Vectors in the Complex Plane
The complex plane is the geometric representation of C, the set of all complex numbers. While R (the set of all real numbers) is contained within a one-dimensional space, a complex number has two parts: the real part and the imaginary part. If z is a complex number, the real part of z is represented as ℜ(z) and the imaginary part as ℑ(z). Each of these two parts take up one of the axes in the coordinate plane. By convention, the x-axis contains the real part and the y-axis contains the imaginary part. Thus, any complex number z can be writted as x + iy, or as the vector (x, y) in the complex plane.
Polar Coordinates in the Complex Plane
As with the real coordinate plane R², we can represent a complex vector with polar coordinates. Based on the definition of the complex exponential function, we choose the following form, where r is the radius and θ is the angle from the positive x-axis.:
z = r e^(iθ)
This is because the complex exponential has the following form:
e^(x + iy) = (e^x)(cos y + i sin y)
This gives us the following x- and y-coordinates, which are familiar from elementary vector analysis:
x = r cos θ
y = r sin θ
We also note that r is the modulus, or length |z| of the complex number z. This will become useful shortly.
Operations in the Complex Plane
Addition and subtraction of vectors in the complex plane works as normal for vectors -- add the x- and y-components of the vectors to find the resultant. This mirrors (as it should) the addition and subtraction of complex numbers, where the real and imaginary parts never mix.
The polar form of the complex number becomes useful when we multiply the vectors. If we have the following two complex numbers:
z = ρ e^(iθ)
w = σ e^(iφ)
Then their product is:
z*w = ρ*σ e^(i(θ + φ))
Thus the modulus of the product is the product of the moduli, and the angle of the product is the sum of the angles of the factors. How elegant! We see also that multiplication by i itself is the equivalent of a 90 degree rotation (counterclockwise) of the number-vector in the plane.
Functions in the Complex Plane
When dealing with real functions, we noticed that a function f(x) = y mapped R into R. Thus, since the input and output of a real function are both one-dimensional, we can represent the input and output in the x-y coordinate plane. Unfortunately, this is not quite the case with complex functions. Representing a complex-valued function of a complex variable would require a 4-dimensional space, which is hard to come by (unless you make a movie of the function.) Integration in the complex plane then necessarily takes the form of a line integral, because a path must be defined along which to integrate. As with other functions of vectors, the x-axis simply will not do. Also, derivatives of complex functions must follow the Cauchy-Riemann Equations, a set of differential equations relating the partial derivatives of complex functions.