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I am reading this paper by Kunz and Shapiro: they state that the integral (Eqs. 3.17-3.19) $$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\frac{1}{4\pi p}\frac{1}{\left[y+i\left(\frac{\lambda^2sq}{2}-\delta\right)\right]^2}$$ can be written as (Eq. 3.22) $$-\frac{b^2}{4\pi p}\sqrt{\frac{a}{b}}\int_0^\infty dt\frac{t}{\sqrt{t+1}}I_1(2\sqrt{ab(t+1)})\exp\left(-t\frac{b^2\lambda^2}{4}\right)$$ using the fact that $\lambda^2sq/2-\delta>0$. $I_1(x)$ is a modified Bessel function of the first kind.
I cannot see how to rewrite the integral in this way. I believe the integration contour can be deformed to an infinitesimal loop about the origin of the complex plane, but I cannot see where to go from there. Any help would be greatly appreciated.

I am reading this paper by Kunz and Shapiro: they state that the integral (Eqs. 3.17-3.19) $$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\frac{1}{4\pi p}\frac{1}{\left[y+i\left(\frac{\lambda^2sq}{2}-\delta\right)\right]^2}$$ can be written as (Eq. 3.22) $$-\frac{b^2}{4\pi p}\sqrt{\frac{a}{b}}\int_0^\infty dt\frac{t}{\sqrt{t+1}}I_1(2\sqrt{ab(t+1)})\exp\left(-t\frac{b^2\lambda^2}{4}\right)$$ using the fact that $\lambda^2sq/2-\delta>0$. $I_1(x)$ is a modified Bessel function of the first kind.
I cannot see how to rewrite the integral in this way. I believe the integration contour can be deformed to an infinitesimal loop about the origin of the complex plane, but I cannot see where to go from there. Any help would be greatly appreciated.

Edit: $a$ and $b$ are positive.

I am reading this paper by Kunz and Shapiro: https://arxiv.org/pdf/cond-mat/9802263.pdf. They state that the integral (Eqs. 3.17-3.19) $$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\frac{1}{4\pi p}\frac{1}{\left[y+i\left(\frac{\lambda^2sq}{2}-\delta\right)\right]^2}$$ can be written as (Eq. 3.22) $$-\frac{b^2}{4\pi p}\sqrt{\frac{a}{b}}\int_0^\infty dt\frac{t}{\sqrt{t+1}}I_1(2\sqrt{ab(t+1)})\exp\left(-t\frac{b^2\lambda^2}{4}\right)$$ using the fact that $\lambda^2sq/2-\delta>0$. $I_1(x)$ is a modified Bessel function of the first kind. I cannot see how to rewrite the integral in this way. I believe the integration contour can be deformed to an infinitesimal loop about the origin of the complex plane, but I cannot see where to go from there. Any help would be greatly appreciated.

I am reading this paper by Kunz and Shapiro: they state that the integral (Eqs. 3.17-3.19) $$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\frac{1}{4\pi p}\frac{1}{\left[y+i\left(\frac{\lambda^2sq}{2}-\delta\right)\right]^2}$$ can be written as (Eq. 3.22) $$-\frac{b^2}{4\pi p}\sqrt{\frac{a}{b}}\int_0^\infty dt\frac{t}{\sqrt{t+1}}I_1(2\sqrt{ab(t+1)})\exp\left(-t\frac{b^2\lambda^2}{4}\right)$$ using the fact that $\lambda^2sq/2-\delta>0$. $I_1(x)$ is a modified Bessel function of the first kind.
I cannot see how to rewrite the integral in this way. I believe the integration contour can be deformed to an infinitesimal loop about the origin of the complex plane, but I cannot see where to go from there. Any help would be greatly appreciated.