If one has an expression of the form,
$$\int \frac{\, _2F_1(a,b;c;z)}{_2F_1(d,e;f;z)}dz,$$
where the arguments are all complex numbers, is it possible to integrate this to get another hypergeometric function, or some other analytic expression?
If one expands the hypergeometric functions into their series representation, it seems straightforward that this expression can be integrated, however possibly the resulting series is no longer convergent. When I tried to evaluate this integral with Mathematica, it won't give me an output, so does this mean that the new series is non-convergent? Or is there some other way I can analyse the series?
In particular, the integral is,
$$\int \frac{\, _2F_1((3 + \alpha), (3 – \sqrt{\beta} + \alpha); 3 +
\alpha; z)}{_2F_1( (1 + \alpha), (1 – \sqrt{\beta} + \alpha ); 1 +
\alpha; z)}dz,$$
where $\alpha$ and $\beta$ are constants.
Thanks!
Best Answer
The ratio of two analytic functions is analytic on any domain where the denominator is not $0$. Also the integral of a power series has the same radius of congergence as the original power series (as can be easily verified by the ratio test).
So yes, this integral exists on any simply-connected domain that does not include a zero of the denominator, and the integral will also be analytic over that domain. However, it is not likely to be another hypergeometric function.