Conversion of definite integral to continued fraction:

Question

Consider integral of the form :

$$\int_a^b f(x)dx$$

$f(x)$ is analytic and real valued for real domain.

Now fix $a$ and $b$ ( most likely $[0,1]$ and $[0,\infty]$ ) .

Can we construct a continued fraction in terms of $f(x)$ , $a$ and $b$ which is equal to the integral ?

Is/are there such type of standard results in literature exists ?

Answer 1

In general, almost certainly not (or at least we are very far from being able to do so). Whatever formula or algorithm of this type you envision would presumably apply in particular to $$ \int_0^1 \frac4{1+x^2}\,dx = \pi, $$ which is dissonant with the fact that we have no special insight into the continued fraction of $\pi$.