About the Project
NIST

multivalued

AdvancedHelp

(0.001 seconds)

1—10 of 32 matching pages

1: 4.1 Special Notation
The main functions treated in this chapter are the logarithm ln z , Ln z ; the exponential exp z , e z ; the circular trigonometric (or just trigonometric) functions sin z , cos z , tan z , csc z , sec z , cot z ; the inverse trigonometric functions arcsin z , Arcsin z , etc. ; the hyperbolic trigonometric (or just hyperbolic) functions sinh z , cosh z , tanh z , csch z , sech z , coth z ; the inverse hyperbolic functions arcsinh z , Arcsinh z , etc. Sometimes in the literature the meanings of ln and Ln are interchanged; similarly for arcsin z and Arcsin z , etc. … sin - 1 z for arcsin z and Sin - 1 z for Arcsin z .
2: 4.24 Inverse Trigonometric Functions: Further Properties
4.24.13 Arcsin u ± Arcsin v = Arcsin ( u ( 1 - v 2 ) 1 / 2 ± v ( 1 - u 2 ) 1 / 2 ) ,
4.24.14 Arccos u ± Arccos v = Arccos ( u v ( ( 1 - u 2 ) ( 1 - v 2 ) ) 1 / 2 ) ,
4.24.15 Arctan u ± Arctan v = Arctan ( u ± v 1 u v ) ,
4.24.16 Arcsin u ± Arccos v = Arcsin ( u v ± ( ( 1 - u 2 ) ( 1 - v 2 ) ) 1 / 2 ) = Arccos ( v ( 1 - u 2 ) 1 / 2 u ( 1 - v 2 ) 1 / 2 ) ,
4.24.17 Arctan u ± Arccot v = Arctan ( u v ± 1 v u ) = Arccot ( v u u v ± 1 ) .
3: 4.8 Identities
4.8.1 Ln ( z 1 z 2 ) = Ln z 1 + Ln z 2 .
This is interpreted that every value of Ln ( z 1 z 2 ) is one of the values of Ln z 1 + Ln z 2 , and vice versa. …
4.8.3 Ln z 1 z 2 = Ln z 1 - Ln z 2 ,
4.8.5 Ln ( z n ) = n Ln z , n ,
4.8.10 exp ( ln z ) = exp ( Ln z ) = z .
4: 4.37 Inverse Hyperbolic Functions
Each of the six functions is a multivalued function of z . Arcsinh z and Arccsch z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . …
5: 5.10 Continued Fractions
5.10.1 Ln Γ ( z ) + z - ( z - 1 2 ) ln z - 1 2 ln ( 2 π ) = a 0 z + a 1 z + a 2 z + a 3 z + a 4 z + a 5 z + ,
6: 4.38 Inverse Hyperbolic Functions: Further Properties
4.38.15 Arcsinh u ± Arcsinh v = Arcsinh ( u ( 1 + v 2 ) 1 / 2 ± v ( 1 + u 2 ) 1 / 2 ) ,
4.38.16 Arccosh u ± Arccosh v = Arccosh ( u v ± ( ( u 2 - 1 ) ( v 2 - 1 ) ) 1 / 2 ) ,
4.38.17 Arctanh u ± Arctanh v = Arctanh ( u ± v 1 ± u v ) ,
4.38.18 Arcsinh u ± Arccosh v = Arcsinh ( u v ± ( ( 1 + u 2 ) ( v 2 - 1 ) ) 1 / 2 ) = Arccosh ( v ( 1 + u 2 ) 1 / 2 ± u ( v 2 - 1 ) 1 / 2 ) ,
4.38.19 Arctanh u ± Arccoth v = Arctanh ( u v ± 1 v ± u ) = Arccoth ( v ± u u v ± 1 ) .
7: 5.17 Barnes’ G -Function (Double Gamma Function)
5.17.4 Ln G ( z + 1 ) = 1 2 z ln ( 2 π ) - 1 2 z ( z + 1 ) + z Ln Γ ( z + 1 ) - 0 z Ln Γ ( t + 1 ) d t .
In this equation (and in (5.17.5) below), the Ln ’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). …
5.17.5 Ln G ( z + 1 ) 1 4 z 2 + z Ln Γ ( z + 1 ) - ( 1 2 z ( z + 1 ) + 1 12 ) Ln z - ln A + k = 1 B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k .
8: 4.2 Definitions
The general logarithm function Ln z is defined by …This is a multivalued function of z with branch point at z = 0 . … Most texts extend the definition of the principal value to include the branch cutIn all other cases, z a is a multivalued function with branch point at z = 0 . … This result is also valid when z a has its principal value, provided that the branch of Ln w satisfies …
9: 4.7 Derivatives and Differential Equations
4.7.2 d d z Ln z = 1 z ,
4.7.4 d n d z n Ln z = ( - 1 ) n - 1 ( n - 1 ) ! z - n .
4.7.6 w ( z ) = Ln ( f ( z ) ) +  constant .
When a z is a general power, ln a is replaced by the branch of Ln a used in constructing a z . …
10: 4.23 Inverse Trigonometric Functions
4.23.4 Arccsc z = Arcsin ( 1 / z ) ,
4.23.5 Arcsec z = Arccos ( 1 / z ) ,
4.23.6 Arccot z = Arctan ( 1 / z ) .
Each of the six functions is a multivalued function of z . Arctan z and Arccot z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . …