Hi!
David Bailey (http://www.davidhbailey.com/) showed Viviane, Travis, and myself
the
following oddity yesterday.
Take the integral
\int_0^1 \int_0^1 |e^{2\pi i x} + e^{2\pi i y}| dx dy
The answer should be 4/\pi.
Both Mathematica and Maple give 0 as an answer. Unfortunately, Sage/Maxima
also gives 0:
sage: f = lambda x,y : simplify((e^(2*pi*I*x)+e^(2*pi*I*y)).abs())
sage: integral(integral(f(x,y),(x,0,1)),(y,0,1))
0
Pulling out e^{2\pi i x} to simplify the integral to a one dimensional integral,
Sage can solve this numerically:
sage: g = lambda x : (1+e^(2*pi*I*x)).abs()
sage: numerical_integral(g,0,1)
(1.2732395447351625, 1.4155343563970746e-14)
sage: n(4/pi)
1.27323954473516
but not symbolically:
sage: integral(g,(x,0,1))
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-68-233115ecebe7> in <module>()
----> 1 integral(g,(x,Integer(0),Integer(1)))
/Applications/sage/local/lib/python2.7/site-packages/sage/misc/functional.pyc
in integral(x, *args, **kwds)
765 else:
766 from sage.symbolic.ring import SR
--> 767 return SR(x).integral(*args, **kwds)
768
769 integrate = integral
...
TypeError:
Best,
Anne
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