# Numerical Integration in Laplace domain

I need to calculate two different integrals containing a Bessel function in the Laplace domain. I have tried different kinds of methods but didn't have any luck. I don’t know how to treat the Laplace variable (s) in the equations. This is really bugging me out! And I still couldn't find a way to compute this efficiently…

$$\int _0^{40}\text{BesselK}\left[0,\sqrt{\xi {}^{\wedge}2+s}\right]d\xi \int _0^{-10\sqrt{\frac{n^2 \pi ^2}{100}+s}}\text{BesselK}\left[0,\sqrt{(-\text{Log}[\tau ])^{2}+\left(\frac{n^2 \pi ^2}{100}+s\right)}\right] \times \text{Cos}\left[n \pi \frac{50+\left(15-\frac{\text{Log}[\tau ]}{\sqrt{\frac{n^2 \pi ^2}{100}+s}}\right)\text{Cot}[30^{\circ}]}{100}\right]\frac{1}{\tau }d\tau$$

Can anyone provide an insight?

@LeonidKovalev, these is my code to generate my results:

eq1[s_?NumericQ] := NIntegrate[BesselK[0, Sqrt[u^2 + s]], {u, 0, 40}]; eq2[s_?NumericQ] := NIntegrate[BesselK[0, Sqrt[u^2 + s]], {u, 0, -10}]; eq3[s_?NumericQ, n_?NumericQ] := NIntegrate[ BesselK[0, Sqrt[(-Log[t])^2 + ((n^2 [Pi]^2)/100 + s)]] Cos[ n [Pi] (50 + (15 - Log[t]/Sqrt[(n^2 [Pi]^2)/100 + s]) Cot[ 30 Degree])/100] 1/t, {t, 0, -10 Sqrt[(n^2 [Pi]^2)/100 + s]}];

eq4[s_?NumericQ, n_?NumericQ] := NIntegrate[ BesselK[0, Sqrt[(-Log[t])^2 + ((n^2 [Pi]^2)/100 + s)]] Cos[ n [Pi] (50 + (15 - Log[t]/Sqrt[(n^2 [Pi]^2)/100 + s]) Cot[ 30 Degree])/100] 1/t, {t, 0, 40 Sqrt[(n^2 [Pi]^2)/100 + s]}];

PD[s_] = (1/(s^(3/2) 100 Sin[30 Degree]) (eq1[s] - eq2[s]) + 2/(s 100 Sin[30 Degree]) Sum[ 1/Sqrt[(n^2 [Pi]^2)/100 + s]* Cos[n [Pi]/2]*(eq3[s, n] - eq4[s, n]), {n, 1, 10}]);

SetOptions[NIntegrate, Method -> "ClenshawCurtisRule", WorkingPrecision -> 5, MaxRecursion -> 12, AccuracyGoal -> 2];

Ni = 8; V[i_, NN_] = (-1)^(i + NN/2) !( *UnderoverscriptBox[([Sum]), (k = Floor[ *FractionBox[(1 + i), (2)]]), (Min[i, *FractionBox[(NN), (2)]])] *FractionBox[( *SuperscriptBox[(k), (1 + *FractionBox[(NN), (2)])]\ (((2\ k))!)), ((((i - k))!)\ *SuperscriptBox[(((k!))), (2)]\ ((((-i) + 2\ k))!)\ ((((-k) + *FractionBox[(NN), (2)]))!))]); (Algoritmo de Inversão \ numérica (Stehfest(1970))) PD1[tD_] = (Log[2] !( *UnderoverscriptBox[([Sum]), (i = 1), (Ni)](PD[ *FractionBox[(i\ Log[2]), (tD)]]\ V[i, Ni])))/tD;

LogLogPlot[{PD1[y], y*PD1'[y]}, {y, 0.1, 1000000}, PlotStyle -> {{Black}, {Dashed, Black}}, Frame -> True]

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Unlikely, if the code remains as unreadable as it is now. – user31373 Jun 9 '12 at 18:12
That's weird... Looks fine to me. I'm trying change it... – Bruno Rangel Jun 9 '12 at 19:09
I don't know what am i doing wrong... I tried to upload a image, but I need 10 reputations for this. – Bruno Rangel Jun 9 '12 at 19:21
OK, you got 10 rep points now. :) – user31373 Jun 9 '12 at 19:41
@LeonidKovalev, thanks! – Bruno Rangel Jun 9 '12 at 20:29