Why does the asymptotic expansion of the real-valued Kummer function contain complex terms?

Question

Working on a problem in spectral theory, I need to study the asymptotics of a confluent hypergeometric function (here $(a)_0=1$ and $(a)_s=a(a+1)\cdots(a+s-1)$ denote the Pochhammer symbol) $$ \mathbf{M}(a,b,z)=\sum_{s=0}^{+\infty}\frac{(a)_s}{\Gamma(b+s)s!}z^s,\quad\text{as}\ z\to+\infty. $$ In my case $-1<a<0$ and $b=1$, and I'm only interested in real $z$.

I had a look in Abramowitz–Stegun (13.5.1, where $M(a,b,z)=\Gamma(b)\mathbf{M}(a,b,z)$), and found that, as $z\to+\infty$, we have the expansion $$ \mathbf{M}(a,1,z)\sim \frac{e^{i\pi a}}{\Gamma(1-a)}z^{-a}\sum_{s=0}^{+\infty}\frac{\bigl((a)_s\bigr)^2}{s!(-z)^s} + \frac{e^z z^{a-1}}{\Gamma(a)}\sum_{s=0}^{+\infty}\frac{\bigl((1-a)_s\bigr)^2}{s!z^s}. $$ What is worrying me is the factor $e^{i\pi a}$ in the first term. It is complex (in fact, non-real), and everything else in the expansion is real for positive $z$. Also, from the definition of $\mathbf{M}$ we see that it should be real for positive $z$. I have also had a look in 13.7.2, where the same expansion is given. It is also the same in the book Asymptotics and special functions by Frank Olver, and I get the same from Mathematica. Thus, I believe that the expansion above is correct.

In the asymptotic expansion above the term with the $e^{i\pi a}$ factor is small in comparison with the second one. In fact, some sources hint that it can be neglected (compare 13.7.1). As it happens, I want to keep that term, even if it is small. Thus, I think I can state my questions as follows:

1. Why is the real-valued function having complex terms in its asymptotic expansion?
2. I'm only considering real $z$. Will the expansion of $\mathbf{M}(a,1,z)$ above still be valid if I replace $e^{i\pi a}$ with its real part, $\cos(\pi a)$?
3. Could it be that the imaginary part somehow cancels? (I don't see how it could.)

Answer 1

When z is real and tends to $\infty$ (that is, $z>0$) the second part of this double series is dominant, that is, is exponentially bigger that the first part and then, in asymptotic sense, you can avoid it to obtain

$$\mathbf{M}(a,1,z)\sim \frac{e^z z^{a-1}}{\Gamma(a)}\sum_{s=0}^{+\infty}\frac{\bigl((1-a)_s\bigr)^2}{s!z^s}$$

so, the complex terms in its asymptotic expansion dissapear. Note that when $z$ is real and negative, that is, $z\to-\infty$, is the first part that is dominant and taking account that $z^{-a}=e^{i\,\pi\,a}(-z)^{-a}$, the imaginary part also dissapear

$$\mathbf{M}(a,1,z)\sim \frac{1}{\Gamma(1-a)}(-z)^{-a}\sum_{s=0}^{+\infty}\frac{\bigl((a)_s\bigr)^2}{s!(-z)^s}$$

where here $-z$ is positive.

Answer 2

The statement is that $$ {\bf M}(a,1,z) \sim \frac{{\mathrm{e}^z z^{a - 1} }}{{\Gamma (a)}}\sum\limits_{n = 0}^\infty {\frac{{((1 - a)_n )^2 }}{{n!z^n }}} + \frac{{\mathrm{e}^{ \pm \pi \mathrm{i}a} z^{ - a} }}{{\Gamma (1 - a)}}\sum\limits_{n = 0}^\infty {\frac{{((a)_n )^2 }}{{n!( - z)^n }}}, $$ as $z\to \infty$ in the sectors $ - \frac{\pi }{2} + \delta \le \pm \arg z \le \frac{{3\pi }}{2} - \delta$. The region of validity is in the sense of Poincaré, i.e., it ignores the Stokes lines $\arg z =0$, $\arg z =\pm \pi$ and extends the expansion up to the anti Stokes lines $\arg z = \mp \frac{\pi }{2}$ and $\arg z =\pm \frac{3\pi }{2}$. A more precise statement is that these expansions hold whenever $0<\pm \arg z < \pi$. On the Stokes line $\arg z=0$, the correct expansion is the average of the expansions on the two sides. Thus, in summary $$ {\bf M}(a,1,z) \sim \frac{{\mathrm{e}^z z^{a - 1} }}{{\Gamma (a)}}\sum\limits_{n = 0}^\infty {\frac{{((1 - a)_n )^2 }}{{n!z^n }}} + S(\theta )\frac{{z^{ - a} }}{{\Gamma (1 - a)}}\sum\limits_{n = 0}^\infty {\frac{{((a)_n )^2 }}{{n!( - z)^n }}} $$ as $z \to \infty$, with $\theta =\arg z$ and the Stokes multiplier $$ S(\theta)=\begin{cases} \mathrm{e}^{-\pi \mathrm{i} a} & \text{if }\; -\pi < \theta <0, \\[0.25em] \cos(\pi a) & \text{if }\; \theta = 0, \\[0.25em] \mathrm{e}^{\pi \mathrm{i} a} & \text{if }\; 0 < \theta < \pi. \end{cases} $$ For an exponentially improved, optimally truncated asymptotic expansion on the line $\arg z=0$, see this paper.