CODEWITHSUNDEEP
wolfram-mathematica
user1886681
user1886681

Reputation: 177

Mathematica's Integrate just spits out the input...Does not solve Integral

Just looking for some mathematica help. When I try to computer this integral:

Integrate[Cos[t]/(1 + b^2 t^2 - (2*b*c*t)/a + c^2/a^2)^(3/2), { t, -Infinity, Infinity}, Assumptions -> {a, b, c} \[Element] Reals]

Mathematica just spits out:

Integrate[Cos[t]/(1 + c^2/a^2 - (2 b c t)/a + b^2 t^2)^(3/2), {t, -\[Infinity], \[Infinity]}, Assumptions -> (a | b | c) \[Element] Reals]

How would I go about evaluating this integral?

Upvotes: 4

Views: 1935

Answers (1)

Vitaliy Kaurov
Vitaliy Kaurov

Reputation: 1300

Simplify. First of all you do not need 3 independent parameters there, just two. Then this integral is equivalent to (via change of variables and re-scaling of parameters)

Integrate[Cos[(a - b t)]/(1 + t^2)^(3/2), {t, -Infinity, Infinity}, 
          Assumptions -> {a, b} \[Element] Reals]

with answer:

2 Abs[b] BesselK[1, Abs[b]] Cos[a]

The constants a, b are different from original ones but can be expressed through them if you do simple re-scaling and change of variable. And here is your beautiful function in the parameter space:

Plot3D[2 Abs[b] BesselK[1, Abs[b]] Cos[a], {a, -5, 5}, {b, -5, 5}, 
 PlotRange -> All, Mesh -> All, ColorFunction -> "DarkRainbow", 
 MeshStyle -> Opacity[.1], AxesLabel -> {a, b}]

enter image description here

Upvotes: 5

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