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Mill ratio for complementary error function

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1: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
§7.18(i) Definition
§7.18(iii) Properties
Hermite Polynomials
2: 7.2 Definitions
§7.2(i) Error Functions
erf z , erfc z , and w ( z ) are entire functions of z , as is F ( z ) in the next subsection.
Values at Infinity
( z ) , C ( z ) , and S ( z ) are entire functions of z , as are f ( z ) and g ( z ) in the next subsection. …
§7.2(iv) Auxiliary Functions
3: 23.15 Definitions
§23.15 Definitions
In §§23.1523.19, k and k ( ) denote the Jacobi modulus and complementary modulus, respectively, and q = e i π τ ( τ > 0 ) denotes the nome; compare §§20.1 and 22.1. …
Elliptic Modular Function
Dedekind’s Eta Function (or Dedekind Modular Function)
4: 9.12 Scorer Functions
§9.12 Scorer Functions
where … - Gi ( x ) is a numerically satisfactory companion to the complementary functions Ai ( x ) and Bi ( x ) on the interval 0 x < . …
Functions and Derivatives
5: 5.12 Beta Function
§5.12 Beta Function
Euler’s Beta Integral
5.12.2 0 π / 2 sin 2 a - 1 θ cos 2 b - 1 θ d θ = 1 2 B ( a , b ) .
5.12.5 0 π / 2 ( cos t ) a - 1 cos ( b t ) d t = π 2 a 1 a B ( 1 2 ( a + b + 1 ) , 1 2 ( a - b + 1 ) ) , a > 0 .
Pochhammer’s Integral
6: 11.9 Lommel Functions
§11.9 Lommel Functions
§11.9(ii) Expansions in Series of Bessel Functions
For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b). …
7: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
§20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
§20.2(iii) Translation of the Argument by Half-Periods
§20.2(iv) z -Zeros
8: 14.20 Conical (or Mehler) Functions
§14.20 Conical (or Mehler) Functions
§14.20(ii) Graphics
For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … For extensions to complex arguments (including the range 1 < x < ), asymptotic expansions, and explicit error bounds, see Dunster (1991). …
9: 5.15 Polygamma Functions
§5.15 Polygamma Functions
The functions ψ ( n ) ( z ) , n = 1 , 2 , , are called the polygamma functions. In particular, ψ ( z ) is the trigamma function; ψ ′′ , ψ ( 3 ) , ψ ( 4 ) are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … For B 2 k see §24.2(i). …
10: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
§14.19(ii) Hypergeometric Representations
§14.19(iv) Sums
§14.19(v) Whipple’s Formula for Toroidal Functions