Le samedi 19 juillet 2014 18:25:42 UTC+2, Anne Schilling a écrit :
>
> 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.
>
Ahem. I do not understand your expected result (but my complex analysis is
rusty)...). In the notebook (sage 6.3beta6) :
var("x,y,t,u,v")
assume(x,"real",y,"real",t,"real",u,"real",v,"real",u>0,v>0,t>0)
foo(x,y)=abs(e^(2*i*pi*x)+e^(2*i*pi*y))
bar(u,v)=integrate(integrate(foo(x,y),y,0,v),x,0,u)
show(bar(u,v).simplify_full())
gee(t)=bar(t,t)
limit(gee(t),t=1)
The expression for bar is atrocious. bit the limit is not :
1/2*I/pi
By the way :
gee(1)
gives :
-1/2*I/pi
indeed. But:
integrate(integrate(foo(x,y),y,0,1),x,0,1)
gives :
0
--
Emmanuel Charpentier
>
> 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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