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31: 27.18 Methods of Computation: Primes
An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). …
32: Roderick S. C. Wong
33: 32.2 Differential Equations
be a nonlinear second-order differential equation in which F is a rational function of w and d w / d z , and is locally analytic in z , that is, analytic except for isolated singularities in . … in which a ( z ) , b ( z ) , c ( z ) , d ( z ) , and ϕ ( z ) are locally analytic functions. …
34: 4.14 Definitions and Periodicity
4.14.7 cot z = cos z sin z = 1 tan z .
35: 33.2 Definitions and Basic Properties
§33.2(i) Coulomb Wave Equation
§33.2(ii) Regular Solution F ( η , ρ )
F ( η , ρ ) is a real and analytic function of ρ on the open interval 0 < ρ < , and also an analytic function of η when < η < . … As in the case of F ( η , ρ ) , the solutions H ± ( η , ρ ) and G ( η , ρ ) are analytic functions of ρ when 0 < ρ < . Also, e i σ ( η ) H ± ( η , ρ ) are analytic functions of η when < η < . …
36: 33.14 Definitions and Basic Properties
§33.14(i) Coulomb Wave Equation
§33.14(ii) Regular Solution f ( ϵ , ; r )
f ( ϵ , ; r ) is real and an analytic function of r in the interval < r < , and it is also an analytic function of ϵ when < ϵ < . …
§33.14(iii) Irregular Solution h ( ϵ , ; r )
h ( ϵ , ; r ) is real and an analytic function of each of r and ϵ in the intervals < r < and < ϵ < , except when r = 0 or ϵ = 0 . …
37: 1.9 Calculus of a Complex Variable
Analyticity
A function f ( z ) is analytic in a domain D if it is analytic at each point of D . A function analytic at every point of is said to be entire. …
38: 14.31 Other Applications
Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …
39: 28.19 Expansions in Series of me ν + 2 n Functions
Let q be a normal value (§28.12(i)) with respect to ν , and f ( z ) be a function that is analytic on a doubly-infinite open strip S that contains the real axis. …
40: 33.13 Complex Variable and Parameters
These functions may also be continued analytically to complex values of ρ , η , and . …