# 希沃特积分

Sievert Integral

${\displaystyle S(x,\theta )=\int _{0}^{\theta }e^{-xsec(\phi )}d\phi }$

## 关系式

${\displaystyle S(x,\theta )\approx {\sqrt {\frac {\pi }{2x}}}e^{-x}erf({\sqrt {\frac {x}{2}}}\theta )}$

${\displaystyle S(x,\pi /2)=BesselK(1,x)}$

## 级数展开

• ${\displaystyle S(x,{\frac {\pi }{5}})\approx {.628-.674*x+.364*x^{2}-.131*x^{3}+0.357e-1*x^{4}-0.780e-2*x^{5}+0.143e-2*x^{6}-0.225e-3*x^{7}+0.310e-4*x^{8}-0.368e-5*x^{9}+O(x^{1}0)}}$
• ${\displaystyle S(3,\theta )\approx {0.498e-1*\theta -0.249e-1*\theta ^{3}+0.498e-2*\theta ^{5}+0.862e-3*\theta ^{7}-0.104e-3*\theta ^{9}-0.857e-4*\theta ^{1}1-0.222e-4*\theta ^{1}3-0.155e-5*\theta ^{1}5+O(\theta ^{1}7)}}$

## 参考文献

1. ^ R. M. Sievert Die Intensitätsverteilung der primären γ-Strahlung in der Nähe medizinischer Radiumpräparate (нем.) // Acta Radiologica. — 1921. — Т. 1. — № 1. — С. 89-128.
2. ^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 27", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 1001, ISBN 978-0486612720