I have question in the analyticity of Jost function on the $p$-plane. The chapter 12 of the book Scattering Theory by John R. Taylor states (p.218): The Jost function is defined as
$$ f_{l}(p)=1+\frac{1}{p}\int^{\infty}_0dr\hat{h}^+_l(pr)U(r)\phi_lp(r),\quad p\in\mathbb{C},\ \tag{12.7} $$
The analyticity region of Jost function is $Im(p)>0.$ $\phi_lp(r)$ is the regular solution and $\hat{h}^+_l(pr)$ is the Riccati-Hankel function.
The author then takes the integration region from the positive real axis to the r-plane, as $r=\rho\ e^{i\theta}$ and integrate $\rho$ from 0 to infinity (p.221).
$$ f_{l}(p)=1+\frac{1}{p}\int^{\infty\ e^{i\theta}}_0dr\hat{h}^+_l(pr)U(r)\phi_lp(r),\quad p\in\mathbb{C},\ \tag{12.12} $$
The analyticity region of Jost function is $Im(pe^{i\theta})>0$
Then the author says since the two functions coincide in the region of overlap, each is an analytic continuation of another.
However, when we integrate along $r=\rho\ e^{i\theta}$ the Jost function must be a function of $\theta$, which means they are not equal in the region of overlap. The statement of analyticity continuation should not be valid. I can not understand how the analytic continuation works.
The function \begin{equation} f_{l}(p) = 1+\frac{1}{p}\int^{\infty}_0dr\hat{h}^+_l(pr)U(r)\phi_lp(r), \tag{1} \end{equation} analytic for $\Im(p)>0$, and the function
\begin{equation} f_{l}(p)_{\theta} = 1+\frac{1}{p}\lim_{\rho \rightarrow \infty}\int^{\rho e^{i\theta}}_0dr\hat{h}^+_l(pr)U(r)\phi_lp(r) \tag{2} \end{equation} analytic for $\Im(pe^{i\theta})>0$ are equal on the union of their domains of analyticity because you can use contour integration to deform one into the other. So the $\theta$ dependence only enters the definition of the function via its domain of analyticity.