Proving Complex Conjugate is Real: Euler's Identity

In summary, the conversation discusses a homework problem involving proving the resulting equation is real given the expression \alphaexp(i\varpit) +\alpha*exp(-i\varpit). The attempted solution involved expanding the exponentials into sines and cosines, but a more efficient solution involved writing \alpha as a complex number and considering the sum of its complex conjugates.
  • #1
JPBenowitz
144
2

Homework Statement



So we are given [itex]\alpha[/itex]exp(i[itex]\varpi[/itex]t) +[itex]\alpha[/itex]*exp(-i[itex]\varpi[/itex]t) and are asked to prove the resulting equation is real.

Homework Equations



[itex]\alpha[/itex] + [itex]\alpha[/itex]* = 2Re([itex]\alpha[/itex]) and Euler's Identity

The Attempt at a Solution



I tried expanding out the exp's to cosines and isines but couldn't reach the solution.
 
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  • #2
What if you also write ##\alpha = a + i b##, where a and b are both real numbers? Can you do the problem then?
 
  • #3
Mute said:
What if you also write ##\alpha = a + i b##, where a and b are both real numbers? Can you do the problem then?

I figured it out but I didn't do it that way. That seems awfully more tedious than usual.
 
  • #4
JPBenowitz said:
I figured it out but I didn't do it that way. That seems awfully more tedious than usual.

It's more tedious than the quick solution, yes, but it was more along the lines of the approach you had tried to take by expanding the exponentials into sines and cosines, so I opted to guide you along that direction, in case the problem wanted you to show it explicitly.
 
  • #5
Why are you writing a bar over the omega? If omega is real then it's sort of obviously true. Because you are adding two complex conjugates. If omega isn't real then it's not even true.
 
Last edited:

Related to Proving Complex Conjugate is Real: Euler's Identity

1. What is Euler's identity?

Euler's identity is a mathematical equation that relates five fundamental mathematical constants: 0, 1, π, e, and i (the imaginary unit). It is written as e = -1.

2. What does it mean for a complex number to be conjugate?

A complex number is written in the form a + bi, where a and b are real numbers and i is the imaginary unit. The conjugate of this complex number is written as a - bi, where the sign of the imaginary term is changed. In other words, the conjugate of a complex number is the same number with the sign of the imaginary part flipped.

3. How is Euler's identity related to the complex conjugate?

Euler's identity states that e = -1. This can also be written as e-iπ = 1. Taking the complex conjugate of both sides, we get (e-iπ)* = (1)*, which simplifies to e = 1. This shows that the complex conjugate of e is itself, proving that it is a real number.

4. Why is proving the complex conjugate real important?

Proving that the complex conjugate of e is a real number is important because it is a fundamental result in complex analysis. It shows that the imaginary part of a complex number can be "cancelled out" by taking its conjugate, resulting in a real number. This is useful in many applications, such as in solving differential equations and analyzing electrical circuits.

5. How is Euler's identity used in real-world applications?

Euler's identity is used in a variety of real-world applications, including signal processing, electrical engineering, and quantum mechanics. It is also used in the study of periodic functions and Fourier series. In addition, Euler's identity has been used to prove the irrationality of π and the transcendence of e, which are important mathematical results.

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