I'm trying to write a generic function to convolve two functions (in one
dimension, for now) and came across a few issues I don't know how to solve.
Rather than give up, maybe I can start a discussion. Some Sympy follows:
a , b = symbols('a b', positive=True)
f = exp(-a*x)
g = exp(-b*x)
integrate(f.subs(x,tau)*g.subs(x,t-tau), (tau,0,t))
1. I had to compute the range of integration myself (tau, 0, t). This
isn't a fatal problem, but piecewise doesn't have enough information to
compute it simply. (My long term solution is to propose a replacement for
the piecewise function, but I'm not there yet.)
The answer is below:
Piecewise((t*exp(-b*t), Eq(a, b)), (1/(a*exp(b*t) - b*exp(b*t)) - 1/(a*exp(a*t)
- b*exp(a*t)), True))
The answer is correct, but there are problems trying to incorporate it into
a larger function.
2. It is not obvious in advance the answer will depend on whether a = b.
Is there some way to assume in advance that a=b or a!=b?
3. The second condition, True, isn't helpful. To understand what True
means, one has to keep track of the previous conditions. Is there some way
to replace the True with a != b.
4. This is a simpler question, but I can't figure it out. How do I
manipulate the second answer to my preferred form?
(exp(-b*t)-exp(-a*t))/(a-b)
Is there a simple way to do it programmatically?
Thanks
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