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Re: Help with identity
Posted:
May 2, 2013 4:06 PM
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On Thursday, April 25, 2013 5:17:47 AM UTC-7, Mike Trainor wrote:
> Would greatly appreciate any pointers to proving the following
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> identity that I came across in Bateman's book on partial
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> differential equations:
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> sinh(x)/(cosh(x) - cos(y) = 1 +
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> sum{n, 1, inf} [exp(-nx)*cos(ny)]
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> It is clearly a simple Fourier expansion in y, but how does
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> one get the coefficients -- in other words do the integral.
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> OTOH, it is trival to prove that the RHS can be summed
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> to give the LHS.
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> The one out I see it expressing either the trignometric
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> or hyperbolic functions as the other type by using an
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> imaginary arguments, but I could not get anywhere.
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> Also 0<= x < inf and 0 <= y <= 2 pi
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> tia
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> mt
Maybe nobody ever did the integrals to get the series; maybe in the distance past somebody obtained the series, perhaps as part of a larger project, and then asked themselves: hmmmm ... I wonder if the series can be expressed in closed form? Then maybe they discovered the function you have started with.
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