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Fresnel integrals

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1: 7.2 Definitions
§7.2(iii) Fresnel Integrals
( z ) , C ( z ) , and S ( z ) are entire functions of z , as are f ( z ) and g ( z ) in the next subsection.
Values at Infinity
lim x C ( x ) = 1 2 ,
§7.2(iv) Auxiliary Functions
2: Sidebar 7.SB1: Diffraction from a Straightedge
The intensity distribution follows | ( x ) | 2 , where is the Fresnel integral (See 7.3.4). Fresnel integrals have many applications in optics. …
3: 7.5 Interrelations
§7.5 Interrelations
7.5.2 C ( z ) + i S ( z ) = 1 2 ( 1 + i ) - ( z ) .
7.5.6 e ± 1 2 π i z 2 ( g ( z ) ± i f ( z ) ) = 1 2 ( 1 ± i ) - ( C ( z ) ± i S ( z ) ) .
7.5.8 C ( z ) ± i S ( z ) = 1 2 ( 1 ± i ) erf ζ .
4: 7.4 Symmetry
§7.4 Symmetry
C ( - z ) = - C ( z ) ,
S ( - z ) = - S ( z ) ,
C ( i z ) = i C ( z ) ,
S ( i z ) = - i S ( z ) .
5: 7.3 Graphics
See accompanying text
Figure 7.3.3: Fresnel integrals C ( x ) and S ( x ) , 0 x 4 . Magnify
See accompanying text
Figure 7.3.4: | ( x ) | 2 , - 8 x 8 . Fresnel (1818) introduced the integral ( x ) in his study of the interference pattern at the edge of a shadow. He observed that the intensity distribution is given by | ( x ) | 2 . Magnify
6: 7.25 Software
§7.25(iv) C ( x ) , S ( x ) , f ( x ) , g ( x ) , x
§7.25(v) C ( z ) , S ( z ) , z
§7.25(vi) ( x ) , G ( x ) , U ( x , t ) , V ( x , t ) , x
§7.25(vii) ( z ) , G ( z ) , z
7: 7.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the error function erf z ; the complementary error functions erfc z and w ( z ) ; Dawson’s integral F ( z ) ; the Fresnel integrals ( z ) , C ( z ) , and S ( z ) ; the Goodwin–Staton integral G ( z ) ; the repeated integrals of the complementary error function i n erfc ( z ) ; the Voigt functions U ( x , t ) and V ( x , t ) . Alternative notations are Q ( z ) = 1 2 erfc ( z / 2 ) , P ( z ) = Φ ( z ) = 1 2 erfc ( - z / 2 ) , Erf z = 1 2 π erf z , Erfi z = e z 2 F ( z ) , C 1 ( z ) = C ( 2 / π z ) , S 1 ( z ) = S ( 2 / π z ) , C 2 ( z ) = C ( 2 z / π ) , S 2 ( z ) = S ( 2 z / π ) . …
8: 7.20 Mathematical Applications
§7.20(ii) Cornu’s Spiral
Let the set { x ( t ) , y ( t ) , t } be defined by x ( t ) = C ( t ) , y ( t ) = S ( t ) , t 0 . …
See accompanying text
Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by x = C ( t ) , y = S ( t ) , t [ 0 , ) . Magnify
7.20.1 1 σ 2 π - x e - ( t - m ) 2 / ( 2 σ 2 ) d t = 1 2 erfc ( m - x σ 2 ) = Q ( m - x σ ) = P ( x - m σ ) .
9: 7.13 Zeros
§7.13(iii) Zeros of the Fresnel Integrals
Similarly for S ( z ) . …
Table 7.13.3: Complex zeros x n + i y n of C ( z ) .
n x n y n
Table 7.13.4: Complex zeros x n + i y n of S ( z ) .
n x n y n
§7.13(iv) Zeros of ( z )
10: 7.6 Series Expansions
§7.6(i) Power Series
7.6.4 C ( z ) = n = 0 ( - 1 ) n ( 1 2 π ) 2 n ( 2 n ) ! ( 4 n + 1 ) z 4 n + 1 ,
7.6.6 S ( z ) = n = 0 ( - 1 ) n ( 1 2 π ) 2 n + 1 ( 2 n + 1 ) ! ( 4 n + 3 ) z 4 n + 3 ,
7.6.10 C ( z ) = z n = 0 j 2 n ( 1 2 π z 2 ) ,
7.6.11 S ( z ) = z n = 0 j 2 n + 1 ( 1 2 π z 2 ) .