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1: 1.6 Vectors and Vector-Valued Functions
Note: The terminology open and closed sets and boundary points in the ( x , y ) plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … and S be the closed and bounded point set in the ( x , y ) plane having a simple closed curve C as boundary. …
2: 1.9 Calculus of a Complex Variable
Point Sets in
An open set in is one in which each point has a neighborhood that is contained in the set. … 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 . …
3: 4.13 Lambert W -Function
W 0 ( z ) is a single-valued analytic function on ( , e 1 ] , real-valued when z > e 1 , and has a square root branch point at z = e 1 . …The other branches W k ( z ) are single-valued analytic functions on ( , 0 ] , have a logarithmic branch point at z = 0 , and, in the case k = ± 1 , have a square root branch point at z = e 1 0 i respectively. …
4: 1.5 Calculus of Two or More Variables
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. …
5: 2.3 Integrals of a Real Variable
Assume also that 2 p ( α , t ) / t 2 and q ( α , t ) are continuous in α and t , and for each α the minimum value of p ( α , t ) in [ 0 , k ) is at t = α , at which point p ( α , t ) / t vanishes, but both 2 p ( α , t ) / t 2 and q ( α , t ) are nonzero. …
6: Mathematical Introduction
These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …
complex plane (excluding infinity).
( a , b ) open interval in , or open straight-line segment joining a and b in .
( a , b ] or [ a , b ) half-closed intervals.
lim inf least limit point.
[ a j , k ] or [ a j k ] matrix with ( j , k ) th element a j , k or a j k .
7: 15.6 Integral Representations
In (15.6.2) the point 1 / z lies outside the integration contour, t b 1 and ( t 1 ) c b 1 assume their principal values where the contour cuts the interval ( 1 , ) , and ( 1 z t ) a = 1 at t = 0 . …
8: 28.33 Physical Applications
As ω runs from 0 to + , with b and f fixed, the point ( q , a ) moves from to 0 along the ray given by the part of the line a = ( 2 b / f ) q that lies in the first quadrant of the ( q , a ) -plane. …
9: 1.16 Distributions
Let ϕ be a function defined on an open interval I = ( a , b ) , which can be infinite. The closure of the set of points where ϕ 0 is called the support of ϕ . If the support of ϕ is a compact set1.9(vii)), then ϕ is called a function of compact support. … The set of tempered distributions is denoted by 𝒯 . … Here 𝜶 ranges over a finite set of multi-indices, P ( 𝐱 ) is a multivariate polynomial, and P ( 𝐃 ) is a partial differential operator. …
10: 1.4 Calculus of One Variable
If f ( x ) is continuous at each point c ( a , b ) , then f ( x ) is continuous on the interval ( a , b ) and we write f C ( a , b ) . … where the supremum is over all sets of points x 0 < x 1 < < x n in the closure of ( a , b ) , that is, ( a , b ) with a , b added when they are finite. …