Comment #13 on issue 2440 by [email protected]: Equal Integrals compare different when using different variables
http://code.google.com/p/sympy/issues/detail?id=2440
I think that perhaps they should compare unequal. Making them equal would be inconvenient. For example, pretend for a moment that issue 2297 were fixed, so that we could take two integrals depending on different variables and easily put one inside the other. Suppose also that we could easily do a change of variables to polar coordinates.
For this example, we want to use SymPy to prove the Gaussian formula integral(exp(-x**2), (x, -oo, oo)) = sqrt(pi). To compute this, you suppose that a = integral(exp(-x**2), (x, -oo, oo)) and take a**2. To do this, you need to take for each a a different integration variable. So you really want to take Integral(exp(-x**2), (x, -oo, oo))*Integral(exp(-y**2), (y, -oo, oo)). You then put one integral inside the other, combine exponentials, convert to polar coordinates, and compute the integral using anti-derivatives. See http://en.wikipedia.org/wiki/Gaussian_integral for this same proof written out in more detail.
But if we compare "equal" integrals as equal, then Integral(exp(-x**2), (x, -oo, oo))*Integral(exp(-y**2), (y, -oo, oo)) will be automatically converted into Integral(exp(-x**2), (x, -oo, oo))**2 (or Integral(exp(-y**2), (y, -oo, oo))**2, depending on the canonical order the result in in the Mul). So the above transformation will be impossible, unless you do it manually (but the whole point is to do things using only given transformation functions, so that you know you are not making any mistakes).
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