View source for 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]].

==Proof==

==References==
{{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Relationship between Sievert integral and exponential integral E}}: $27.4.2$

[[Category:Theorem]]
[[Category:Unproven]]