Difference between revisions of "Relationship between Sievert integral and exponential integral E"
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(Created page with "==Theorem== The following formula holds for $x \geq 0$ and $0 < \theta < \dfrac{\pi}{2}$: $$S(x,\theta)=\displaystyle\int_0^{\frac{\pi}{2}} e^{-x\sec(\phi)} \mathrm{d}\phi - \...") |
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$$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),$$ | $$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),$$ | ||
where $S$ denotes the [[Sievert integral]], $e^{*}$ denotes the [[exponential]], $\sec$ denotes [[secant]], $\alpha_k$ is defined by $\alpha_0 := 1$ and for $k=1,2,3,\ldots$, $\alpha_k := \dfrac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2k+1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2k}$, $\cos$ denotes [[cosine]], and $E_{2k+2}$ denotes the [[exponential integral E]]. | where $S$ denotes the [[Sievert integral]], $e^{*}$ denotes the [[exponential]], $\sec$ denotes [[secant]], $\alpha_k$ is defined by $\alpha_0 := 1$ and for $k=1,2,3,\ldots$, $\alpha_k := \dfrac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2k+1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2k}$, $\cos$ denotes [[cosine]], and $E_{2k+2}$ denotes the [[exponential integral E]]. | ||
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| + | ==Proof== | ||
==References== | ==References== | ||
Revision as of 02:06, 21 December 2016
Theorem
The following formula holds for $x \geq 0$ and $0 < \theta < \dfrac{\pi}{2}$: $$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
1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $27.4.2$
