About the Project

auxiliary functions

AdvancedHelp

(0.003 seconds)

1—10 of 30 matching pages

1: 6.1 Special Notation
β–Ί
2: 7.10 Derivatives
β–Ί
d g ⁑ ( z ) d z = Ο€ ⁒ z ⁒ f ⁑ ( z ) 1 .
3: 7.5 Interrelations
§7.5 Interrelations
β–Ί
7.5.3 C ⁑ ( z ) = 1 2 + f ⁑ ( z ) ⁒ sin ⁑ ( 1 2 ⁒ Ο€ ⁒ z 2 ) g ⁑ ( z ) ⁒ cos ⁑ ( 1 2 ⁒ Ο€ ⁒ z 2 ) ,
β–Ί β–Ί
7.5.10 g ⁑ ( z ) ± i ⁒ f ⁑ ( z ) = 1 2 ⁒ ( 1 ± i ) ⁒ e ΞΆ 2 ⁒ erfc ⁑ ΞΆ .
β–Ί
7.5.11 | β„± ⁑ ( x ) | 2 = f 2 ⁑ ( x ) + g 2 ⁑ ( x ) , x 0 ,
4: 6.4 Analytic Continuation
β–Ί
6.4.6 f ⁑ ( z ⁒ e ± Ο€ ⁒ i ) = Ο€ ⁒ e βˆ“ i ⁒ z f ⁑ ( z ) ,
β–Ί
6.4.7 g ⁑ ( z ⁒ e ± Ο€ ⁒ i ) = βˆ“ Ο€ ⁒ i ⁒ e βˆ“ i ⁒ z + g ⁑ ( z ) .
β–Ί
5: 7.4 Symmetry
β–Ί
g ⁑ ( z ) = 2 ⁒ sin ⁑ ( 1 4 ⁒ Ο€ + 1 2 ⁒ Ο€ ⁒ z 2 ) g ⁑ ( z ) .
6: 7.2 Definitions
β–Ί
§7.2(iv) Auxiliary Functions
β–Ί
7.2.10 f ⁑ ( z ) = ( 1 2 S ⁑ ( z ) ) ⁒ cos ⁑ ( 1 2 ⁒ Ο€ ⁒ z 2 ) ( 1 2 C ⁑ ( z ) ) ⁒ sin ⁑ ( 1 2 ⁒ Ο€ ⁒ z 2 ) ,
β–Ί
7.2.11 g ⁑ ( z ) = ( 1 2 C ⁑ ( z ) ) ⁒ cos ⁑ ( 1 2 ⁒ Ο€ ⁒ z 2 ) + ( 1 2 S ⁑ ( z ) ) ⁒ sin ⁑ ( 1 2 ⁒ Ο€ ⁒ z 2 ) .
7: 6.11 Relations to Other Functions
8: 7.24 Approximations
β–Ί
§7.24(i) Approximations in Terms of Elementary Functions
β–Ί
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf ⁑ x , erfc ⁑ x and the auxiliary functions f ⁑ ( x ) and g ⁑ ( x ) .

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

  • 9: 7.22 Methods of Computation
    β–Ί
    §7.22(i) Main Functions
    10: 6.20 Approximations
    β–Ί
  • 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).

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

  • β–Ί
  • 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 | < Ο€ the scheme can be used in backward direction.