Following (Le Ru, Etchegoin, p. 571), and (Novotny, Hecht pp. 335–360), the enhancement factor for the total decay rate for a dipole perpendicular to the interface is

$$M_{\text{tot}}^{\perp }=1+\frac{3}{2}{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\{\frac{q^3}{\sqrt{1-q^2}}{r}^p(q)\exp \left(2i{k}_{1}d\sqrt{1-q^2}\right)\}\text{d}q$$

The integrand diverges as q → 1, it is therefore advantageous to perform the substitution $u:=\sqrt{1 - q^2}$. In order to maintain a real path of integration, the integral is first split into a radiative region (0 ≤ q ≤ 1, $u:=\sqrt{1 - q^2}\geq 0$), and an evanescent region (1 ≤ q ≤ ∞, $-i u:=\sqrt{q^2 - 1}\geq 0$). After some algebraic manipulation, we obtain, $$M_{\text{tot}}^{\perp }=1+\frac{3}{2}(I_1+I_2)$$ where $$\begin{array}{rl}{I}_1+I_2=& {\int }_0^1[1-u^2]\cdot \mathrm{\Re }\{r^p(\sqrt{1-u^2})\exp \left(2id{k}_{1}u\right)\}\text{d}u\\ & +{\int }_0^{\mathrm{\infty }}[1+u^2]\cdot \exp (-2d{k}_{1}u)\cdot \mathrm{\Im }\{r^p(\sqrt{1+u^2})\}\text{d}u\end{array}$$ Similarly, for the parallel dipole $$M_{\text{tot}}^{\parallel }=1+\frac{3}{4}{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\{[\frac{r^s(q)}{\sqrt{1-q^2}}-r^p(q)\sqrt{1-q^2}]\cdot q\cdot \exp \left(2i{k}_{1}d\sqrt{1-q^2}\right)\}\text{d}q$$ which can be rewritten as, $$M_{\text{tot}}^{\parallel }=1+\frac{3}{4}(I_1^{\parallel }+I_2^{\parallel })$$ where $$\begin{array}{rl}{I}_1^{\parallel }+I_2^{\parallel }=& {\int }_0^1\mathrm{\Re }\{[r^s(\sqrt{1-u^2})-u^2\cdot r^p(\sqrt{1-u^2})]\exp \left(2id{k}_{1}u\right)\}\text{d}u\\ & +{\int }_0^{\mathrm{\infty }}\exp (-2d{k}_{1}u)\cdot \mathrm{\Im }\{r^s(\sqrt{1+u^2})+u^2\cdot r^p(\sqrt{1+u^2})\}\text{d}u\end{array}$$