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image Is there a proof for this? Share It Thread Tools Search this Thread image
Old Mar25-09, 12:30 PM                  #1
Heirot

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Is there a proof for this?

Hello,

let's say we have two functions of two variables: f(x,y) and g(x,y). Say we know that the sum / integral over all y's of f^n * g does not depend on x for every natural number n (and zero). Does that mean that f and g both don't depend on x?

Thanks
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Old Mar25-09, 01:44 PM                  #2
HallsofIvy

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Re: Is there a proof for this?

Not necessarily. Let g(x,y)= 0 for all x and y. Then f^n*g (I assume you mean composition) is f^n(0) for all x and y so the sum/integral is a constant no matter what f is. If f^n*g is ordinary multiplication of functions, then f^n*g= 0 for all x and y and again, the integral is a constant no matter what f is.
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Old Mar25-09, 02:40 PM                  #3
Office_Shredder

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Re: Is there a proof for this?

If it's composition, then g better be a vector
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Old Mar25-09, 05:32 PM                  #4
Heirot

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Re: Is there a proof for this?

Sorry for not being clear - f(x,y) and g(x,y) are scalar functions and * is ordinary multiplication. f^n is then f multiplied n times by itself. Now, if we don't assume the trivial null solution, f(x,y)=0 or g(x,y)=0, does my statement hold?
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