Hello,
I have tried to do the inverse Laplace transform of ‘2/((3*s + 1)*(s**2 + 1))’
and I got a really ugly result. Fortunately it’s easy to get a much better
solution.
It wouldn’t surprise me to know that I’m doing something wrong... If not, I
hope it could improve in someway how Sympy does the inverse Laplace transform.
Here is the Jupyter session showing how I got that ugly solution:
+*In[1]:*+
[source, ipython3]
----
from sympy import *
t = symbols("t", real=True)
s = symbols("s")
y = 2/((3*s + 1)*(s**2 + 1))
inverse_laplace_transform(y, s, t)
----
+*Out[1]:*+
----
2*(-(I*sin(t) - cos(t))*gamma(-2*I)*gamma(1/3 - I)/(gamma(1 - 2*I)*gamma(4/3 -
I)) + (I*sin(t) + cos(t))*gamma(2*I)*gamma(1/3 + I)/(gamma(1 + 2*I)*gamma(4/3 +
I)) + exp(-t/3)*gamma(-1/3 - I)*gamma(-1/3 + I)/(gamma(2/3 - I)*gamma(2/3 +
I)))*Heaviside(t)/3
----
The solution could be improved a lot if we perform a partial fraction
decomposition and then do the inverse Laplace transform:
+*In[2]:*+
[source, ipython3]
----
yapart = apart(expand(y))
yapart
----
+*Out[2]:*+
----
-(3*s - 1)/(5*(s**2 + 1)) + 9/(5*(3*s + 1))
----
+*In[3]:*+
[source, ipython3]
----
inverse_laplace_transform(yapart.args[0], s, t) + \
inverse_laplace_transform(yapart.args[1], s, t)
----
+*Out[3]:*+
----
(sin(t) - 3*cos(t))*Heaviside(t)/5 + 3*exp(-t/3)*Heaviside(t)/5
----
Maybe it could be a good idea if the inverse Laplace transform performed this
decomposition.
Javier
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