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plotting - How to plot the result of this singular integral?


Please I open a new post here after this one : https://mathematica.stackexchange.com/a/59203/10158



Now I want to plot the function f(a,b) as a function of b for different values of a : a=0.5 , a=10 and a=100


f(a,b)=1−ab2K2(a)∂2∂a2[1a1√1−b2∫∞1e−ax√x2−1(1+x√1−b2)(x−1/√1−b2)dx]


where K2(a) is the modified Bessel function of the second kind and ∂2∂a2 is the 2nd derivative with respect to a.


Mathematica input form:


1 - ((a b^2/BesselK[2, a]) D[(1/a) (1/Sqrt[1 - b^2]) Integrate[E^-ax/(Sqrt[x^2 - 1] (1 + x Sqrt[1 - b^2]) (x - 1/Sqrt[1 - b^2])), {x,1, Infinity}], {a, 2}])

how do it please?


Thank's.



Answer



First I assume that the OP wants to compute the Cauchy principal value of the integral, since the integrand has a simple pole in the interior of the interval of integration. Dealing with such singularities is described in the tutorial NIntegrate Integration Strategies.



I let the derivative be computed in the process of defining f. The symbols a and b are highlighted in red by the syntax highlighter because they appear as function arguments and as symbols being blocked. It indicates a warning of a potential scoping conflict. But Block is evaluated before the definition of f is set, so the warning may be ignored in this case. The differentiation results in an expression that contains a linear combination of three NIntegrate integrals. I combine these, more or less manually, with pattern replacements that implement the linearity properties of the integral. This reduces computation time by about 40%.


ClearAll[f];

f[a_, b_] /; a > 0 && 0 < b < 1 :=
Evaluate[Block[{a, b, NIntegrate},
1 - ((a b^2/BesselK[2, a]) *
D[(1/a) (1/Sqrt[1 - b^2]) *
NIntegrate[Exp[-a x]/(Sqrt[x^2 - 1] (1 + x Sqrt[1 - b^2]) (x - 1/Sqrt[1 - b^2])),
{x, 1, 1/Sqrt[1 - b^2], Infinity},
Method -> "PrincipalValue"],

{a, 2}]) /.
HoldPattern[Times[NIntegrate[int_, args__], coeff__]] :>
NIntegrate[Times[int, coeff], args] /.
t : Plus[_NIntegrate, __NIntegrate] :>
NIntegrate[First /@ t, Sequence @@ t[[1, 2 ;;]]]
]]

For the plot, I set a few options for NIntegrate suitable for plotting. Somehow some of the coefficients get set to MachinePrecision which causes a precision warning; therefore I turned the warning off.


Module[{opts = Options[NIntegrate], plot, offQ},
SetOptions[NIntegrate, {WorkingPrecision -> 12, AccuracyGoal -> 3, MaxRecursion -> 12}];

offQ = Head[NIntegrate::precw] === $Off;
Off[NIntegrate::precw];

plot = Plot[
Evaluate@Table[f[a, b], {a, {0.5`20, 10, 100}}], {b, 0, 1},
PlotPoints -> 15, WorkingPrecision -> 12];

SetOptions[NIntegrate, opts];
If[! offQ, On[NIntegrate::precw]];
plot

]

Mathematica graphics


Evaluation is still rather slow, the plot above taking about 28-29 seconds.


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