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auxiliary functions for Fresnel integrals

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1: 7.5 Interrelations
§7.5 Interrelations
7.5.3 C ( z ) = 1 2 + f ( z ) sin ( 1 2 π z 2 ) - g ( z ) cos ( 1 2 π z 2 ) ,
7.5.10 g ( z ) ± i f ( z ) = 1 2 ( 1 ± i ) e ζ 2 erfc ζ .
7.5.11 | ( x ) | 2 = f 2 ( x ) + g 2 ( x ) , x 0 ,
2: 7.4 Symmetry
g ( - z ) = 2 sin ( 1 4 π + 1 2 π z 2 ) - g ( z ) .
3: 7.22 Methods of Computation
§7.22(i) Main Functions
4: 7.10 Derivatives
d g ( z ) d z = π z f ( z ) - 1 .
5: 7.2 Definitions
§7.2(iv) Auxiliary Functions
7.2.10 f ( z ) = ( 1 2 - S ( z ) ) cos ( 1 2 π z 2 ) - ( 1 2 - C ( z ) ) sin ( 1 2 π z 2 ) ,
7.2.11 g ( z ) = ( 1 2 - C ( z ) ) cos ( 1 2 π z 2 ) + ( 1 2 - S ( z ) ) sin ( 1 2 π z 2 ) .
6: 7.7 Integral Representations
§7.7(ii) Auxiliary Functions
7.7.10 f ( z ) = 1 π 2 0 e - π z 2 t / 2 t ( t 2 + 1 ) d t , | ph z | 1 4 π ,
7.7.11 g ( z ) = 1 π 2 0 t e - π z 2 t / 2 t 2 + 1 d t , | ph z | 1 4 π ,
Mellin–Barnes Integrals
7: 7.24 Approximations
§7.24(i) Approximations in Terms of Elementary Functions
8: 7.12 Asymptotic Expansions
§7.12(ii) Fresnel Integrals
7.12.2 f ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m ,
7.12.3 g ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m + 1 ( π z 2 / 2 ) 2 m + 1 ,
7.12.4 f ( z ) = 1 π z m = 0 n - 1 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m + R n ( f ) ( z ) ,
7.12.5 g ( z ) = 1 π z m = 0 n - 1 ( - 1 ) m ( 1 2 ) 2 m + 1 ( π z 2 / 2 ) 2 m + 1 , + R n ( g ) ( z ) ,
9: 7.14 Integrals
7.14.5 0 e - a t C ( t ) d t = 1 a f ( a π ) , a > 0 ,
7.14.6 0 e - a t S ( t ) d t = 1 a g ( a π ) , a > 0 ,
10: Bibliography F
  • V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function w ( z ) = e - z 2 ( 1 + 2 i π - 1 / 2 0 z e t 2 d t ) for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.
  • H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis and J. C. Caslin (1973) Table errata; Complex zeros of Fresnel integrals. Math. Comp. 27 (121), pp. 219.
  • A. Fresnel (1818) Mémoire sur la diffraction de la lumière. Mém. de l’Académie des Sciences, pp. 247–382.
  • Y. Fukui and T. Horiguchi (1992) Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.