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1: 36.4 Bifurcation Sets
§36.4(i) Formulas
Critical Points for Cuspoids
Critical Points for Umbilics
This is the codimension-one surface in 𝐱 space where critical points coalesce, satisfying (36.4.1) and … This is the codimension-one surface in 𝐱 space where critical points coalesce, satisfying (36.4.2) and …
2: 25.12 Polylogarithms
Other notations and names for Li 2 ( z ) include S 2 ( z ) (Kölbig et al. (1970)), Spence function Sp ( z ) (’t Hooft and Veltman (1979)), and L 2 ( z ) (Maximon (2003)). In the complex plane Li 2 ( z ) has a branch point at z = 1 . The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches. …
See accompanying text
Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
3: 28.7 Analytic Continuation of Eigenvalues
The only singularities are algebraic branch points, with a n ( q ) and b n ( q ) finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. …The branch points are called the exceptional values, and the other points normal values. … For a visualization of the first branch point of a 0 ( i q ^ ) and a 2 ( i q ^ ) see Figure 28.7.1. …
4: 3.1 Arithmetics and Error Measures
A nonzero normalized binary floating-point machine number x is represented as … …
IEEE Standard
Rounding
5: 4.2 Definitions
This is a multivalued function of z with branch point at z = 0 . The principal value, or principal branch, is defined by … 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 …
6: Mark J. Ablowitz
ODEs which do not have moveable branch point singularities. …
7: 4.3 Graphics
Corresponding points share the same letters, with bars signifying complex conjugates. …In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . …
See accompanying text
Figure 4.3.3: ln ( x + i y ) (principal value). There is a branch cut along the negative real axis. Magnify 3D Help
8: 1.10 Functions of a Complex Variable
Then a is a branch point of F ( z ) . For example, z = 0 is a branch point of z . … (a) By introducing appropriate cuts from the branch points and restricting F ( z ) to be single-valued in the cut plane (or domain). (b) By specifying the value of F ( z ) at a point z 0 (not a branch point), and requiring F ( z ) to be continuous on any path that begins at z 0 and does not pass through any branch points or other singularities of F ( z ) . …
9: 10.2 Definitions
This solution of (10.2.1) is an analytic function of z , except for a branch point at z = 0 when ν is not an integer. … Whether or not ν is an integer Y ν ( z ) has a branch point at z = 0 . … Each solution has a branch point at z = 0 for all ν . …
Branch Conventions
10: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
§36.12(i) General Theory for Cuspoids
The branch for 𝐱 ( 𝐲 ) is such that 𝐱 is real when 𝐲 is real. … Branches are chosen so that Δ is real and positive if the critical points are real, or real and negative if they are complex. … For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).