Analysis: Real- and Complex-Valued Functions of a Real Variable

Functions are mappings from some domain to some range. The domain might be the real line (denoted ${\cal R}^{1}$), such as time, or the real plane (denoted ${\cal R}^{2}$), such as an optical image. The range refers to the mapped value or values associated with all the points in the domain. For example, the function might associate to each point on the line or the plane just another real value (a scalar, such as temperature), or an ordered set of real values (a vector). A weather map showing wind velocity at each point in Britain exemplifies a vector-valued function of the real plane; and so on.

Functions may also associate a complex-valued quantity to each point in the domain. Complex variables are denoted ${\cal Z} = a + i b$ where $i = \sqrt{-1}$, and a is the real part and b is the imaginary part of ${\cal Z}$. For example, the Fourier Transform of a musical melody associates a complex variable to every possible frequency, each of which is represented by a point in the (real-valued) frequency domain.

The complex conjugate of ${\cal Z}$ is denoted by the asterisk (*), and it simply requires changing the sign of the imaginary part. Thus, the complex conjugate of ${\cal Z} = a + i b$ is: ${\cal Z}^{*} = a - i b$.

The modulus of a complex variable ${\cal Z}$ is $\sqrt{a^{2}+b^{2}}$ and it is denoted by $\Vert{\cal Z}\Vert$. It is easy to see that $\Vert{\cal Z}\Vert = \sqrt{{\cal Z}{\cal Z}^{*}}$.

The angle of a complex variable ${\cal Z} = a + i b$is $\tan^{-1}(\frac{b}{a})$and it is denoted $\angle {\cal Z}$.
A very important relation that we will use later is: ${\cal Z} = \Vert{\cal Z}\Vert \exp(i \angle {\cal Z})$. This can be regarded simply as converting the complex variable ${\cal Z}$from its ``Cartesian" form a + i b (where the real part a and the imaginary part b form orthogonal axes defining the complex plane), to polar form $(r,\theta)$ in which r is the modulus, or length $\Vert{\cal Z}\Vert$ of the complex variable, and $\theta$ is its angle $\angle {\cal Z} = \tan^{-1}(\frac{b}{a})$.

These relations and constructions are central to Fourier analysis and harmonic analysis, which in turn are the mathematical cornerstone of all of electrical engineering involving linear devices; optics; holography; broadcast communications; electronic filter theory; acoustics; quantum mechanics; wave phenomena; much of mechanical engineering, and most of physics! Indeed, the great Nobel Laureate in Physics, Julian Schwinger, once said: ``There are only two problems that we can solve in Physics. One is the simple harmonic oscillator [described in terms of the above complex variables]; and the second problem reduces to that one."