# Dictionary:Complex number

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

A number with both real and imaginary parts, such as

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

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

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

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

For example, if we consider a general quadratic equation

$az^{2}+bz+c=0$ where $a,b,c$ are real numbers. The roots of this equation may be obtained via the quadratic formula

$z={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.$ Here, we see that when the quantity (known as the discriminant of the quadratic) $b^{2}-4ac<0$ the result will be

$z={\frac {-b\pm i{\sqrt {|4ac-b^{2}|}}}{2a}}={\frac {-b}{2a}}\pm i{\frac {\sqrt {|4ac-b^{2}|}}{2a}}$ where we use the mathematician's convention of using $i={\sqrt {-1}}.$ 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 $P(z)$ of degree $n$ with complex coefficients has $n$ 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 $z=x+iy$ where $x$ is the real part and $y$ is called the imaginary part (called so, reflecting a time when mathematicians were uncomfortable with quantities involving ${\sqrt {-1}}.$ )

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 $x$ as the real axis and $y$ as the imaginary axis.

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

$z=x+iy=r[\cos(\theta )+i\sin(\theta )].$ Here [/itex] where $r={\sqrt {x^{2}+y^{2}}}=|z|.$ and $r=|z|$ is called the modulus of $z.$ The modulus may be written as $r={\sqrt {(x+iy)(x-iy)}}={\sqrt {z{\overline {z}}}}=|z|^{2}$ where ${\overline {z}}=x-iy$ is called the complex conjugate of $z.$ Another polar representation is the following identities

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

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

$\cos(\theta )\pm i\sin(\theta )=e^{\pm i\theta }.$ We begin by formally writing the Taylor series representations of $\cos(\theta )$ and $i\sin(\theta )$ and sum the resulting series

$\cos(\theta )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\theta ^{2k}}{(2k)!}}=\sum _{k=0}^{\infty }{\frac {(i)^{2k}\theta ^{2k}}{(2k)!}}=\sum _{k=0}^{\infty }{\frac {(i\theta )^{2k}}{(2k)!}},$ where $k=0,1,2,3,...$ We may write the Taylor series form of $i\sin(\theta )$ $i\sin(\theta )=i\sum _{l=0}^{\infty }{\frac {(-1)^{l}\theta ^{2l+1}}{(2l+1)!}}=\sum _{l=0}^{\infty }{\frac {(i)^{2l+1}\theta ^{2l+1}}{(2l+1)!}}=\sum _{l=0}^{\infty }{\frac {(i\theta )^{2l+1}}{(2l+1)!}},$ where $l=0,1,2,3,...$ . In both of the previous cases, free use has been made of the identity $-1=i^{2}.$ The sum of these two series yields the series representation of the exponential function

$\cos(\theta )+i\sin(\theta )=\sum _{n=0}^{\infty }{\frac {(i\theta )^{n}}{n!}}=e^{i\theta },$ where $n=0,1,2,3,...$ .

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

$f(z)=u(x,y)+iv(x,y).$ 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.