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1: 2.1 Definitions and Elementary Properties
Let X be a point set with a limit point c . …
2.1.16 f ( x ) a 0 + a 1 ( x - c ) + a 2 ( x - c ) 2 + , x c in X ,
If the set X in §2.1(iii) is a closed sector α ph x β , then by definition the asymptotic property (2.1.13) holds uniformly with respect to ph x [ α , β ] as | x | . …Suppose u is a parameter (or set of parameters) ranging over a point set (or sets) U , and for each nonnegative integer n
2: 1.5 Calculus of Two or More Variables
A function is continuous on a point set D if it is continuous at all points of D . … … For f ( x , y ) defined on a point set D contained in a rectangle R , let … where D is the image of D * under a mapping ( u , v ) ( x ( u , v ) , y ( u , v ) ) which is one-to-one except perhaps for a set of points of area zero. …Again the mapping is one-to-one except perhaps for a set of points of volume zero. …
3: 1.9 Calculus of a Complex Variable
An open set in is one in which each point has a neighborhood that is contained in the set. A point z 0 is a limit point (limiting point or accumulation point) of a set of points S in (or ) if every neighborhood of z 0 contains a point of S distinct from z 0 . … A domain D , say, is an open set in that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … Conversely, if at a given point ( x , y ) the partial derivatives u / x , u / y , v / x , and v / y exist, are continuous, and satisfy (1.9.25), then f ( z ) is differentiable at z = x + i y . …
1.9.49 R = lim inf n | a n | - 1 / n .
4: 18.2 General Orthogonal Polynomials
Let X be a finite set of distinct points on , or a countable infinite set of distinct points on , and w x , x X , be a set of positive constants. …when X is a finite set of N + 1 distinct points. …
5: 2.8 Differential Equations with a Parameter
in which u is a real or complex parameter, and asymptotic solutions are needed for large | u | that are uniform with respect to z in a point set D in or . … Corresponding to each positive integer n there are solutions W n , j ( u , ξ ) , j = 1 , 2 , that depend on arbitrarily chosen reference points α j , are C or analytic on Δ , and as u
6: 21.3 Symmetry and Quasi-Periodicity
The set of points m 1 + Ω m 2 form a g -dimensional lattice, the period lattice of the Riemann theta function. …
7: 1.6 Vectors and Vector-Valued Functions
and S be the closed and bounded point set in the ( x , y ) plane having a simple closed curve C as boundary. … Suppose S is a piecewise smooth surface which forms the complete boundary of a bounded closed point set V , and S is oriented by its normal being outwards from V . …
8: 4.13 Lambert W -Function
See accompanying text
Figure 4.13.1: Branches Wp ( x ) and Wm ( x ) of the Lambert W -function. A and B denote the points - 1 / e and e , respectively, on the x -axis. Magnify
9: 26.13 Permutations: Cycle Notation
An element of 𝔖 n with a 1 fixed points, a 2 cycles of length 2 , , a n cycles of length n , where n = a 1 + 2 a 2 + + n a n , is said to have cycle type ( a 1 , a 2 , , a n ) . …
10: 22.19 Physical Applications
As a 1 / β from below the period diverges since a = ± 1 / β are points of unstable equilibrium. … For an initial displacement with 1 / β | a | < 2 / β , bounded oscillations take place near one of the two points of stable equilibrium x = ± 1 / β . …As a 2 / β from below the period diverges since x = 0 is a point of unstable equlilibrium. …