西弗積分 (Sievert Integral)是用於模擬放射性 劑量的一種特殊函數 ，為瑞典 醫學物理師羅爾夫·馬克西米連·西弗 （瑞典語 ：Rolf Maximilian Sievert ）所推得，輻射轉移 定義如下：[1] [2]
Sievert Integral
S
(
x
,
θ
)
=
∫
0
θ
e
−
x
s
e
c
(
ϕ
)
d
ϕ
{\displaystyle S(x,\theta )=\int _{0}^{\theta }e^{-xsec(\phi )}d\phi }
關係式
S
(
x
,
θ
)
≈
π
2
x
e
−
x
e
r
f
(
x
2
θ
)
{\displaystyle S(x,\theta )\approx {\sqrt {\frac {\pi }{2x}}}e^{-x}erf({\sqrt {\frac {x}{2}}}\theta )}
S
(
x
,
π
/
2
)
=
B
e
s
s
e
l
K
(
1
,
x
)
{\displaystyle S(x,\pi /2)=BesselK(1,x)}
級數展開
S
(
x
,
π
5
)
≈
.628
−
.674
∗
x
+
.364
∗
x
2
−
.131
∗
x
3
+
0.357
e
−
1
∗
x
4
−
0.780
e
−
2
∗
x
5
+
0.143
e
−
2
∗
x
6
−
0.225
e
−
3
∗
x
7
+
0.310
e
−
4
∗
x
8
−
0.368
e
−
5
∗
x
9
+
O
(
x
1
0
)
{\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)}}
S
(
3
,
θ
)
≈
0.498
e
−
1
∗
θ
−
0.249
e
−
1
∗
θ
3
+
0.498
e
−
2
∗
θ
5
+
0.862
e
−
3
∗
θ
7
−
0.104
e
−
3
∗
θ
9
−
0.857
e
−
4
∗
θ
1
1
−
0.222
e
−
4
∗
θ
1
3
−
0.155
e
−
5
∗
θ
1
5
+
O
(
θ
1
7
)
{\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)}}
參考文獻
^ R. M. Sievert Die Intensitätsverteilung der primären γ-Strahlung in der Nähe medizinischer Radiumpräparate (нем.) // Acta Radiologica. — 1921. — Т. 1. — № 1. — С. 89-128.
^ 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
