Return to Complex Analysis.
A number with both real and imaginary parts, such as
where , is called the real part, and is called the imaginary part of is called a complex number. [The symbol j is also used to indicate and is the notation favored by electrical engineers, as the symbol is often reserved for representing electric current]. The modulus or magnitude of the above complex number is and the angle indicating its direction with respect to the positive real axis is
A graph of a complex function or quantity (such as a frequency spectrum) is shown in Figure C-10.
Understanding complex numbers and functions of a complex variable.
In the computation of the roots of polynomial equations, additive quantities which are scaled by often appear.
Example: a quadratic equation
For example, if we consider a general quadratic equation
where are real numbers. The roots of this equation may be obtained via the quadratic formula
Here, we see that when the quantity (known as the discriminant of the quadratic) the result will be
where we use the mathematician's convention of using
The point that is demonstrated here is that the relation between real and imaginary parts is additive.
The fundamental theorem of algebra
The generalization of the quadratic result yields the fundamental theorem of algebra which states that every polynomial of degree with complex coefficients has complex roots. Because real numbers are subsumed into the complex numbers (being considered as having imaginary part 0), this describes all polynomials.
The polar representation and the Argand plane
In general we consider a complex number where is the real part and is called the imaginary part (called so, reflecting a time when mathematicians were uncomfortable with quantities involving )
An amateur mathematician named Jean-Robert Argand is credited with being the first person to publish a representation of complex numbers geometrically as defining a plane with the horizontal axis as the real axis and as the imaginary axis.
If we consider the angle as the angle turned in a counter-clockwise direction from the positive real axis, then the natural polar representation results
Here and is called the modulus of
The modulus may be written as where is called the complex conjugate of
Another polar representation is the following identities
We begin by formally writing the Taylor series representations of and and sum the resulting series
where We may write the Taylor series form of
where . In both of the previous cases, free use has been made of the identity
The sum of these two series yields the series representation of the exponential function
You will likely not see this argument in a textbook, as it is a plausibility argument rather than a proof, because it depends on having the machinery of convergence of series in place.
Functions of a complex variable
If a function is a mapping from the field of complex numbers to the field of complex numbers, then must, itself be the sum of a purely real valued function and a purely imaginary valued function thus,
A certain class of complex valued function, known as an Analytic function is of particular importance in the application of complex-valued functions to problems in the physical sciences.
Return to Complex Analysis.
- Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
- Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.