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31: Errata
We now include Markov’s Theorem. …
  • Chapter 5 Addition

    Equation (5.2.9).

  • Chapter 10 Additions

    Equations (10.22.78), (10.22.79).

  • Additions

    Equation (16.16.5_5).

  • Additions

    Equation (4.13.5_3) (suggested by Warren Smith on 2023-08-10).

  • 32: 1.10 Functions of a Complex Variable
    Picard’s Theorem
    §1.10(iv) Residue Theorem
    In addition, …
    Rouché’s Theorem
    Lagrange Inversion Theorem
    33: 10.22 Integrals
    Hankel’s inversion theorem is given by
    10.22.77 f ( y ) = 0 g ( x ) J ν ( x y ) ( x y ) 1 2 d x .
    Sufficient conditions for the validity of (10.22.77) are that 0 | f ( x ) | d x < when ν 1 2 , or that 0 | f ( x ) | d x < and 0 1 x ν + 1 2 | f ( x ) | d x < when 1 < ν < 1 2 ; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62). …
    10.22.78 f ( x ) = 0 ( x t ) 1 2 J ν ( x t ) Y ν ( a t ) Y ν ( x t ) J ν ( a t ) J ν 2 ( a t ) + Y ν 2 ( a t ) a ( y t ) 1 2 ( J ν ( y t ) Y ν ( a t ) Y ν ( y t ) J ν ( a t ) ) f ( y ) d y d t , a > 0 .
    10.22.79 f ( x ) = 0 ( x t ) 1 2 c J ν ( x t ) + t 2 ν J ν ( x t ) c 2 + 2 c cos ( ν π ) t 2 ν + t 4 ν 0 ( y t ) 1 2 ( c J ν ( y t ) + t 2 ν J ν ( y t ) ) f ( y ) d y d t , 0 < ν < 1 , c > 0 .
    34: 1.12 Continued Fractions
    Pringsheim’s Theorem
    Van Vleck’s Theorem
    The continued fraction converges iff, in addition, …
    35: 2.1 Definitions and Elementary Properties
    For example, if f ( z ) is analytic for all sufficiently large | z | in a sector 𝐒 and f ( z ) = O ( z ν ) as z in 𝐒 , ν being real, then f ( z ) = O ( z ν 1 ) as z in any closed sector properly interior to 𝐒 and with the same vertex (Ritt’s theorem). This result also holds with both O ’s replaced by o ’s. … These include addition, subtraction, multiplication, and division. …
    36: 18.17 Integrals
    For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. …
    37: 4.13 Lambert W -Function
    4.13.5_3 ( 1 + W 0 ( z ) ) 2 = 1 2 n = 1 n n 2 n ! ( z ) n , | z | < e 1 .
    4.13.10 W k ( z ) ξ k ln ξ k + n = 1 ( 1 ) n ξ k n m = 1 n [ n n m + 1 ] ( ln ξ k ) m m ! ,
    38: 2.7 Differential Equations
    For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
    Liouville–Green Approximation Theorem
    Suppose in addition | f 1 / 2 ( x ) d x | is unbounded as x a 1 + and x a 2 . …