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Intriguing Integral Illation (Posted on 2013-03-02) Difficulty: 3 of 5
     1 1
    ∫ ∫ {x/y}{y/x} dxdy
    0 0
where {n}= n - floor(n)

No Solution Yet Submitted by K Sengupta    
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exact solution | Comment 3 of 6 |
let f(n)=floor(n) for brevity
(x/y-f(x/y))*(y/x-f(y/x) expanding
now there are three cases to consider
x<y then f(x/y)=0 and it simplifies to
now let f(y/x)=k with k some integer greater than 1
this implies that
now y is further restricted by the range [0,1] thus we need
kx<=1 or x<=1/k
now here there are two ranges to split x into
x in [0,1/(k+1)]
in this range y is in [kx,(k+1)x]
and x in [1/(k+1),1/k]
in this range y is in [kx,1]
thus we have two integrals
using mathematica to evaluate these integrals gives us
a nice big equation too complicated for me to type here.  I take this expression and sum it for k=1 to infinity and get the value

next case is if x=y
in this case f(x/y)=f(y/x)=1 and the expression simplifies to zero

finally, we have x>y
by symmetry this is the same as x<y and thus again we get the value

adding these two values get the final value of the integral as
which is approximately
which seems to agree with Charlies numerical solution

  Posted by Daniel on 2013-03-03 01:28:12
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