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11: 4.15 Graphics
See accompanying text
Figure 4.15.2: Arcsin x and Arccos x . … Magnify
12: 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
13: 22.14 Integrals
22.14.2 cn ( x , k ) d x = k - 1 Arccos ( dn ( x , k ) ) ,
22.14.5 sd ( x , k ) d x = ( k k ) - 1 Arcsin ( - k cd ( x , k ) ) ,
22.14.6 nd ( x , k ) d x = k - 1 Arccos ( cd ( x , k ) ) .
14: 5.9 Integral Representations
5.9.10 Ln Γ ( z ) = ( z - 1 2 ) ln z - z + 1 2 ln ( 2 π ) + 2 0 arctan ( t / z ) e 2 π t - 1 d t ,
5.9.11 Ln Γ ( z + 1 ) = - γ z - 1 2 π i - c - i - c + i π z - s s sin ( π s ) ζ ( - s ) d s ,
15: 5.11 Asymptotic Expansions
5.11.1 Ln Γ ( z ) ( z - 1 2 ) ln z - z + 1 2 ln ( 2 π ) + k = 1 B 2 k 2 k ( 2 k - 1 ) z 2 k - 1
5.11.8 Ln Γ ( z + h ) ( z + h - 1 2 ) ln z - z + 1 2 ln ( 2 π ) + k = 2 ( - 1 ) k B k ( h ) k ( k - 1 ) z k - 1 ,
16: 6.4 Analytic Continuation
6.4.1 E 1 ( z ) = Ein ( z ) - Ln z - γ ;
17: 14.25 Integral Representations
where the multivalued functions have their principal values when 1 < z < and are continuous in ( - , 1 ] . …
18: 19.21 Connection Formulas
19.21.7 ( x - y ) R D ( y , z , x ) + ( z - y ) R D ( x , y , z ) = 3 R F ( x , y , z ) - 3 y 1 / 2 x - 1 / 2 z - 1 / 2 ,
19.21.8 R D ( y , z , x ) + R D ( z , x , y ) + R D ( x , y , z ) = 3 x - 1 / 2 y - 1 / 2 z - 1 / 2 ,
19.21.10 2 R G ( x , y , z ) = z R F ( x , y , z ) - 1 3 ( x - z ) ( y - z ) R D ( x , y , z ) + 3 x 1 / 2 y 1 / 2 z - 1 / 2 , z 0 .
19: 14.1 Special Notation
Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. …
20: 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). …