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11: 1.9 Calculus of a Complex Variable
Point Sets in
Any point whose neighborhoods always contain members and nonmembers of D is a boundary point of D . When its boundary points are added the domain is said to be closed, but unless specified otherwise a domain is assumed to be open. … A region is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called interior points. …
12: 3.7 Ordinary Differential Equations
Write τ j = z j + 1 - z j , j = 0 , 1 , , P , expand w ( z ) and w ( z ) in Taylor series (§1.10(i)) centered at z = z j , and apply (3.7.2). …
§3.7(iii) Taylor-Series Method: Boundary-Value Problems
The remaining two equations are supplied by boundary conditions of the form … It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …
13: Mathematical Introduction
Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9. This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries. … This means that the variable x ranges from 0 to 1 in intervals of 0. …
14: 10.41 Asymptotic Expansions for Large Order
§10.41(ii) Uniform Expansions for Real Variable
10.41.7 η = ( 1 + z 2 ) 1 2 + ln z 1 + ( 1 + z 2 ) 1 2 ,
10.41.8 p = ( 1 + z 2 ) - 1 2 ,
§10.41(iii) Uniform Expansions for Complex Variable
The curve E 1 B E 2 in the z -plane is the upper boundary of the domain K depicted in Figure 10.20.3 and rotated through an angle - 1 2 π . …
15: 32.11 Asymptotic Approximations for Real Variables
§32.11 Asymptotic Approximations for Real Variables
with boundary condition … and with boundary condition …
16: 11.13 Methods of Computation
For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for K ν ( z ) and M ν ( z ) . … For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … For M ν ( x ) both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …
17: 10.21 Zeros
The parameter t may be regarded as a continuous variable and ρ ν , σ ν as functions ρ ν ( t ) , σ ν ( t ) of t . … In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points ± 1 are the boundaries of K , that is, the eye-shaped domain depicted in Figure 10.20.3. … Lastly, there are two conjugate sets, with n zeros in each set, that are asymptotically close to the boundary of K as n . … In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points ± 1 is the lower boundary of K . … The only other set comprises n zeros that are asymptotically close to the lower boundary of K as n . …
18: 1.10 Functions of a Complex Variable
§1.10 Functions of a Complex Variable
§1.10(i) Taylor’s Theorem for Complex Variables
Let D be a bounded domain with boundary D and let D ¯ = D D . … (Or more generally, a simple contour that starts at the center and terminates on the boundary.) …
19: 22.3 Graphics
§22.3(i) Real Variables: Line Graphs
See accompanying text
Figure 22.3.2: k = 0.7 , - 3 K x 3 K , K = 1.8456 . For cn ( x , k ) the curve for k = 1 / 2 = 0.70710 is a boundary between the curves that have an inflection point in the interval 0 x 2 K ( k ) , and its translates, and those that do not; see Walker (1996, p. 146). Magnify
§22.3(ii) Real Variables: Surfaces
See accompanying text
Figure 22.3.15: dn ( x , k ) for k = 1 - e - n , n = 0 to 20, - 5 π x 5 π . Magnify 3D Help
§22.3(iii) Complex z ; Real k
20: Bibliography H
  • S. P. Hastings and J. B. McLeod (1980) A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.
  • P. Holmes and D. Spence (1984) On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.
  • L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
  • T. E. Hull and A. Abrham (1986) Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.