Difference between revisions of "Relationship between Sievert integral and exponential integral E"

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==Theorem==
 
==Theorem==
 
The following formula holds for $x \geq 0$ and $0 < \theta < \dfrac{\pi}{2}$:
 
The following formula holds for $x \geq 0$ and $0 < \theta < \dfrac{\pi}{2}$:
−
$$S(x,\theta)=\displaystyle\int_0^{\frac{\pi}{2}} e^{-x\sec(\phi)} \mathrm{d}\phi - \displaystyle\sum_{k=0}^{\infty} \alpha_k (\cos(\theta))^{2k+1} E_{2k+2} \left( \dfrac{x}{\cos(\theta)} \right),$$
+
$$S(x,\theta)=S\left(x,\dfrac{\pi}{2} \right) - \displaystyle\sum_{k=0}^{\infty} \alpha_k (\cos(\theta))^{2k+1} E_{2k+2} \left( \dfrac{x}{\cos(\theta)} \right),$$
−
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]], $\alpha_k$ 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==
 
==Proof==

Latest revision as of 02:11, 21 December 2016

Theorem

The following formula holds for $x \geq 0$ and $0 < \theta < \dfrac{\pi}{2}$: $$S(x,\theta)=S\left(x,\dfrac{\pi}{2} \right) - \displaystyle\sum_{k=0}^{\infty} \alpha_k (\cos(\theta))^{2k+1} E_{2k+2} \left( \dfrac{x}{\cos(\theta)} \right),$$ where $S$ denotes the Sievert integral, $\alpha_k$ 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): $27.4.2$

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