Relationship between Sievert integral and exponential integral E

Theorem

The following formula holds for $x \geq 0$ and $0 < \theta < \dfrac{\pi}{2}$: $$S(x,\theta)=\displaystyle\int_0^{\frac{\pi}{2}} e^{-x\sec(\phi)} \mathrm{d}\phi - \displaystyle\sum_{k=0}^{\infty} \alpha_k (\cos(\theta))^{2k+1} E_{2k+2} \left( \dfrac{x}{\cos(\theta)} \right),$$ where $S$ denotes the Sievert integral, $e^{*}$ denotes the exponential, $\sec$ denotes secant, $\alpha_k$ is defined by $\alpha_0 := 1$ and for $k=1,2,3,\ldots$, $\alpha_k := \dfrac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2k+1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2k}$, $\cos$ denotes cosine, and $E_{2k+2}$ denotes the exponential integral E.

References

1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $27.4.2$