Pole Contribution in Method of Steepest Descent

While studying the method of steepest descent, I've come across this exercise

The problem reads: Evaluate the dominant term of the asymptotic expansion of the following integral

$$I_N(q)=\int_{-\infty}^{+\infty} \frac{1}{z-i}\,e^{-N(z^2-2iqz)}\,dz\qquad q\in\mathbb{R}$$ for $$N\rightarrow \infty$$

Now, the integrand has a saddle point in $$z_0=iq$$ and also has a pole for $$z_1=i$$ For $$q<1$$, I have no problem finding the dominant term by simply applying the method of steepest descent.

For $$q>1$$ however, I know that I have to take into account the pole of the integrand but I don't know how to do that.

Also, most books on the subject that I've read through only go over the "simple" case in which the integrand has no singularities, so I would appreciate if you have any suggestions (books, articles, etc) on the subject.

• I'm rusty on this subject, can you specify the path that you are using when $q<1$? This should help illuminate the issues that arise when $q>1$.
– Ian
Sep 4 '20 at 16:26

When $$q>1$$, write \begin{align*} & \int_{ - \infty }^{ + \infty } {\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} dz} \\ & \!= \int_{ - \infty }^{ + \infty } {\!\!\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} dz} - \int_{iq - \infty }^{iq + \infty } {\!\!\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} dz} + \int_{iq - \infty }^{iq + \infty } {\!\!\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} dz} \\ & \!= 2\pi i\mathop {{\mathop{\rm Res}\nolimits} }\limits_{z = i} \left[ {\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} } \right] + \int_{iq - \infty }^{iq + \infty } {\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} dz} \\ & \!= 2\pi ie^{N(1 - 2q)} + \int_{iq - \infty }^{iq + \infty } {\frac{1}{{z - i}}e^{ - N(z^2 - 2iqz)} dz} . \end{align*} Now apply the method of steepest descents to the final integral.