I am frustrated by the asymptotic expansion of the free surface green function when epsilon (or H for I2)tends to zero and with a singularity K(K is a constant). Can anyone help me derive the formula posted in the picture, or give me some suggestions? Thanks very much!

$$I1 = \int_{0}^\infty \frac{e^{-\varepsilon x}} { x-K } dx$$

another integral is (similar to the proceeding one, part of finite depth green function, regular part has omitted),

$$I2 = P.V.\int_{0}^\infty k^{2n}Cosh{(kV)}e^{-kH}(\frac{1} {kSinh(kH)-Cosh(kH) }) dk$$

where, $$k^{2n}$$ comes from the series of the Bessel function of the first kind, H is the water depth, V is a coordinate refer to vertical position. The integral is divergent when H tends to zero. How to derive the asymptotic expansion when H tends to zero.