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11: 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. …
12: About Color Map
Surface visualizations in the DLMF represent functions of the form z = f ( x , y ) by the height z or the magnitude, | z | , for complex functions, over the x × y plane. We use color to augment these vizualizations, either to reinforce the recognition of the height, or to convey an extra dimension to represent the phase of complex valued functions. … By painting the surfaces with a color that encodes the phase, ph f , both the magnitude and phase of complex valued functions can be displayed. …
13: 8.5 Confluent Hypergeometric Representations
8.5.1 γ ( a , z ) = a 1 z a e z M ( 1 , 1 + a , z ) = a 1 z a M ( a , 1 + a , z ) , a 0 , 1 , 2 , .
8.5.2 γ ( a , z ) = e z 𝐌 ( 1 , 1 + a , z ) = 𝐌 ( a , 1 + a , z ) .
8.5.3 Γ ( a , z ) = e z U ( 1 a , 1 a , z ) = z a e z U ( 1 , 1 + a , z ) .
8.5.4 γ ( a , z ) = a 1 z 1 2 a 1 2 e 1 2 z M 1 2 a 1 2 , 1 2 a ( z ) .
8.5.5 Γ ( a , z ) = e 1 2 z z 1 2 a 1 2 W 1 2 a 1 2 , 1 2 a ( z ) .
14: 10.13 Other Differential Equations
10.13.1 w ′′ + ( λ 2 ν 2 1 4 z 2 ) w = 0 , w = z 1 2 𝒞 ν ( λ z ) ,
10.13.2 w ′′ + ( λ 2 4 z ν 2 1 4 z 2 ) w = 0 , w = z 1 2 𝒞 ν ( λ z 1 2 ) ,
10.13.5 z 2 w ′′ + ( 1 2 r ) z w + ( λ 2 q 2 z 2 q + r 2 ν 2 q 2 ) w = 0 , w = z r 𝒞 ν ( λ z q ) ,
10.13.9 z 2 w ′′′ + 3 z w ′′ + ( 4 z 2 + 1 4 ν 2 ) w + 4 z w = 0 , w = 𝒞 ν ( z ) 𝒟 ν ( z ) ,
10.13.10 z 3 w ′′′ + z ( 4 z 2 + 1 4 ν 2 ) w + ( 4 ν 2 1 ) w = 0 , w = z 𝒞 ν ( z ) 𝒟 ν ( z ) ,
15: 4.15 Graphics
See accompanying text
Figure 4.15.6: arccsc x and arcsec x . …(Both functions are complex when 1 < x < 1 .) Magnify
§4.15(iii) Complex Arguments: Surfaces
The corresponding surfaces for arccos ( x + i y ) , arccot ( x + i y ) , arcsec ( x + i y ) can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).
16: 22.7 Landen Transformations
22.7.2 sn ( z , k ) = ( 1 + k 1 ) sn ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 sn 2 ( z / ( 1 + k 1 ) , k 1 ) ,
22.7.3 cn ( z , k ) = cn ( z / ( 1 + k 1 ) , k 1 ) dn ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 sn 2 ( z / ( 1 + k 1 ) , k 1 ) ,
22.7.4 dn ( z , k ) = dn 2 ( z / ( 1 + k 1 ) , k 1 ) ( 1 k 1 ) 1 + k 1 dn 2 ( z / ( 1 + k 1 ) , k 1 ) .
22.7.6 sn ( z , k ) = ( 1 + k 2 ) sn ( z / ( 1 + k 2 ) , k 2 ) cn ( z / ( 1 + k 2 ) , k 2 ) dn ( z / ( 1 + k 2 ) , k 2 ) ,
22.7.8 dn ( z , k ) = ( 1 k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) + k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) .
17: 20.12 Mathematical Applications
§20.12(ii) Uniformization and Embedding of Complex Tori
Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …
18: 10.36 Other Differential Equations
10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w ( ( z 2 + ν 2 ) 2 + z 2 ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .
19: 4.6 Power Series
4.6.1 ln ( 1 + z ) = z 1 2 z 2 + 1 3 z 3 , | z | 1 , z 1 ,
4.6.2 ln z = ( z 1 z ) + 1 2 ( z 1 z ) 2 + 1 3 ( z 1 z ) 3 + , z 1 2 ,
4.6.3 ln z = ( z 1 ) 1 2 ( z 1 ) 2 + 1 3 ( z 1 ) 3 , | z 1 | 1 , z 0 ,
4.6.4 ln z = 2 ( ( z 1 z + 1 ) + 1 3 ( z 1 z + 1 ) 3 + 1 5 ( z 1 z + 1 ) 5 + ) , z 0 , z 0 ,
4.6.5 ln ( z + 1 z 1 ) = 2 ( 1 z + 1 3 z 3 + 1 5 z 5 + ) , | z | 1 , z ± 1 ,
20: 22.6 Elementary Identities
22.6.1 sn 2 ( z , k ) + cn 2 ( z , k ) = k 2 sn 2 ( z , k ) + dn 2 ( z , k ) = 1 ,
22.6.2 1 + cs 2 ( z , k ) = k 2 + ds 2 ( z , k ) = ns 2 ( z , k ) ,
22.6.3 k 2 sc 2 ( z , k ) + 1 = dc 2 ( z , k ) = k 2 nc 2 ( z , k ) + k 2 ,
22.6.4 k 2 k 2 sd 2 ( z , k ) = k 2 ( cd 2 ( z , k ) 1 ) = k 2 ( 1 nd 2 ( z , k ) ) .
22.6.5 sn ( 2 z , k ) = 2 sn ( z , k ) cn ( z , k ) dn ( z , k ) 1 k 2 sn 4 ( z , k ) ,