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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane.
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number.
On the region consisting of complex numbers that are not negative real numbers or 0, the function is the analytic continuation of the natural logarithm. The values on the negative real line can be obtained as limits of values at nearby complex numbers with positive imaginary parts.
Figure 1. This Argand diagram represents the complex number lying on a plane.For each point on the plane, arg is the function which returns the angle . In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in ...
In fact, the same proof shows that Euler's formula is even valid for all complex numbers x. A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used ...
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space ), which the Mathematics Subject ...