In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called...

of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the...

In mathematics, the complex projective plane, usually denoted ⁠ P 2 ( C ) {\displaystyle \mathbb {P} ^{2}(\mathbb {C} )} ⁠ or ⁠ C P 2 , {\displaystyle...

Poincaré half-plane model. Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to...

standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their...

in adding more structure, one may view the plane as a 1-dimensional complex manifold, called the complex line. Many fundamental tasks in mathematics...

regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point ⁠ z...

complex ones. The hyperbolic unit j is not a real number but an independent quantity. The collection of all such z is called the split-complex plane....

These logarithms are equally spaced along a vertical line in the complex plane. A complex-valued function log : U → C {\displaystyle \log \colon U\to \mathbb...

sphere. The extended Euclidean plane can be identified with the extended complex plane, so that equations of complex numbers can be used to describe...

In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases...

getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions...

meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its...

represent physical positions, like an affine plane or complex plane. The most basic example is the flat Euclidean plane, an idealization of a flat surface in...

additional examples. In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers z such that | z...

the complex plane, the function e i x {\displaystyle e^{ix}} for real values of x {\displaystyle x} traces out the unit circle in the complex plane. Both...

generalized to accept complex numbers as arguments. This reveals relations between multiplication of complex numbers, rotations in the complex plane, and trigonometry...

to a real plane, not a real line. The "complex plane" commonly refers to the graphical representation of the complex line on the real plane, and is thus...

and the line joining the origin and z, represented as a point in the complex plane, shown as φ {\displaystyle \varphi } in Figure 1. By convention the...

multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers T = { z ∈ C...

sinusoidal plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. It...

A\cos(\omega t+\theta ).} Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a phasor...

algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous...

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all...

\infty } can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting...

The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can...

geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation x 2 − y 2 = 1. {\displaystyle x^{2}-y^{2}=1...

thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite...

2π to this angle works as well.) In the complex plane, which is a special interpretation of a Cartesian plane, i is the point located one unit from the...

d}}t,\ \qquad \Re (z)>0\,.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic...