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Question: Self-convolving an expression containing elliptic integrals

Question:Self-convolving an expression containing elliptic integrals

JoyDivisionMan 85 Maple 2025
integration elliptic-integral convolution
4 hours ago
0

Hello Friends,

I have a function f(t) which I would like to convolve with itself.  I have no problem doing that typically, but my function contains two elliptic integrals (EllipticE and EllipticK), which clearly complicates matters.

My work in included.  As you can see, the plot on the (0,2) interval is continuous, and the area under the curve sums to unity via numerical integration.

> 

with(Statistics)
> 

f := proc (t) options operator, arrow; piecewise(t <= 0, 0, 0 < t and t < 2, (4/3)*t*((4+t^2)*EllipticE(sqrt(1-4/t^2))-8*EllipticK(sqrt(1-4/t^2)))/Pi^2, 2 <= t, 0) end proc

proc (t) options operator, arrow; piecewise(t <= 0, 0, 0 < t and t < 2, (4/3)*t*((4+t^2)*EllipticE(sqrt(1-4/t^2))-8*EllipticK(sqrt(1-4/t^2)))/Pi^2, 2 <= t, 0) end proc

 (1)
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plot(f(t), t = 0 .. 2)

 
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evalf(int(f(t), t = 0 .. 2))

1.000000000+0.*I

 (2)

Download CircleDerivation2.mw

Are there any sort of transformations that I can employ which will permit me to convole this function with itself?  I realize a closed-form solution is probably not likely, but something approaching closed-form would be nice.  Via simulation, I have learned that the self-convolved function will have a continuous boundary on the (0, 2*sqrt(2)) interval.
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