# Dictionary:Complex number

{{#category_index:C|complex number}}

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

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z=x+iy=re^{i\theta} }**,

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=\sqrt{-1} }**
. [The symbol *j* is also used to indicate **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}}**
and
is notation favored
by electrical engineers, as the symbol * Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i}
* is often reserved for representing electric current]. The

**modulus**or magnitude of the above complex number is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=\sqrt{x^2+y^2} }**and the angle indicating its direction with respect to the positive real axis is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1} }**
often appear. For example, if we consider a general quadratic equation

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a z^2 + b z + c = 0 }**

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a, b, c }**
are real numbers. The roots of this equation may be obtained via the quadratic formula

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = \frac{ - b \pm \sqrt{ b^2 - 4 a c }}{ 2 a } .}**

Here, we see that when the quantity **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b^2 - 4 ac < 0 }**
the result will be

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = \frac{ - b \pm i \sqrt{ |4 a c - b^2|}}{ 2 a } = \frac{-b}{2a} \pm i \frac{\sqrt{|4 a c - b^2|}}{2a} }**

where we use the mathematician's convention of using **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i = \sqrt{-1}. }**

### The polar representation and the Argand plane

In general we consider a complex number **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = x + i y }**
where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x }**
is the *real part* and
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y }**
is called the *imaginary part* (called so, reflecting a time when mathematicians were uncomfortable with
quantities involving **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x }**
as the *real axis* and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y}**
as the *imaginary axis.*

If we consider the angle **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta }**
as the angle turned in a counter-clockwise direction from the positive real axis, then
the natural polar representation results

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = x + i y = r( \cos(\theta) + i \sin(\theta) ) .}**

Here </math> where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{x^2 + y^2} = |z|. }**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = |z|}**
is called the *modulus* of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z .}**

The modulus may be written as **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{(x + i y)(x -iy)} = \sqrt{z \overline{z}} = |z|^2 }**
where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overline{z} = x - i y }**
is called the **complex conjugate** of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z .}**

Another polar representation is the following identities

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = r( \cos(\theta) + i \sin(\theta) ) = r e^{i \theta}, }**

and

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overline{z} = r( \cos(\theta) - i \sin(\theta) ) = r e^{-i \theta}, }**

implying that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos(\theta) \pm i \sin(\theta) = e^{\pm i \theta}. }**

We begin by formally writing the Taylor series representations of
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos (\theta) }**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin (\theta) }**
and sum the resulting series

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=0,1,2,3,... }**
We may write the Taylor series form of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin(\theta) }**

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle l=0,1,2,3,...}**
. In both of the previous cases, free use has been made of the identity **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1 = i^2.}**

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

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos(\theta) + i \sin (\theta) = \sum_{n=0}^{\infty} \frac{( i \theta)^{n} }{ n! } = e^{ i \theta }, }**

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(z) }**
is a mapping from the field of complex numbers to the field of complex numbers, then **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(z) }**
must, itself be the sum of a purely real valued function **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x,y) }**
and a purely imaginary valued function
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i v(x,y), }**
thus,

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(z) = u(x,y) + i v(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.