I changed the inverse_laplace_transform function in
sympy.integrals.transforms to first break the given function in partial
fractions and calculate the inverse_laplace_transform as the sum of
inverse_laplace_transform of the partial fractions.
def inverse_laplace_transform(F, s, t, plane=None, **hints):
def part_frac_list(ex):
fracs = []
parts = apart_list(ex)
try:
for x in parts[2]:
fracs.append(assemble_partfrac_list((parts[0],parts[1],[x])))
except:
return fracs
return fracs
parts_fracs = part_frac_list(F)
return sum([InverseLaplaceTransform(x.refine(), s, t,
plane).doit(**hints) for x in parts_fracs])
Now the the function appears to work better
>>> inverse_laplace_transform(2/(s**2*(s**2 + 1)),s,t)
2*t*Heaviside(t) - 2*sin(t)*Heaviside(t)
The integration algorithm can be improved in the same by breaking the
function into partial fractions.
The answer for inverse_laplace_transform(ln(s/(s-1)),s,t) is -(1-exp(t))/t.
It can be evaluated by the observing t*f(t) = -F'(s) where F(s) is the
laplace transform f(t).
F(s) = ln(s/(s-1))
F'(s) = (s - 1)*(-s/(s - 1)**2 + 1/(s - 1))/s
inverse_laplace_transform(F'(s)) = t*f(t) = (-exp(t) + 1)
Hence,
f(t) = -(1-exp(t))/t
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