Solving Complex Exponentials: Deduce & Explain Relation w/ Vector Diagram

  • Thread starter kevi555
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In summary, the conversation was about deducing the relationship between z and dz, given z=Ae^{i\theta}. The derivative of z was found to be dz=i*A*e^(i*Θ)*dΘ, which can be represented in a vector diagram where i is equal to a 90 degree angle. The equation z=A*cosΘ+i*A*sinΘ was also mentioned and used to help understand the derivative equation.
  • #1
kevi555
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Hi,

If [tex]z=Ae^{i\theta}[/tex], deduce that [tex] dz = iz d\theta [/tex], and explain the relation in a vector diagram.

I know that [tex] z = x + iy [/tex] but I don't know if that's going to help. Any hints or tips would be appreciated! Thanks!
 
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  • #2
The derivative of z:

dz=i*A*e^(i*Θ)*dΘ

You have to ask yourself what are i,j: are they directions (r=xi+yj) or are they complex numbers ( x=a+i*b y=c+j*d).


Secondly, graphing on a polar plot one direction is for the real (x-axis) and the other for the imaginary (y-axis).
 
Last edited:
  • #3
Please note, I made an error in my first post...it is now correct.
 
  • #4
This may help:

z=A*cosΘ+i*A*sinΘ
 
  • #5
Alright, I can picture the [tex]Acos\theta[/tex] and [tex]Asin\theta[/tex] on the horizontal and vertical axes, respectively. I don't understand how the z is still in the derivative equation?
 
Last edited:
  • #6
dz is the change in z. i in a vector diagram is equal to 90deg, right. So when you take the derivative of a vector you shift 90deg.
 
  • #7
kevi555 said:
Alright, I can picture the [tex]Acos\theta[/tex] and [tex]Asin\theta[/tex] on the horizontal and vertical axes, respectively. I don't understand how the z is still in the derivative equation?

Philosophaie gave you a big hint. Look at the equation in his first post real carefully. What is "z" equal to? You already told us. =)
 

1. What are complex exponentials?

Complex exponentials are mathematical expressions in the form of eix, where e represents the base of the natural logarithm, i represents the imaginary unit, and x is a real number. They are commonly used in the field of engineering and physics to represent sinusoidal functions.

2. How do you solve complex exponentials?

To solve a complex exponential, you can use Euler's formula: eix = cosx + isinx. This allows you to break down the complex exponential into its real and imaginary parts. From there, you can use trigonometric identities and algebraic manipulation to simplify the expression.

3. What is the relation between complex exponentials and vector diagrams?

Complex exponentials are closely related to vector diagrams because they can be represented as vectors on a complex plane. The magnitude of the vector corresponds to the amplitude of the sinusoidal function, while the angle of the vector represents the phase shift.

4. How do you deduce the relation between complex exponentials and vector diagrams?

To deduce the relation between complex exponentials and vector diagrams, you can use the properties of complex numbers and trigonometry. By representing the complex exponential as a vector on a complex plane, you can see how the real and imaginary parts of the expression correspond to the x and y coordinates of the vector.

5. What are some practical applications of complex exponentials and vector diagrams?

Complex exponentials and vector diagrams are commonly used in fields such as electrical engineering, signal processing, and quantum mechanics. They are used to model and analyze periodic phenomena, such as electric currents and quantum states. They are also used in the design of electronic circuits and communication systems.

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