I think there are two basic cases to consider:
1) delta functions with *linear* arguments.
This is (I believe) by a large margin the most common case. For this
deltaintegrate should simply be fixed; one only needs to correctly
implement integral(f(x) delta(a*x+b), (x, c, d)) and this shouldn't be
hard. (Note that this will also yield the multi-dimensional cases, by
repeated integration.)
2) delta functions with more complicated arguments
Getting this to work is (I believe) much more work. It is non-trivial to
even define this.
Tom
On 20.12.2011 16:21, Matthew Rocklin wrote:
Hi Everyone,
I'd like to compute multivariate integrals that contain Dirac
Deltafunctions. I.e. expressions like
integrate(exp(-(x**2+y**2))/pi * delta(2*x+3*y), (x,-oo, oo), (y,-oo, oo))
The deltaintegrate function inside sympy fails to compute these
correctly, see issue 2630.
<http://code.google.com/p/sympy/issues/detail?id=2630>
Wikipedia says
<http://en.wikipedia.org/wiki/Dirac_delta_functions#Properties_in_n_dimensions>
that you can compute general expressions of this form as follows:
\int_{\mathbb{R}^n} f(\mathbf{x}) \, \delta(g(\mathbf{x})) \,
d\mathbf{x} =
\int_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x})
(hopefully the above image makes it through, if not go here)
http://upload.wikimedia.org/wikipedia/en/math/3/3/f/33fbbed28ec715257d268faefc9e0e9f.png
How hard would it be to compute the right hand side in sympy? In
particular I'm confused by how to express the domain and what they mean
by \delta\sigma(x)
Or, if there is a better way of going about this I'm happy to hear it.
-Matt
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