Euler’s gamma functions and exponential integrals

Question

I am trying to solve these integrals,
where $ (C_i,S_i) ≡ (\cos λp_i, \sin λp_i)$ I have done $D_{SS}$. But in $D_{SC}$, I encounter some divergences. The author says this in the paper:

The integration of λ in Eqs. (C15)-(C18) seems straightforward as one can express the trigonometric functions to exponential functions and then convert the integrals to Euler’s gamma functions. It is worth mentioning, however, that one should handle the branch cut singularities in the gamma functions carefully

Could any of you point me to some resources regarding solving these types of integrals? Where can I read up on exponential integrals and gamma functions?
Apologies if this is not the right place to ask this question.

Answer 1

If you intend to follow the process described in your quotation, you will want

• the exponential forms of sine and cosine (alternatively, (3) here and (4) here, or alt. ref.)
• partial fraction decomposition (probably, although one might sneak around this requirement), and
• the gamma function, et seq. (alt. ref., alt. ref.)

For instance, using the exponential forms of sine and cosine in the integrand of $D_{SC}$, we obtain $$ D_{SC} = \frac{4}{\pi} \int_0^{\infty} \frac{-1}{16 \lambda^2} \left( \mathrm{e}^{-\mathrm{i}p_1 \lambda} - \mathrm{e}^{\mathrm{i}p_1 \lambda} \right) \left( \mathrm{e}^{-\mathrm{i}p_2 \lambda} - \mathrm{e}^{\mathrm{i}p_2 \lambda} \right) \left( \left( \mathrm{e}^{-\mathrm{i}p_3 \lambda} + \mathrm{e}^{\mathrm{i}p_3 \lambda} \right) - \frac{\mathrm{i}\left( \mathrm{e}^{-\mathrm{i}p_3 \lambda} - \mathrm{e}^{\mathrm{i}p_3 \lambda} \right)}{p_3 \lambda} \right) \left( \left( \mathrm{e}^{-\mathrm{i}p_4 \lambda} + \mathrm{e}^{\mathrm{i}p_4 \lambda} \right) - \frac{\mathrm{i}\left( \mathrm{e}^{-\mathrm{i}p_4 \lambda} - \mathrm{e}^{\mathrm{i}p_4 \lambda} \right)}{p_4 \lambda} \right) \,\mathrm{d}\lambda $$

The remainder of this process is a herculean effort. Are you sure you wouldn't be happier evaluating these integrals using the Cauchy integral formula?