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1: 10.17 Asymptotic Expansions for Large Argument
10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t ) exp ( | ν 2 1 4 | 𝒱 z , ± i ( t 1 ) ) ,
where 𝒱 denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that | t | changes monotonically. Bounds for 𝒱 z , i ( t ) are given by
10.17.15 𝒱 z , i ( t ) { | z | , 0 ph z π , χ ( ) | z | , 1 2 π ph z 0  or  π ph z 3 2 π , 2 χ ( ) | z | , π < ph z 1 2 π  or  3 2 π ph z < 2 π ,
The bounds (10.17.15) also apply to 𝒱 z , i ( t ) in the conjugate sectors. …
2: 10.40 Asymptotic Expansions for Large Argument
10.40.11 | R ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ( t ) exp ( | ν 2 1 4 | 𝒱 z , ( t 1 ) ) ,
where 𝒱 denotes the variational operator2.3(i)), and the paths of variation are subject to the condition that | t | changes monotonically. Bounds for 𝒱 z , ( t ) are given by
10.40.12 𝒱 z , ( t ) { | z | , | ph z | 1 2 π , χ ( ) | z | , 1 2 π | ph z | π , 2 χ ( ) | z | , π | ph z | < 3 2 π ,
3: 1.4 Calculus of One Variable
1.4.33 𝒱 a , b ( f ) = sup j = 1 n | f ( x j ) f ( x j 1 ) | ,
If 𝒱 a , b ( f ) < , then f ( x ) is of bounded variation on ( a , b ) . In this case, g ( x ) = 𝒱 a , x ( f ) and h ( x ) = 𝒱 a , x ( f ) f ( x ) are nondecreasing bounded functions and f ( x ) = g ( x ) h ( x ) . …
1.4.34 𝒱 a , b ( f ) = a b | f ( x ) | d x ,
Lastly, whether or not the real numbers a and b satisfy a < b , and whether or not they are finite, we define 𝒱 a , b ( f ) by (1.4.34) whenever this integral exists. …
4: 2.7 Differential Equations
2.7.23 | ϵ j ( x ) | , 1 2 f 1 / 2 ( x ) | ϵ j ( x ) | exp ( 1 2 𝒱 a j , x ( F ) ) 1 , j = 1 , 2 ,
provided that 𝒱 a j , x ( F ) < . …and 𝒱 denotes the variational operator2.3(i)). …
2.7.25 𝒱 a j , x ( F ) = | a j x | 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) g ( t ) f 1 / 2 ( t ) | d t | .
Assuming also 𝒱 a 1 , a 2 ( F ) < , we have …
5: 2.8 Differential Equations with a Parameter
In addition, 𝒱 𝒬 j ( A 1 ) and 𝒱 𝒬 j ( A n ) must be bounded on 𝚫 j ( α j ) . … These results are valid when 𝒱 α 1 , α 2 ( | ξ | 1 / 2 B 0 ) and 𝒱 α 1 , α 2 ( | ξ | 1 / 2 B n 1 ) are finite. … These results are valid when 𝒱 0 , α 2 ( ξ 1 / 2 B 0 ) and 𝒱 0 , α 2 ( ξ 1 / 2 B n 1 ) are finite. … These results are valid when 𝒱 α 1 , 0 ( | ξ | 1 / 2 B 0 ) and 𝒱 α 1 , 0 ( | ξ | 1 / 2 B n 1 ) are finite. …
6: 2.3 Integrals of a Real Variable
In both cases the n th error term is bounded in absolute value by x n 𝒱 a , b ( q ( n 1 ) ( t ) ) , where the variational operator 𝒱 a , b is defined by
2.3.6 𝒱 a , b ( f ( t ) ) = a b | f ( t ) | d t ;
7: Errata
  • Equation (2.7.25)
    2.7.25 𝒱 a j , x ( F ) = | a j x | 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) g ( t ) f 1 / 2 ( t ) | d t |

    The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value for 𝒱 a j , x ( F ) .

  • Equation (2.3.6)
    2.3.6 𝒱 a , b ( f ( t ) ) = a b | f ( t ) | d t

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Juan Luis Varona on 2021-02-08

  • Equation (1.4.34)
    1.4.34 𝒱 a , b ( f ) = a b | f ( x ) | d x

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Tran Quoc Viet on 2020-08-11

  • Equation (10.17.14)
    10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t ) exp ( | ν 2 1 4 | 𝒱 z , ± i ( t 1 ) )

    Originally the factor 𝒱 z , ± i ( t 1 ) in the argument to the exponential was written incorrectly as 𝒱 z , ± i ( t ) .

    Reported 2014-09-27 by Gergő Nemes.

  • 8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Bounded and Unbounded Linear Operators
    Self-Adjoint Operators
    Spectrum of an Operator