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Table 23.2 Power series of the incremental orientation parameter Δα
Δα = κ 0 Δs + k 0 Δs 2
2
Δα 2 = κ 0 Δs 2 + κ 0 κ 0 Δs 3 + κ 0 Δs 4
4
Δα 3 = κ 0 Δs 3 + 3 κ 0 κ 0 Δs 4
+ 3 κκ 0 Δs 5
4
+ κ 0 Δs 6
8
2
Δα 4 = κ 0 Δs 4 + 3 κ 0 κ 0 Δs 5
+ 3 κ 0 κ 0 Δs 6
2
+ κ 0 Δs 8
16
2
Δα 5 = κ 0 Δs 5 + 5 κ 0 κ 0 Δs 6
+ 5 κ 0 κ 0 Δs 7
2
+ 5 κ 0 κ 0 Δs 8
4
+ 5 κ 0 κ 0 Δs 9
16
+ 5 κ 0 Δs 10
32
2
Table 23.3 Powerseriesexpressionsof Fresnel integrals
x
−
x 0 =
s
s 0
cos α 0 − 1!
− 2!
cos α 0 Δα 2 + 3!
sin α 0 Δα 3 +
sin α 0 Δα
sin α 0 Δα 5 + O c Δα 6 d ( s
1
4!
cos α 0 Δα 4 − 5!
−
s 0 )
y
−
y 0 =
s
s 0
sin α 0 + 1!
− 2!
sin α 0 Δα 2 − 3!
sin α 0 Δα 3 +
cos α 0 Δα
cos α 0 Δα 5 + O s Δα 6 d ( s
1
4!
sin α 0 Δα 4 + 5!
−
s 0 )
1 st integrals
s
s
cos α 0 ds =cos α 0 ( s
−
s 0 ) ,
sin α 0 ds =sin α 0 ( s
−
s 0 )
s 0
s 0
2 nd integrals
κ 0 s−s 0
0
Δs 2 dΔs/ 2
sin α 0 Δαds
s 0
Δ sd Δs + κ 0 s−s 0
=sin α 0
0
κ 0 s−s 0
0
Δs 2 dΔs/ 2
s
s 0
Δ sd Δs + κ 0 s−s 0
cos α 0 Δα ds =cos α 0
0
s
s 0
s 0 ) 2 / 2
s 0 ) 3 / 6
− 1!
κ 0
sin α 0 Δα ds =
−
κ 0 sin( s
−
−
sin α 0 ( s
−
s
s 0
s 0 ) 2 / 2+ κ 0
s 0 ) 3 / 6
+ 1!
cos α 0 Δα ds =
−
κ 0 cos( s
−
cos α 0 ( s
−
3 nd integrals
κ 0 s−s 0
Δs 4 dΔs/ 4
s
s 0
Δs 2 dΔs + κ 0 κ 0 s−s 0
Δs 3 dΔs + κ 0 s−s 0
cos α 0 Δα 2 ds =cos α 0
0
0
0
κ 0 s−s 0
Δs 4 dΔs/ 4
s
s 0
Δs 2 dΔs + κ 0 κ 0 s−s 0
Δs 3 dΔs + κ 0 s−s 0
sin α 0 Δα 2 ds =sin α 0
0
0
0
s
s 0
s 0 ) 3 / 6
s 0 ) 4 / 8
− 2!
cos α 0 Δα 2 ds =
κ 0
κ 0 κ 0
−
cos α 0 ( s
−
−
cos α 0 ( s
−
s
s 0
s 0 ) 3 / 6
s 0 ) 4 / 8
− 2!
sin α 0 Δα 2 ds =
κ 0
κ 0 κ 0
−
sin α 0 ( s
−
−
sin α 0 ( s
−
4 th integrals
s
s 0
sin α 0 Δα 3 ds
=sin α 0
κ 0 s−s 0
κ 0 s−s 0
Δs 3 dΔs +3 κ 0
Δs 4 dΔs/ 2
0
0
Δs 6 dΔs/ 8
+3 κ 0 κ 0 s−s 0
Δs 5 dΔs/ 4+ κ 0 s−s 0
0
0
s
s 0
cos α 0 Δα 3 ds
=cos α 0
κ 0 s−s 0
0
Δs 3 dΔs +
Δs 6 dΔs/ 8
κ 0 s−s 0
Δs 4 dΔs/ 2+3 κ 0 κ 0 s−s 0
Δs 5 dΔs/ 4+ κ 0 s−s 0
3 κ 0
0
0
0
s
s 0
+ 3!
sin α 0 Δα 3 ds
= κ 0
s 0 ) 4 / 24 + κ 0
κ 0
sin α 0
( s − s 0 ) 5 / 4+ κ 0 κ 0 sin α 0 ( s − s 0 ) 6 / 48 + κ 0 cos α 0 ( s − s 0 ) 7 / 336
sin α 0 ( s
−
+ 3! s
s 0 ) 4 / 24
cos α 0 Δα 3 ds =
−
κ 0 cos α 0 ( s
−
−
s 0
s 0 ) 5 / 4
s 0 ) 6 / 48
κ 0 cos α 0 ( s
s 0 ) 7 / 336
κ 0 κ 0 cos α 0 ( s
−
−
κ 0 κ
0 cos α 0 ( s
−
−
−
(continued)
 
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