Apr 11, 2017 #6

    Dr Transport

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    $$\frac{1}{\pi}\int_{-\pi}^{\pi}\sin(nx)\sin(mx)dx = \delta_{nm}$$
     
  8. Apr 11, 2017 #7
    Yes I've solved this now- thank you
     
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