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let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $

also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ g(s)=g(1-s) $

then we use some numerical method to evaluate the roots and we check that the first 100 roots agree so can we conclude that $ f(x)= g(x) $ or at least $ f(x)=h(x)g(x) $ they are proportional function

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    @BenMillwood - yes, that is right, but the idea of comparing the first 100 roots of two functions without qualification means that two sets of roots need to be well-ordered - and there might be expected to be some natural relationship between those well-orderings, since nothing more is said. There are easy examples where the question makes sense even if the roots are not isolated (roots at $s=1-\frac 1 n$ and $s=\frac 1 n$ being an example).2012-09-01

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You can almost never conclude that some equation is true by just checking with numerical methods. For instance if:

$ f = \mathbb 1_{\mathbb R \backslash {\mathbb Q}} $

Any computer will give you $f = 0$.

Moreover, in your case, $g = \mathbb 1_{{\mathbb Q}}$ gives a counter-example.

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    +1. Additional conditions are needed, such as, for example, the continuity of the functions and the density of the set of zeroes.2012-09-01