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Is there a proof for this?

  1. 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
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,410
    Staff Emeritus
    Science Advisor

    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.
     
  4. Office_Shredder

    Office_Shredder 4,500
    Staff Emeritus
    Science Advisor
    Gold Member

    If it's composition, then g better be a vector
     
  5. 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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