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 ${\displaystyle i={\sqrt {-1}}}$. [The symbol j is also used to indicate ${\displaystyle {\sqrt {-1}}}$ and is 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 ${\displaystyle b^{2}-4ac<0}$ 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 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 [/itex] where ${\displaystyle r={\sqrt {x^{2}+y^{2}}}=|z|.}$ 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.

References

1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.