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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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integral with simple approximation. But why?

I have the following integral $$g(x_0) = \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^{3/4}}\frac{1}{(1+(x+x_0)^2)^{3/4}}\exp\left(-\frac{2\pi i}{\lambda}\left[\sqrt{1+x^2}-\sqrt{1+(x+x_0)^2} \right] \ri …