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1: 33.25 Approximations
§33.25 Approximations
Cody and Hillstrom (1970) provides rational approximations of the phase shift σ 0 ( η ) = ph Γ ( 1 + i η ) (see (33.2.10)) for the ranges 0 η 2 , 2 η 4 , and 4 η . …
2: Possible Errors in DLMF
One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …
3: 6.15 Sums
6.15.2 n = 1 si ( π n ) n = 1 2 π ( ln π - 1 ) ,
6.15.4 n = 1 ( - 1 ) n si ( 2 π n ) n = π ( 3 2 ln 2 - 1 ) .
4: 33.13 Complex Variable and Parameters
The quantities C ( η ) , σ ( η ) , and R , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
5: 6.14 Integrals
6.14.3 0 e - a t si ( t ) d t = - 1 a arctan a , a > 0 .
6.14.5 0 cos t Ci ( t ) d t = 0 sin t si ( t ) d t = - 1 4 π ,
6.14.6 0 Ci 2 ( t ) d t = 0 si 2 ( t ) d t = 1 2 π ,
6.14.7 0 Ci ( t ) si ( t ) d t = ln 2 .
6: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
Shifted Chebyshev of first kind T n * ( x ) ( 0 , 1 ) ( x - x 2 ) - 1 2 { 1 2 π , n > 0 π , n = 0 { 2 2 n - 1 , n > 0 1 , n = 0 - 1 2 n
Shifted Chebyshev of second kind U n * ( x ) ( 0 , 1 ) ( x - x 2 ) 1 2 1 8 π 2 2 n - 1 2 n
Shifted Legendre P n * ( x ) ( 0 , 1 ) 1 1 / ( 2 n + 1 ) 2 2 n ( 1 2 ) n / n ! - 1 2 n
7: 35.4 Partitions and Zonal Polynomials
Also, | κ | denotes k 1 + + k m , the weight of κ ; ( κ ) denotes the number of nonzero k j ; a + κ denotes the vector ( a + k 1 , , a + k m ) . The partitional shifted factorial is given by
35.4.1 [ a ] κ = Γ m ( a + κ ) Γ m ( a ) = j = 1 m ( a - 1 2 ( j - 1 ) ) k j ,
35.4.2 Z κ ( I ) = | κ | !  2 2 | κ | [ m / 2 ] κ 1 j < l ( κ ) ( 2 k j - 2 k l - j + l ) j = 1 ( κ ) ( 2 k j + ( κ ) - j ) !
8: 18.1 Notation
Classical OP’s
  • Shifted Chebyshev of first and second kinds: T n * ( x ) , U n * ( x ) .

  • Shifted Legendre: P n * ( x ) .

  • Nor do we consider the shifted Jacobi polynomials:
    18.1.2 G n ( p , q , x ) = n ! ( n + p ) n P n ( p - q , q - 1 ) ( 2 x - 1 ) ,
    9: 6.1 Special Notation
    The main functions treated in this chapter are the exponential integrals Ei ( x ) , E 1 ( z ) , and Ein ( z ) ; the logarithmic integral li ( x ) ; the sine integrals Si ( z ) and si ( z ) ; the cosine integrals Ci ( z ) and Cin ( z ) . …
    10: 5.2 Definitions
    5.2.5 ( a ) n = Γ ( a + n ) / Γ ( a ) , a 0 , - 1 , - 2 , .
    5.2.6 ( - a ) n = ( - 1 ) n ( a - n + 1 ) n ,
    5.2.7 ( - m ) n = { ( - 1 ) n m ! ( m - n ) ! , 0 n m , 0 , n > m ,