Solving complex exponentials

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In summary, the conversation discusses a problem involving finding real numbers a and b to satisfy the equation (5e^(j*a))(3 + j*b) = -25. The equation is converted to 5*sqrt(9 + b^2)*e^(j*a + j * arctan(b/3)) = -25, but the speaker is unsure how to proceed. They suggest trying to rewrite the equation in the form c*e^(j*θ), where c is a positive real number.
  • #1
Bob Busby
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I posted this problem here because I would like to know a reliable method for solving such a thing.

(5e^(j*a))(3 + j*b) = -25 Find real numbers a and b satisfying the preceding equation.

I converted it to get 5*sqrt(9 + b^2)*e^(j*a + j * arctan(b/3)) = -25. I don't really see where to go from here. If I separate the real parts I will just get a cos(a + arctan(b/3)) which doesn't help me even if I equate real and imaginary parts. What do I do?
 
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  • #2
Bob Busby said:
I posted this problem here because I would like to know a reliable method for solving such a thing.

(5e^(j*a))(3 + j*b) = -25 Find real numbers a and b satisfying the preceding equation.

I converted it to get 5*sqrt(9 + b^2)*e^(j*a + j * arctan(b/3)) = -25. I don't really see where to go from here. If I separate the real parts I will just get a cos(a + arctan(b/3)) which doesn't help me even if I equate real and imaginary parts. What do I do?

You try

[tex] e^{ja}= cos (a) + j sin(a) \hbox { and }\; 3+jb = \sqrt { 3^2 + b^2 } \; e^{j[tan^{-1} (\frac b 3)]} \hbox {?}[/tex]
 
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  • #3
yungman said:
You try

[tex] e^{ja}= cos (a) + j sin(a) \hbox { and }\; 3+jb = \sqrt { 3^2 + b^2 } \; e^{j[tan^{-1} (\frac b 3)]} \hbox {?}[/tex]

Yes. that's what I tried.
 
  • #4
Bob Busby said:
5*sqrt(9 + b^2)*e^(j*a + j * arctan(b/3)) = -25. I don't really see where to go from here.
How about rewriting that equation with -25 in the form c*e^(j*θ), where c is a positive real number?
 

1. What are complex exponentials?

Complex exponentials are mathematical expressions in the form of ex, where e is a constant approximately equal to 2.71828, and x is any complex number (a number with a real and imaginary component).

2. What is the purpose of solving complex exponentials?

Solving complex exponentials is important in many areas of science and engineering, as it allows us to model and analyze various physical phenomena, such as electrical circuits, waves, and growth rates.

3. How do I solve a complex exponential equation?

To solve a complex exponential equation, you can use the properties of exponents and logarithms, as well as basic algebraic techniques. It may also be helpful to convert the complex exponential into its polar form, where ex = r(cosθ + i sinθ), with r and θ representing the magnitude and angle of the complex number, respectively.

4. Are there any special cases when solving complex exponentials?

Yes, there are a few special cases to consider when solving complex exponentials. One is when the exponent is a pure imaginary number, in which case the solution will involve trigonometric functions. Another special case is when the base is a negative number, which can be solved using the properties of logarithms.

5. Can complex exponentials be graphed?

Yes, complex exponentials can be graphed on the complex plane, where the real and imaginary components are represented on the x and y axes, respectively. The graph will typically take the form of a spiral, with the magnitude increasing as the angle increases.

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