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A graph of a real function can be drawn in two dimensions because there are two represented variables, and .However, complex numbers are represented by two variables and therefore two dimensions; this means that representing a complex function (more precisely, a complex-valued function of one complex variable:) requires the visualization of four dimensions.
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 graph of a real function of a real variable is the set of ordered pairs (x,y) such that y = f(x). This is the ordinary two-dimensional Cartesian graph studied in school algebra. Every complex number has both a real part and an imaginary part, so one complex variable is two-dimensional and a BBC pair of complex variables is four-dimensional. A ...
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 ...
A complex-valued function :, defined on some subset of the set of nonzero complex numbers, satisfying = for all in . Such complex logarithm functions are analogous to the real logarithm function ln : R > 0 → R {\displaystyle \ln \colon \mathbb {R} _{>0}\to \mathbb {R} } , which is the inverse of the real exponential function and hence ...
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.