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1: 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 cut
4.2.26 z a = exp ( a Ln z ) , z 0 .
In all other cases, z a is a multivalued function with branch point at z = 0 . …
2: 1.10 Functions of a Complex Variable
§1.10(vi) Multivalued Functions
Let F ( z ) be a multivalued function and D be a domain. … Branches can be constructed in two ways: …
Example
3: 14.25 Integral Representations
where the multivalued functions have their principal values when 1 < z < and are continuous in ( - , 1 ] . …
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: 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 . …
4.23.31 w = Arcsin z = ( - 1 ) k arcsin z + k π ,
6: 14.1 Special Notation
Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. …
7: 14.21 Definitions and Basic Properties
When z is complex P ν ± μ ( z ) , Q ν μ ( z ) , and Q ν μ ( z ) are defined by (14.3.6)–(14.3.10) with x replaced by z : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when z ( 1 , ) , and by continuity elsewhere in the z -plane with a cut along the interval ( - , 1 ] ; compare §4.2(i). …
8: 4.8 Identities
4.8.1 Ln ( z 1 z 2 ) = Ln z 1 + Ln z 2 .
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 .
9: Mathematical Introduction
These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …
complex plane (excluding infinity).
F ( z 0 e 2 k π i ) multivalued functions. More generally, F ( ( z 0 - a ) e 2 k π i + a ) . See §1.10(vi).
10: 19.11 Addition Theorems
Hence, care has to be taken with the multivalued functions in (19.11.5). …