# Is there a way to compute the following definite integrals?

While evaluating the final wavefunction of a free spin-0 particle in one dimension in terms of Green's functions (propagators) and considering the initial conditions as Gaussian wavepackets, I arrived at an expression that consists of two definite integrals each of which has to be evaluated separately. The integration is with respect to the variable $z$ (the relative position). The two integrals $I_{1}$ and $I_{2}$ are:

I1 = Integrate[(7.2288*
BesselJ[1, Sqrt[-2 + 2*Sqrt[2]]*Sqrt[200*Sqrt[2] - z^2]]*
Sin[x + z])/(E^(1.25*(x + z)^2)*Sqrt[200*Sqrt[2] - z^2]),
{z, -16.8179, 16.8179}]


and

I2 = Integrate[
-((0.0181*
BesselJ[0, Sqrt[-2 + 2*Sqrt[2]]*Sqrt[200*Sqrt[2] - z^2]]*
BesselK[1, Sqrt[0.04 + (x + z)^2]]*Sin[x + z])/
Sqrt[0.04 + (x + z)^2]),
{z, -16.8179, 16.8179}]


A guidance on how to go about evaluating this further would be appreciated.

• this will need to be done numerically. Specify x and use NIntegrate – george2079 Apr 15 '17 at 18:05

Maybe so:

I1[x_?NumericQ] := NIntegrate[(7.2288 Exp[-1.25 (x + z)^2] BesselJ[1,
Sqrt[-2 + 2 Sqrt[2]] Sqrt[200 Sqrt[2] - z^2]] Sin[x + z])/Sqrt[
200 Sqrt[2] - z^2], {z, -16.8179, 16.8179}]

ListLinePlot@Table[{x, I1[x]}, {x, -5, 5, 1/5}]


I2[x_?NumericQ] := NIntegrate[-((0.0181 BesselJ[0,
Sqrt[-2 + 2 Sqrt[2]] Sqrt[200 Sqrt[2] - z^2]] BesselK[1, Sqrt[
0.04 + (x + z)^2]] Sin[x + z])/Sqrt[0.04 + (x + z)^2]), {z, -16.8179, 16.8179}]

ListLinePlot@Table[{x, I2[x]}, {x, -5, 5, 1/5}]