Sentence examples for be the complex plane from inspiring English sources

Let C be the complex plane.

Corollary 4 Let S 2 be the complex plane ℂ compactified at infinity and S 1 be the unit circle.

Let (mathcal{C}) be the complex plane, and, for (zinmathcal{C}), (x=operatorname{Re} z), and (y=operatorname{Im} z).

There is, however, an example [34] of a discontinuous pseudocontractive mapping T with a unique fixed point for which the Mann iteration process does not always converge to the fixed point of T. Let H be the complex plane and K : = { z ∈ H : | z | ≤ 1 }.

The following example illustrates the fact that the corollary, and therefore Theorem 3, require the hypothesis that the map f is smooth, by exhibiting a non-smooth map f : ( S 2, S 1 ) → ( S 2, S 1 ) homotopic to the suspension of ϕ that has no fixed points on S 2 ∖ S 1. Example 1 Let S 2 = C ∪ be the complex plane ℂ compactified at infinity and S 1 be the unit circle.

Let C be the complex plane, D = { z ∈ C : | z | < 1 } the open unit disk and H (D ) the class of all analytic functions on D. Let φ be an analytic self-map of D and ψ ∈ H (D ).

The underlying manifold we work on is the complex plane ({mathbb C}).

The set R β, n ( U ) is the complex plane with slits along the half-lines Re w = 0 and | Im w | ≥ γ = n 1 + 2 β / n.

Remark 1 The function R c defined by (1.4) is univalent in U, where R c ( 0 ) = c, and R c ( U ) = R ( U ) is the complex plane with slits along the half-lines given by R ( w ) = 0 and | I ( w ) | ≧ N. Lemma 4 (Totoi [21]).

Throughout this paper, let H = { τ ∈ C | Im τ > 0 } be the complex upper half-plane.

And since we are mapping the "complex plane" to the "Cartesian plane", with the x axis representing "real" and the y axis representing "imaginary", we will also refer to this as (x2-y2, 2xy).