Status: Accepted
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Labels: Type-Defect Priority-Medium Integration Simplify
New issue 3037 by [email protected]: integrate(exp(I*k*x)/(k**2 - 2*lamda
+ 1), (k, -oo, oo)) should be expressible in closed form
http://code.google.com/p/sympy/issues/detail?id=3037
In [105]: print integrate(exp(I*k*x)/(k**2 - 2*lamda + 1), (k, -oo, oo))
Piecewise((meijerg(((1/2,), ()), ((1/2, 0, 1/2), ()),
exp_polar(-I*pi)*polar_lift(x)**2*polar_lift(-2*lamda +
1)/4)/(2*sqrt(pi)*sqrt(polar_lift(-2*lamda + 1))) + meijerg(((1/2,), ()),
((1/2, 0, 1/2), ()), exp_polar(I*pi)*polar_lift(x)**2*polar_lift(-2*lamda +
1)/4)/(2*sqrt(pi)*sqrt(polar_lift(-2*lamda + 1))),
And(Or(And(Abs(periodic_argument(exp_polar(I*pi)*polar_lift(x)**2, oo)) ==
pi, Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi),
And(Or(And(Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) != -pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) != pi),
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi),
Abs(periodic_argument(exp_polar(I*pi)*polar_lift(x)**2, oo)) < pi),
And(Abs(periodic_argument(exp_polar(I*pi)*polar_lift(x)**2, oo)) < pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi)),
Or(And(Or(And(Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) != -pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) != pi),
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi),
Abs(periodic_argument(exp_polar(-I*pi)*polar_lift(x)**2, oo)) < pi),
And(Abs(periodic_argument(exp_polar(-I*pi)*polar_lift(x)**2, oo)) == pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi),
And(Abs(periodic_argument(exp_polar(-I*pi)*polar_lift(x)**2, oo)) < pi,
Abs(periodic_argument(1/polar_lift(-2*lamda + 1), oo)) < pi)))),
(Integral(exp(I*k*x)/(k**2 - 2*lamda + 1), (k, -oo, oo)), True))
In [106]: integrate(exp(I*k*x)/(k**2 - 2*lamda + 1), (k, -oo, oo),
conds='none')
Out[106]:
⎛ │ -ⅈ⋅π 2
⎞ ⎛ │ ⅈ⋅π 2 ⎞
╭─╮3, 1 ⎜ 1/2 │ ℯ ⋅polar_lift (x)⋅polar_lift(-2⋅λ + 1)⎟
╭─╮3, 1 ⎜ 1/2 │ ℯ ⋅polar_lift (x)⋅polar_lift(-2⋅λ + 1)⎟
│╶┐ ⎜ │ ─────────────────────────────────────────⎟
│╶┐ ⎜ │ ────────────────────────────────────────⎟
╰─╯1, 3 ⎝1/2, 0, 1/2 │ 4 ⎠
╰─╯1, 3 ⎝1/2, 0, 1/2 │ 4 ⎠
──────────────────────────────────────────────────────────────────── +
───────────────────────────────────────────────────────────────────
___
______________________ ___
______________________
2⋅╲╱ π ⋅╲╱ polar_lift(-2⋅λ +
1) 2⋅╲╱ π ⋅╲╱ polar_lift(-2⋅λ + 1)
In [107]:
But this should be expressible in closed form. I think it should be
pi*exp(-sqrt(1 - 2*lamda)*abs(x))/sqrt(1 - 2*lamda), for at least lamda >
1/2 and x real (you can compute the answer using residue calculus using an
appropriate contour).
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