
#1
Mar2509, 11:30 AM

P: 151

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
Mar2509, 12:44 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,900

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.




#3
Mar2509, 01:40 PM

Mentor
P: 4,499

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




#4
Mar2509, 04:32 PM

P: 151

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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