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1: 1.4 Calculus of One Variable
Functions of Bounded Variation
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. This definition also applies when f ( x ) is a complex function of the real variable x . …
2: 10.40 Asymptotic Expansions for Large Argument
10.40.6 I ν ( z ) K ν ( z ) 1 2 z ( 1 - 1 2 μ - 1 ( 2 z ) 2 + 1 3 2 4 ( μ - 1 ) ( μ - 9 ) ( 2 z ) 4 - ) ,
10.40.7 I ν ( z ) K ν ( z ) - 1 2 z ( 1 + 1 2 μ - 3 ( 2 z ) 2 - 1 2 4 ( μ - 1 ) ( μ - 45 ) ( 2 z ) 4 + ) ,
§10.40(iii) Error Bounds for Complex Argument and Order
where 𝒱 denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that | t | changes monotonically. …
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: 8.23 Statistical Applications
§8.23 Statistical Applications
4: 10.77 Software
§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions
§10.77(v) Bessel FunctionsReal Order and Complex Argument (including Hankel Functions)
§10.77(vii) Bessel FunctionsComplex Order and Real Argument
§10.77(viii) Bessel FunctionsComplex Order and Argument
5: 14.31 Other Applications
§14.31(iii) Miscellaneous
Legendre functions P ν ( x ) of complex degree ν appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …
6: 30.6 Functions of Complex Argument
§30.6 Functions of Complex Argument
The solutions …
30.6.3 𝒲 { Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) } = ( - 1 ) m ( n + m ) ! ( 1 - z 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ,
7: 12.8 Recurrence Relations and Derivatives
12.8.1 z U ( a , z ) - U ( a - 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.2 U ( a , z ) + 1 2 z U ( a , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.3 U ( a , z ) - 1 2 z U ( a , z ) + U ( a - 1 , z ) = 0 ,
12.8.4 2 U ( a , z ) + U ( a - 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 .
12.8.5 z V ( a , z ) - V ( a + 1 , z ) + ( a - 1 2 ) V ( a - 1 , z ) = 0 ,
8: 5.23 Approximations
§5.23(iii) Approximations in the Complex Plane
9: 22.17 Moduli Outside the Interval [0,1]
22.17.1 p q ( z , k ) = p q ( z , - k ) ,
22.17.2 sn ( z , 1 / k ) = k sn ( z / k , k ) ,
22.17.3 cn ( z , 1 / k ) = dn ( z / k , k ) ,
22.17.4 dn ( z , 1 / k ) = cn ( z / k , k ) .
§22.17(ii) Complex Moduli
10: 4.29 Graphics
§4.29(ii) Complex Arguments
The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …