Can we use Pade approximation to accurately integrate well-behaved functions?

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The discussion centers on the application of Padé approximation for integrating well-behaved functions, specifically focusing on the relationship between the integral of a function f(x) and a rational function Q(x). The key assertion is that for smooth and differentiable functions, such as sin(x) or exp(x^3), the integral of Q(x) can be computed exactly, allowing for an approximation of the integral of f(x) to be nearly zero. This method provides a powerful tool for evaluating integrals that are otherwise difficult to compute directly.

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Sangoku
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can we find for a well-behaved f(x) a Rational (Padé approximation) so

[tex]\int_{0}^{\infty}dx f(x) - \int_{0}^{\infty}dx Q(x) \approx 0[/tex] ??


Where Q(x) is a rational function, the main idea is that the integral for Q(x) can be performed exactly , whereas the initial integral of f(x) not.
 
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What do you mean by "well behaved"?
 
smooth and differentiable function, for example sin(x) or exp(x^3) ... so it is many times differentiable.
 

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