Dictionary:Complex number

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{{#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 complex z plane."


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.


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.