Equation (10.4) looks like a very good example indeed. I have enough to write M(a,b,z) in the most general case (a,b,z are complex). Next, I transform 10.4 to an ODE that I solve using Boost, which can handle complex coefficients. BTW do you solve 10.4 using numerical quadrature?
Yes, the code is in the book, middle of pg 461 -- I just use Mathematica's NIntegrate. There is an example calling routine on the previous page. Note that [$](a,b,c) = (\omega,\theta,\sigma)[$].
CHF looks likes a benign integrand and even if not I reckon NIntegate is able to see that fact? (I'm guessing). The graphs of CHF in A&S look OK.
I see that the complex-valued Fresnel integral can be written in terms of CHF. I have solved the former integral by turning it into an ODE and solving bot as a complex-valued ODE and as a pair of real-valued ODEs, I get the same results, also the examples in A&S.
So, at some stage I want to try 10.4 as an ODE and check against Mathematica.
// Nice analysis in chapter 10, Alan.
I would like to call it Hypergeometric1F1
to avoid confusion with [$]M(a,b,\rho)[$] which is a different ball-game.
Q: For 10.4 we don't need the complex case for H1F1 (just yet) and we can start with doubles
as pilot case?