I would like to compute the following integral $I(k)$ in Mathematica to check if the result equals something I 'feel' is correct but I have no experience with contour integrals in Mathematica.

I want to compute $$ I(k) = 1 + \frac{k}{2}\int_\gamma \frac{1}{\sqrt{1-t}} J_1(k\sqrt{1-t}) dt, \quad \quad \quad \quad (1)$$ where $k > 0$, $J_1$ is the Bessel function of the first kind and where $\gamma$ is a contour starting at $1$ and ending at $-\infty$ which does not contain the origin.

I have a feeling this should result in something like: $$I(k) = H_0^{(1)}(k). \quad \quad \quad \quad \quad (2)$$ where $H_0^{1}$ is the Bessel function of the first kind of order $0$.

In fact, the similarity can be seen by plotting an approximation of the contour: \begin{align} I_{approx}(k) & = 1 - \frac{k}{2}\int_{-99999}^{-0.1} \frac{1}{\sqrt{1-t}} J_1(k\sqrt{1-t}) dt \\ & = 1 + \frac{1}{2} (2 I_0((316.228 i) k) - 2 I_0((1.04881 i) k)) \end{align} Here is the code:

1 - (k/2) Integrate[(1/Sqrt[1 - t]) BesselJ[1, k Sqrt[1 - t]], {t, -99999, -0.1}]

Here are plots for comparison:

Plot of $I_{approx}(k)$ for $k=10$ to $200$

Plot of $H_0^{(1)}(k)$ for $k=10$ to $200$

Clearly the frequency and magnitude of the oscillations are qualitatively very alike. Maybe if I was computing the full integral the plots would match and show (at least emperically) that $I(k) = H_0^{(1)}(k)$. So how can I perform the full contour integral in Mathematica?

I would like to compute the following integral $I(k)$ in Mathematica to check if the result equals something I 'feel' is correct but I have no experience with contour integrals in Mathematica.

I want to compute $$ I(k) = \frac{k}{4\pi}\int_\gamma \log(t) \frac{1}{\sqrt{1-t}} J_1(k\sqrt{1-t}) dt,$$ where $k > 0$, $J_1$ is the Bessel function of the first kind and where $\gamma$ is a contour starting at $1$ and ending at $-\infty$ which does not contain the origin.

I have a feeling this should have some relation to the function $H_0^{(1)}(k)$ which is the is the Hankel function of the first kind of order $0$.

Edit: I noticed I was missing the $\log(t)$ term in the integral after a comment by Carl.