How does one compute the following integral?
∫∞0√xsin(x)1+x2dx
I have tried extending x to the complex plane then evaluating the following contour integral
∮C√xeix1+x2dx
with the contour C running along the whole real axis and then upper semicircle. I obtain
∫∞0√x(cos(x)+sin(x))1+x2dx=πe√2
but not the original integral.
Answer
For information :
∫∞0√xsin(yx)1+x2dx is the Fourier Sine Transform of √x1+x2 In the Harry Bateman's Tables of Integral Transforms an even more general formula can be seen on page 71, Eq.28 , the Fourier Sine Transform of x2ν(x2+a2)−μ−1 :
12a2ν−2μΓ(1+ν)Γ(μ−ν)Γ(μ+1)y1F2(ν+1;ν+1−μ,3/2;a2y2/4)+4ν−μ−1√πΓ(ν−μ)Γ(μ−ν+3/2)y2μ−2ν+11F2(μ+1;μ−ν+3/2,μ−ν+1;a2y2/4)
Withy=1;a=1;ν=1/4;μ=0→∫∞0√xsin(x)1+x2dx=
=12Γ(5/4)Γ(−1/4)1F2(5/4;5/4,3/2;1/4)+4−3/4√πΓ(5/4)Γ(5/4)1F2(1;5/4,3/4;1/4)
The Generalized Hypergeometric 1F2 function (don't confuse with the well-known 2F1) reduces to functions of lower level in the particular cases :
1F2(5/4;5/4,3/2;1/4)=sinh(1)
1F2(1;5/4,3/4;1/4)=√π4e(e2erfi(1)+erf(1))
and after simplification :
∫∞0√xsin(x)1+x2dx=−π√2sinh(1)+π2√2e(e2erfi(1)+erf(1))
∫∞0√xsin(x)1+x2dx=π2√2e(e2erfi(1)+erf(1)+1−e2)
∫∞0√xsin(x)1+x2dx=π2√2e(−e2erfc(1)+erf(1)+1)
For the Hypergeometric 1F2 function, see : http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F2/02/
About the functions erf, erfi, erfc, see : http://mathworld.wolfram.com/Erf.html , http://mathworld.wolfram.com/Erfi.html , http://mathworld.wolfram.com/Erfc.html
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