Exploring Complex Exponentials: \theta = n(tan^{-1}(\frac{b}{a}))

In summary: That is, the distance from (0,0) to (cos(\theta)+ i sin(\theta)^n is 1, the same as that from (0,0) to (a+bi)^n. Since (cos(\theta)+ i sin(\theta)^n is in the first quadrant, its angle is just n\theta. That is, the angle of (a+bi)^n is n\theta.In summary, the absolute value of a complex number a+bi is equal to the square root of the sum of the squares of its real and imaginary parts. The argument or angle \theta of a complex number is represented by the inverse tangent of b/a. The absolute value of a complex
  • #1
Gregg
459
0
[tex] | a + bi | = \sqrt{(a)^2+(b)^2} [/tex]
[tex] \theta = tan^{-1}(\frac{b}{a}) [/tex]

[tex] |(a + bi)^n| = (\sqrt{(a)^2+(b)^2})^n[/tex]
[tex] \theta = n(tan^{-1}(\frac{b}{a}))[/tex]

[tex] (a+bi) = (\sqrt{(a)^2+(b)^2})^n(cos(n(tan^{-1}(\frac{b}{a})) + isin(n(tan^{-1}(\frac{b}{a}))))[/tex]

Why is [tex]\theta = n(tan^{-1}(\frac{b}{a}) [/tex] ? For example when I have

[tex]5 + i[/tex]

[tex]\theta = tan^{-1}(\frac{1}{5}) = 0.1974...[/tex]

[tex](5+i)^2 = 25 + 10i -1[/tex]

[tex](5+i)^2 = 24 + 10i[/tex]

[tex]\alpha = tan^{-1}(\frac{10}{24}) = 0.3948...[/tex]

[tex]2(0.1974) =0.3948[/tex]

Why is it that the exponent of the vector can be used to get the angle of the resultant by simply multiplying it with the tan function? Also in the first part of that:

[tex] (\sqrt{(a)^2+(b)^2})^n(cos(n(tan^{-1}(\frac{b}{a})) + isin(n(tan^{-1}(\frac{b}{a}))))[/tex]

Just to make sure, it is only

[tex] isin(n(tan^{-1}(\frac{b}{a}))))[/tex]

and not simply

[tex] (\sqrt{(a)^2+(b)^2})^n(cos(n(tan^{-1}(\frac{b}{a})) + sin(n(tan^{-1}(\frac{b}{a}))))[/tex]

because one of the components is complex?
 
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  • #2
you are talk about putting complex numbers in "polar form". If you represent a+ bi as the point (a, b) in Cartesian coordinates and the polar form is [itex](r, \theta)[/itex] (That is, distance from (0,0) to (a,b) is r and the line from (0,0) to (a,b) is [itex]\theta[/itex], it is easy to see that [itex]a= rcos(\theta)[/itex] and [itex]b= rsin(\theta)[/itex] so that [itex]a+ bi= r (cos(\theta)+ i sin(\theta))[/itex]. Then [/itex](a+bi)^2= r^2(cos^2(\theta)-sin^2(\theta)+ i(2sin(\theta)cos(\theta)))[/itex] and recognise that [itex]cos^2(\theta)- sin^2(\theta)= cos(2\theta)[/itex] and [/itex]2cos(\theta)sin(\theta)= sin(2\theta)[/itex]. The more general formula follows from the identites for [itex]sin(\theta+ \phi)[/itex] and [itex]cos(\theta+ \phi)[/itex].

Even simpler is to note that [itex]cos(\theta)+ i sin(\theta)= e^{i\theta}[/itex] so that [itex](cos(\theta)+ i sin(\theta)^n= (e^{i\theta})^n= e^{ni\theta}[/itex].
 

1. What is the purpose of exploring complex exponentials?

The purpose of exploring complex exponentials is to understand and analyze the behavior of exponential functions with complex inputs and outputs. This concept is important in various fields of science and engineering, such as physics, electrical engineering, and signal processing.

2. What does the symbol θ represent in the equation θ = n(tan^{-1}(\frac{b}{a}))?

In this equation, θ represents the phase angle, which is a measure of the shift or displacement of a complex exponential function from its original position on the complex plane. It is often measured in radians or degrees.

3. How does the value of a affect the graph of the complex exponential function?

The value of a affects the amplitude of the complex exponential function. It determines the vertical scaling factor of the function, which can stretch or compress its graph along the imaginary axis.

4. Can you explain the significance of the term tan^{-1}(\frac{b}{a}) in the equation?

The term tan^{-1}(\frac{b}{a}) represents the angle between the real and imaginary components of the complex exponential function. This angle is important in determining the phase angle and can also provide information about the frequency and rate of change of the function.

5. How is the parameter n related to the concept of harmonics in complex exponentials?

The parameter n, also known as the harmonic number, represents the number of full cycles of the complex exponential function within a given interval. It is directly related to the concept of harmonics, as different values of n correspond to different frequencies of the function. Higher values of n indicate higher harmonics, which can be used to analyze and synthesize complex signals.

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