# Dictionary:Complex number

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Here we follow standard texts, such as Spiegel (1964) [1] or Levinson and Redheffer (1970). [2]

A number with both real and imaginary parts, such as

${\displaystyle z=x+iy=re^{i\theta }}$,

where ${\displaystyle i={\sqrt {-1}}}$, ${\displaystyle x}$ is called the real part, and ${\displaystyle y}$ is called the imaginary part of ${\displaystyle z}$ is called a complex number. [The symbol j is also used to indicate ${\displaystyle {\sqrt {-1}}}$ and is the notation favored by electrical engineers, as the symbol ${\displaystyle i}$ is often reserved for representing electric current]. The modulus or magnitude of the above complex number is ${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$ and the angle indicating its direction with respect to the positive real axis is

${\displaystyle \theta =\tan ^{-1}{\frac {y}{x}}}$.

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 ${\displaystyle {\sqrt {-1}}}$ often appear.

For example, if we consider a general quadratic equation

${\displaystyle az^{2}+bz+c=0}$

where ${\displaystyle a,b,c}$ are real numbers. The roots of this equation may be obtained via the quadratic formula

${\displaystyle z={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$

Here, we see that when the quantity (known as the discriminant of the quadratic) ${\displaystyle b^{2}-4ac<0}$ the result will be

${\displaystyle z={\frac {-b\pm i{\sqrt {|4ac-b^{2}|}}}{2a}}={\frac {-b}{2a}}\pm i{\frac {\sqrt {|4ac-b^{2}|}}{2a}}}$

where we use the mathematician's convention of using ${\displaystyle i={\sqrt {-1}}.}$

"The complex z plane."

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 ${\displaystyle P(z)}$ of degree ${\displaystyle n}$ with complex coefficients has ${\displaystyle n}$ 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 ${\displaystyle z=x+iy}$ where ${\displaystyle x}$ is the real part and ${\displaystyle y}$ is called the imaginary part (called so, reflecting a time when mathematicians were uncomfortable with quantities involving ${\displaystyle {\sqrt {-1}}.}$)

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 ${\displaystyle x}$ as the real axis and ${\displaystyle y}$ as the imaginary axis.

If we consider the angle ${\displaystyle \theta }$ as the angle turned in a counter-clockwise direction from the positive real axis, then the natural polar representation results

${\displaystyle z=x+iy=r[\cos(\theta )+i\sin(\theta )].}$

Here ${\displaystyle r={\sqrt {x^{2}+y^{2}}}=|z|.}$ and ${\displaystyle r=|z|}$ is called the modulus of ${\displaystyle z.}$

The modulus may be written as ${\displaystyle r={\sqrt {(x+iy)(x-iy)}}={\sqrt {z{\overline {z}}}}=|z|^{2}}$ where ${\displaystyle {\overline {z}}=x-iy}$ is called the complex conjugate of ${\displaystyle z.}$

Another polar representation is the following identities

${\displaystyle z=r(\cos(\theta )+i\sin(\theta ))=re^{i\theta },}$

and

${\displaystyle {\overline {z}}=r(\cos(\theta )-i\sin(\theta ))=re^{-i\theta },}$

implying that

${\displaystyle \cos(\theta )\pm i\sin(\theta )=e^{\pm i\theta }.}$

We begin by formally writing the Taylor series representations of ${\displaystyle \cos(\theta )}$ and ${\displaystyle i\sin(\theta )}$ and sum the resulting series

${\displaystyle \cos(\theta )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\theta ^{2k}}{(2k)!}}=\sum _{k=0}^{\infty }{\frac {(i)^{2k}\theta ^{2k}}{(2k)!}}=\sum _{k=0}^{\infty }{\frac {(i\theta )^{2k}}{(2k)!}},}$

where ${\displaystyle k=0,1,2,3,...}$ We may write the Taylor series form of ${\displaystyle i\sin(\theta )}$

${\displaystyle i\sin(\theta )=i\sum _{l=0}^{\infty }{\frac {(-1)^{l}\theta ^{2l+1}}{(2l+1)!}}=\sum _{l=0}^{\infty }{\frac {(i)^{2l+1}\theta ^{2l+1}}{(2l+1)!}}=\sum _{l=0}^{\infty }{\frac {(i\theta )^{2l+1}}{(2l+1)!}},}$

where ${\displaystyle l=0,1,2,3,...}$ . In both of the previous cases, free use has been made of the identity ${\displaystyle -1=i^{2}.}$

The sum of these two series yields the series representation of the exponential function

${\displaystyle \cos(\theta )+i\sin(\theta )=\sum _{n=0}^{\infty }{\frac {(i\theta )^{n}}{n!}}=e^{i\theta },}$

where ${\displaystyle n=0,1,2,3,...}$.

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 ${\displaystyle f(z)}$ is a mapping from the field of complex numbers to the field of complex numbers, then ${\displaystyle f(z)}$ must, itself be the sum of a purely real valued function ${\displaystyle u(x,y)}$ and a purely imaginary valued function ${\displaystyle iv(x,y),}$ thus,

${\displaystyle f(z)=u(x,y)+iv(x,y).}$

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.