Sunday, November 13, 2016

integration - How to evaluate the following integral.




P(S)=e(a2+c2+λ b2)δ[S4b2+(ac)2] da db bc.



where λ is a constant. How to Evaluate above integral ?.



I tried!, but could reach up to mid way only.



On transforming P(S) to the three dimensional the spherical polar co-ordinates using 2b=rcosθ,a=rsinθcosϕ,c=rsinθsinϕ as



P(S)=0π02π0er2(λ(cos2θ)/4+sin2θ(cos2ϕ+sin2ϕ))δ[Srg(θ,ϕ)] r2drsinθ dθ dϕ.




Crashing the delta function in above, we get a θ,ϕ integral



P(S)=Aπ/20π0eS2(λ(cos2θ)/4+sin2θ(cos2ϕ+sin2ϕ))/g2(θ,ϕS2|g[θ,ϕ)|3sinθ dθ dϕ,



Where g(θ,ϕ)=1sin2θsin2ϕ. and A is evaluated value (I am taking it as a constant) of r integral.



After that, I took help of mathematica to evaluate it numerically.



It would be really a great help If anyone can help me by evaluating integral



Answer



First, let's deal with the delta function. We will use the delta function identity
δ[f(x)]=nδ(xxn)|f(xn)|,
where xn is a zero of the function f(x) and the sum runs over all zeros. In our case,
f(b)=S4b2+(ac)2,
which has two zeros
b±=±12S2(ac)2,
and the derivative of f(b) can be worked out easily. Applying the above identity gives
δ[S4b2+(ac)2]=S4|b|(δ(bb+)+δ(bb))
Note that the bintegral is nonzero only if b± are real, which implies |ac|S. Thus, the desired integral now reads

P(S)=S4
Introducing new variables x=a-c and y=a+c,
\mathrm{d}a\,\mathrm{d}c=\left|\frac{\partial(a,c)}{\partial(x,y)}\right|\mathrm{d}x\,\mathrm{d}y=\frac{1}{2}\mathrm{d}x\,\mathrm{d}y,
we have
\begin{align} P(S)&=\frac{S}{2}e^{-\frac{\lambda}{4}S^2}\iint_{|x|\leq S}\mathrm{d}x\,\mathrm{d}y~\frac{e^{-\frac{y^2}{2}-\left(\frac{1}{2}-\frac{\lambda}{2}\right)x^2}}{\sqrt{S^2-x^2}}\\ &=\frac{S}{2}e^{-\frac{\lambda}{4}S^2}\color{Blue}{\int_{-\infty}^{\infty}\!\!\!\mathrm{d}y~e^{-\frac{y^2}{2}}}\int_{-S}^{S}\!\!\!\mathrm{d}x~\frac{e^{-\left(\frac{1}{2}-\frac{\lambda}{4}\right)x^2}}{\sqrt{S^2-x^2}}\\ &=\frac{S}{2}e^{-\frac{\lambda}{4}S^2}\color{Blue}{\sqrt{2\pi}}\cdot2\int_0^S\!\!\!\mathrm{d}x~\frac{e^{-\left(\frac{1}{2}-\frac{\lambda}{4}\right)x^2}}{\sqrt{S^2-x^2}}\\ &=\sqrt{2\pi}S\,e^{-\frac{\lambda}{4}S^2}\int_0^S\!\!\!\mathrm{d}x~\frac{e^{-\Lambda\left(x/S\right)^2}}{\sqrt{S^2-x^2}},\qquad\Lambda\equiv\left(\frac{1}{2}-\frac{\lambda}{4}\right)S^2\\ \end{align}
setting x=S\sin\left(\frac{\theta}{2}\right), we have
\begin{align} P(S)&=\sqrt{\frac{\pi}{2}}S\,e^{-\frac{\lambda}{4}S^2}\int_0^{\pi}\!\!\!\mathrm{d}\theta~e^{-\Lambda\sin^2\left(\frac{\theta}{2}\right)}\\ &=\sqrt{\frac{\pi}{2}}S\,e^{-\frac{\lambda}{4}S^2-\frac{\Lambda}{2}}\int_0^{\pi}\!\!\!\mathrm{d}\theta~e^{\frac{\Lambda}{2}\cos\theta}\\ &=\sqrt{\frac{\pi}{2}}S\,e^{-\frac{\lambda}{4}S^2-\frac{\Lambda}{2}}\cdot\pi\,I_0\!\left(\frac{\Lambda}{2}\right), \end{align}
where I_0(x) is the modified Bessel function of the first kind (see Eq. (5) of http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html). With some further simplifications, we have

\boxed{P(S)=\sqrt{\frac{\pi^3}{\!2}}S\exp{\!\left(-\frac{2+\lambda}{8}S^2\right)}\,I_0\!\left(\frac{2-\lambda}{8}S^2\right)}
Cheers!


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