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Simple Integration Orthogonal Sin

  1. Nov 21, 2013 #1

    take the function

    I(m,n) = Integral from 0 to 1 of sin(m*pi*x)*sin(n*pi*x) over dx

    depending from n and m, being +-1, +-2, and so on.
    If I use sin(x)sin(y)=1/2(cos(x-y)-cos(x+y)) I get sin^2(n*pi*x)=1/2-1/2cos(2n*pi*x)

    or I(n,n)=1/2

    because the integral of cosine over full periods is zero.

    Well, the integral for n not equal to m is zero (easy to see, sin is zero for n*pi).

    Now let's integrate via partial integration without knowledge of formulars between
    periodic functions.

    After two integrations I get

    I(m,n) = (m/n)^2 I(m,n)

    which means that if the integration is correct

    1. m=+-n or
    2. I(m,n)=0

    You can choose m and n independently from each other. This means
    that I(m,n)=0 is proven for m not equal +-n.

    This is interesting from a point that for example J. Farlow in "Partial Differential Equations
    for Scientists and Engineers" does not mention the negative sign. He completely skips this

    This is an indirect prove. Is it correct?

  2. jcsd
  3. Nov 22, 2013 #2


    User Avatar
    Science Advisor

    Yes, that is correct.
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