Simplify Tricky Equation for Purely Imaginary C with Complex Constants

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In summary, the conversation discusses solving an equation for a purely imaginary value of C using logarithms and simplifications. The equation is ##A^{-C/2} = (-1)^{1-C}B ##, where A and B are complex valued constants. The conversation ends with the original questioner stating that they have already solved the equation and asking about its origin.
  • #1
thatboi
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Hey all,
I am currently trying to solve the following equation for C:
1659137331786.png

where C is a purely imaginary value, ##F_{+}##, ##F_{-}## and ##G_{+}## and ##G_{-}## are all complex valued constants (so ##G_{+}^{*}## just means complex conjugate of ##G_{+}##. I am not really sure where to start with isolating C, any advice would be greatly appreciated!
 
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  • #2
Take logarithms of both sides and see if you can solve that equation for C.
 
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  • #3
Let ##F_+ /F_- = A## and ##\sqrt{ \dfrac{G_-G_+^*}{G_-^*G_+} } = B##

Your equation is ##A^{-C/2} = (-1)^{1-C}B ##

Always do simplifications and change of variables, to see what is going on.
 
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  • #4
@thatboi , with the two hints given to you above, it is fairly easy to solve for C. Is that working out for you?
 
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  • #5
phyzguy said:
@thatboi , with the two hints given to you above, it is fairly easy to solve for C. Is that working out for you?
Thanks for the hints I have already worked it out!
 
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  • #6
Great!

May I ask where this equation came from?
 

1. What is a purely imaginary number?

A purely imaginary number is a complex number where the real part is equal to 0. It is represented by the form bi, where b is a non-zero real number and i is the imaginary unit (√-1).

2. What are complex constants in an equation?

Complex constants are numbers that contain both a real and imaginary part. They are typically represented by the form a + bi, where a and b are real numbers and i is the imaginary unit.

3. How do you simplify a tricky equation with purely imaginary numbers and complex constants?

To simplify a tricky equation with purely imaginary numbers and complex constants, you can use the rules of complex arithmetic, such as combining like terms and using the distributive property. You can also use the fact that i^2 = -1 to simplify terms with i.

4. Can a purely imaginary number be the solution to a real-world problem?

Yes, a purely imaginary number can be the solution to a real-world problem. For example, in electrical engineering, imaginary numbers are used to represent the phase of a signal, which is crucial for designing and analyzing circuits.

5. Are there any special properties of purely imaginary numbers and complex constants in equations?

Yes, there are several special properties of purely imaginary numbers and complex constants in equations. For example, the product of two purely imaginary numbers is a real number, and the sum of a purely imaginary number and its complex conjugate is always a real number. Additionally, complex constants can be factored out of an equation just like real numbers.

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