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21: 27.12 Asymptotic Formulas: Primes
27.12.1 lim n p n n ln n = 1 ,
27.12.2 p n > n ln n , n = 1 , 2 , .
where the series terminates when the product of the first r primes exceeds x . …
27.12.5 | π ( x ) - li ( x ) | = O ( x exp ( - c ( ln x ) 1 / 2 ) ) , x .
where λ ( α ) depends only on α , and ϕ ( m ) is the Euler totient function27.2). …
22: 13.2 Definitions and Basic Properties
§13.2(vi) Wronskians
Kummer’s Transformations
23: 11.9 Lommel Functions
§11.9 Lommel Functions
§11.9(ii) Expansions in Series of Bessel Functions
If either of μ ± ν equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents S μ , ν ( z ) exactly. …
24: 13.4 Integral Representations
§13.4(i) Integrals Along the Real Line
§13.4(ii) Contour Integrals
The contour of integration starts and terminates at a point α on the real axis between 0 and 1 . … …
25: 1.10 Functions of a Complex Variable
Analytic Functions
§1.10(vi) Multivalued Functions
(Or more generally, a simple contour that starts at the center and terminates on the boundary.) …
§1.10(vii) Inverse Functions
26: 15.6 Integral Representations
§15.6 Integral Representations
In all cases the integrands are continuous functions of t on the integration paths, except possibly at the endpoints. … In (15.6.1) all functions in the integrand assume their principal values. … In (15.6.5) the integration contour starts and terminates at a point A on the real axis between 0 and 1 . … In each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values. …
27: 4.6 Power Series
§4.6(i) Logarithms
4.6.1 ln ( 1 + z ) = z - 1 2 z 2 + 1 3 z 3 - , | z | 1 , z - 1 ,
4.6.2 ln z = ( z - 1 z ) + 1 2 ( z - 1 z ) 2 + 1 3 ( z - 1 z ) 3 + , z 1 2 ,
4.6.3 ln z = ( z - 1 ) - 1 2 ( z - 1 ) 2 + 1 3 ( z - 1 ) 3 - , | z - 1 | 1 , z 0 ,
If a = 0 , 1 , 2 , , then the series terminates and z is unrestricted. …
28: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … Hence unless the series (2.7.8) terminate (in which case the corresponding Λ j is zero) they diverge. … such that …Here F ( x ) is the error-control function
29: Errata
  • Equation (17.9.3)
    17.9.3 ϕ 1 2 ( a , b c ; q , z ) = ( a b z / c ; q ) ( b z / c ; q ) ϕ 2 3 ( a , c / b , 0 c , c q / ( b z ) ; q , q ) + ( a , b z , c / b ; q ) ( c , z , c / ( b z ) ; q ) ϕ 2 3 ( z , a b z / c , 0 b z , b z q / c ; q , q )

    Originally, the second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ϕ 1 2 is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).

    Reported by Roberto S. Costas-Santos on 2019-04-26

  • Paragraph Confluent Hypergeometric Functions (in §10.16)

    Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

  • Equation (19.25.37)

    The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

  • Equation (7.2.3)

    Originally named as a complementary error function, w ( z ) has been renamed as the Faddeeva (or Faddeyeva) function.

  • The Handbook of Mathematical Functions was published, and the Digital Library of Mathematical Functions was released.
    30: 29.3 Definitions and Basic Properties
    §29.3(i) Eigenvalues
    If ν is a nonnegative integer and 2 p ν , then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with 2 m ν . …
    §29.3(iv) Lamé Functions
    §29.3(v) Normalization
    §29.3(vi) Orthogonality