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11: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …
12: 4.3 Graphics
Figure 4.3.2 illustrates the conformal mapping of the strip π < z < π onto the whole w -plane cut along the negative real axis, where w = e z and z = ln w (principal value). …
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Figure 4.3.3: ln ( x + i y ) (principal value). … Magnify 3D Help
13: 14.25 Integral Representations
The principal values of P ν μ ( z ) and 𝑸 ν μ ( z ) 14.21(i)) are given by …where the multivalued functions have their principal values when 1 < z < and are continuous in ( , 1 ] . …
14: 4.1 Special Notation
k , m , n integers.
15: 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). … …
16: 1.14 Integral Transforms
1.14.3 1 2 ( f ( u + ) + f ( u ) ) = 1 2 π F ( x ) e i x u d x ,
where the last integral denotes the Cauchy principal value (1.4.25). …
1.14.41 ( f ) ( x ) = f ( x ) = 1 π f ( t ) t x d t ,
1.14.44 f ( x ) = 1 π f ( u ) u x d u .
17: 6.3 Graphics
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Figure 6.3.3: | E 1 ( x + i y ) | , 4 x 4 , 4 y 4 . Principal value. … Magnify 3D Help
18: 8.2 Definitions and Basic Properties
8.2.2 Γ ( a , z ) = z t a 1 e t d t ,
However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, γ ( a , z ) and Γ ( a , z ) take their principal values; compare §4.2(i). Except where indicated otherwise in the DLMF these principal values are assumed. …
19: 11.3 Graphics
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Figure 11.3.8: | 𝐊 0 ( x + i y ) | (principal value) for 8 x 8 and 3 y 3 . … Magnify 3D Help
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Figure 11.3.9: | 𝐇 1 2 ( x + i y ) | (principal value) for 8 x 8 and 3 y 3 . … Magnify 3D Help
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Figure 11.3.10: | 𝐊 1 2 ( x + i y ) | (principal value) for 8 x 8 and 3 y 3 . … Magnify 3D Help
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Figure 11.3.12: | 𝐊 1 ( x + i y ) | (principal value) for 8 x 8 and 3 y 3 . … Magnify 3D Help
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Figure 11.3.19: | 𝐌 1 2 ( x + i y ) | (principal value) for 3 x 3 and 3 y 3 . … Magnify 3D Help
20: 1.16 Distributions