Complex Numbers and Stereographic Projection

A mathematical expression representing the relationship between exponential, trigonometric, and imaginary units, i.e., eiθ = cos θ + i sin θ. It helps connect complex numbers and trigonometric functions using exponential form.

The solutions of the equation z^n = 1, where n is a positive integer. They represent the vertices of a regular polygon inscribed in the unit circle.

A complex number that can be expressed in the form z = x + iy, where x and y are real numbers. The imaginary unit 'i' is defined as the square root of -1.

The transformation z → w = 1/z, which maps points in the complex plane to their inverses. It results in points close to the origin being mapped far away, and vice versa.

A circular region in the complex plane with a radius of one unit and centered at the origin, represented by the equation |z| = 1. It's significant because it contains the n-th roots of unity.

A point that represents 'infinity' in the complex plane. It's a concept that allows us to explore transformations and analyze functions at extreme values of the complex variable.

A transformation that maps points in the complex plane to other points in the complex plane, defined by the equation z → w = λz + µ, where λ ≠ 0. It's a linear transformation that preserves the basic structure of the plane.

A geometric representation of the complex plane, where points on the sphere correspond to complex numbers.

A method to project points from a sphere onto a plane, typically used to represent complex numbers on the Riemann Sphere.

The point (0, 0, 1) on the sphere, used as the reference point for stereographic projection.

A point on the sphere, represented by coordinates (x1, x2, x3), where the sum of the squares of the coordinates is equal to 1.

The point where the line connecting the North Pole (N) and a point A0 on the sphere intersects the x3 = 0 plane (plane of projection).

A way to express the relationship between the complex number z and the corresponding point A0 on the sphere using the stereographic projection.

A formula that relates the magnitude of the complex number z to the z-coordinate (x3) of the corresponding point on the sphere.

Inversion is a transformation in the complex plane where a point z is mapped to its reciprocal, 1/z. This transformation has the property that points far away from the origin are mapped to points close to the origin, and vice versa.

The extended complex plane, denoted by C∞, is the complex plane C combined with the point at infinity. It provides a more complete representation of the behavior of functions at infinity.

The north pole of the unit sphere in stereographic projection corresponds to the point at infinity in the extended complex plane.

The extended complex plane and the unit sphere with stereographic projection establish a one-to-one correspondence between points on the sphere and points in the extended complex plane.

Riemann's spherical representation provides a geometric visualization of the extended complex plane by using the unit sphere and stereographic projection.

Stereographic projection allows us to visualize the behavior of complex functions near infinity by mapping it to the north pole of the sphere. This representation becomes useful in understanding the behavior of functions at infinity.

Riemann's spherical representation, through stereographic projection, provides a way to understand functions and their behavior at infinity, which is not easily visualized on the flat complex plane

The equation z^n = 1, where n is a positive integer, represents the nth roots of unity. Solving for z, we get n distinct solutions, which are the vertices of a regular polygon inscribed within the unit circle.

A complex number is represented in the form z = x + iy, where x and y are real numbers and 'i' is the imaginary unit (√-1). It can be visualized as a point in the complex plane, with x being the real part and y being the imaginary part.

Calculating all the values of z that satisfy the equation z^n = a, where a is a complex number and n is a positive integer. We essentially find the nth roots of the complex number 'a'.

The Argand plane, also known as the complex plane, is a two-dimensional graph where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Points in this plane represent complex numbers.

When finding all solutions for a complex variable, the solutions can be represented geometrically and will often be equally spaced on a circle centered at the origin of the Argand Plane.

Each solution of z^n = a, where a is a non-zero complex number is called an nth-root of a. The solutions are distributed evenly around a circle in the complex plane.

The solutions for z^n = a are represented by the vertices of a regular polygon with n sides. The polygon's center is at the origin and the radius is equal to the nth root of the absolute value of a.

A regular polygon with n sides inscribed in a circle centered at the origin. Each vertex of the polygon represents a solution to the equation z^n = a, where a is a non-zero complex number.

The transformation z → w = 1/z, which maps points in the complex plane to their inverses. It results in points close to the origin being mapped far away, and vice versa.

Geometrically, the solutions for z^n = a, where a is a non-zero complex number are the vertices of a regular polygon with n sides inscribed in a circle centered at the origin. The radius of the circle is the nth root of the absolute value of a. Each solution represents a unique nth root of the complex number a.

A complex number that can be expressed in the form z = x + iy, where x and y are real numbers. The imaginary unit 'i' is defined as the square root of -1.

This inequality establishes a relationship between the magnitude of the sum of two complex numbers and the sum of their individual magnitudes multiplied by their moduli.

The magnitude of the sum of two complex numbers is less than or equal to the sum of the magnitudes of each complex number multiplied by the magnitudes of the other complex numbers.

The magnitude of the sum of two complex numbers is less than or equal to the sum of the magnitudes of each complex number multiplied by the magnitudes of the other complex numbers.

The magnitude of the sum of two complex numbers is less than or equal to the sum of the magnitudes of each complex number multiplied by the magnitudes of the other complex numbers.

The magnitude of the sum of two complex numbers is less than or equal to the sum of the magnitudes of each complex number multiplied by the magnitudes of the other complex numbers.