# The confluent limit of $_2F_1(a,b,c;\frac{1}{z})$

It is known that the confluent hypergeometric function $_1F_1$ is obtained after the following confluent limit:

$$_1F_1\left[ \begin{matrix} a \\ b \end{matrix},z \right]= \lim_{b \to \infty} ~ _2F_1\left[ \begin{matrix} a ~ , ~ b \\ c \end{matrix}~,~\frac{z}{b} \right].$$

Now, start from the connection formulae from $\frac{1}{z}$ to $z$ for $_2F_1$. In a suitable definition of the parameters, and when z is in the upper half plane (if I'm correct), it reads as

$_2F_1\left[ \begin{matrix} a,1+a-c \\ 1+a-b \end{matrix}; \frac{1}{z}\right]=e^{-i \pi a}z^{a}\frac{\Gamma(1+a-b)\Gamma(1-c)}{\Gamma(1-b)\Gamma(1+a-c)}~_2F_1\left[ \begin{matrix} a,b \\ c \end{matrix};z \right]+e^{i\pi(1+a-c)}z^{1+a-c} \frac{\Gamma(1+a-b)\Gamma(c-1)}{\Gamma(a)\Gamma(c-b)}~_2F_1\left[ \begin{matrix} 1+a-c,1+b-c \\ 2-c \end{matrix}; z\right].$

If we take naive confluent limit $z \to z/b$, and $b \to \infty$, after some computations we get

$z^{-a}~_2F_0\left[\begin{matrix}a,1+a-c\end{matrix};-\frac{1}{z} \right]=\frac{\Gamma(1-c)}{\Gamma(1+a-c)}~_1F_1\left[\begin{matrix} a \\ c \end{matrix};z \right]+z^{1-c}\frac{\Gamma(c-1)}{\Gamma(a)}~_1F_1\left[\begin{matrix} 1+a-c \\ 2-c \end{matrix};z \right]$.

For me it does not make sense because the right-hand side of this equations corresponds to the definition of the U-function $U(a,c;z)$, while the left hand side corresponds to its asymptotic expansion, which is an asymptotic series. From the Internet, this asymptotic expansion is valid for $| \text{arg}~z|<\frac{3\pi}{2}$.

My question is: how to make sense of this procedure, and how to make this "Stokes ray" $| \text{arg}~z|<\frac{3\pi}{2}$ appear? I could not find an interesting reference.