About the Project
NIST

auxiliary functions

AdvancedHelp

(0.001 seconds)

1—10 of 30 matching pages

1: 6.1 Special Notation
2: 7.10 Derivatives
d g ( z ) d z = π z f ( z ) - 1 .
3: 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 ,
4: 6.4 Analytic Continuation
6.4.6 f ( z e ± π i ) = π e i z - f ( z ) ,
6.4.7 g ( z e ± π i ) = π i e i z + g ( z ) .
5: 7.4 Symmetry
g ( - z ) = 2 sin ( 1 4 π + 1 2 π z 2 ) - g ( z ) .
6: 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 ) .
7: 6.11 Relations to Other Functions
8: 7.24 Approximations
§7.24(i) Approximations in Terms of Elementary Functions
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf x , erfc x and the auxiliary functions f ( x ) and g ( x ) .

  • Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions f ( x ) and g ( x ) for x 3 (15D).

  • 9: 7.22 Methods of Computation
    §7.22(i) Main Functions
    10: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.