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...

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

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

applicable to all complex numbers; see § Complex plane for the extension of exp ⁡ x {\displaystyle \exp x} to the complex plane. Using the power series...

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...

other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional...

everywhere within some neighbourhood of z0 in the complex plane. Given a complex-valued function f of a single complex variable, the derivative of f at a point...

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...

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

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....

the complex projective plane, usually denoted P2(C) or CP2, is the two-dimensional complex projective space. It is a complex manifold of complex dimension...

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 plane was thought of as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex...

definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function...

two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has two complex dimensions...

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...

identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often...

ex to the complex plane. The exponential function f ( z ) = e z {\displaystyle f(z)=e^{z}} is the unique differentiable function of a complex variable...

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

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. When...

mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration...

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...

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...

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...

cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed...

function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple...

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...

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...

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...

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...