Complex exponentials (simplify the expression)

In summary, in order to simplify the expression ei6x(1+e-i10x)/(1+ei2x), you need to use the Euler formula e^{ix} = \cos x + i \sin x. By rewriting the expression in terms of cosines and sines, you can use the formula 1/2(e^x +e^-x) = cosx to simplify it.
  • #1
Luongo
120
0
1. Simplify ei6x(1+e-i10x)/(1+ei2x)
2. i have no idea how to simplify this its supposed to be in terms of cosines
3. i don't how i can simplify this such that i can use the 1/2(e^x +e^-x) = cosx formula
 
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  • #2
You need the Euler formula [tex]e^{ix} = \cos x + i \sin x[/tex].
 
  • #3
fzero said:
You need the Euler formula [tex]e^{ix} = \cos x + i \sin x[/tex].

i tried that, i got garbage it MUST be in terms of cosines
 
  • #4
You might want to show some work to see where your problem is. (Also you might find [tex]e^{6ix} = e^{5ix} e^{ix}[/tex] useful.)
 
  • #5
fzero said:
You might want to show some work to see where your problem is. (Also you might find [tex]e^{6ix} = e^{5ix} e^{ix}[/tex] useful.)


well, I've multiplied it out i got e^6x+e^-4x and then i did e^2x * e^4x + e^-4x
in the numerator to try to get it in cosine form but i can't get the e^2 out of there so i really have no idea how, can someone show me the steps of simplifying this
 
  • #6
You also want to simplify the denominator, so you don't want to multiply the numerator through by the whole [tex]e^{6ix}[/tex] factor. (Also you're leaving out the factors of i in your exponentials, which is a bit confusing, but you shouldn't do it in anything you turn into be graded.)
 
  • #7
fzero said:
You might want to show some work to see where your problem is. (Also you might find [tex]e^{6ix} = e^{5ix} e^{ix}[/tex] useful.)


i got it! you're a genius, how on Earth did you see that??
 
  • #8
Exponent rules :D...
 
  • #9
Luongo said:
i got it! you're a genius, how on Earth did you see that??

You know that

[tex]\frac{ e^{ia} + e^{-ia}}{2} = \cos a[/tex]

so if you see

[tex] 1 + e^{ib}[/tex]

you want to rewrite that as

[tex] e^{ib/2} ( e^{-ib/2} + e^{ib/2} ).[/tex]
 
  • #10
fzero said:
You know that

[tex]\frac{ e^{ia} + e^{-ia}}{2} = \cos a[/tex]

so if you see

[tex] 1 + e^{ib}[/tex]

you want to rewrite that as

[tex] e^{ib/2} ( e^{-ib/2} + e^{ib/2} ).[/tex]


thanks a lot i appreciate it i was hung up on this question for a while :redface:
 
  • #11
Luongo said:
thanks a lot i appreciate it i was hung up on this question for a while :redface:


for simplifying in terms of sines... can i use the same formula except the negative sign is between the 2 expos?
 
  • #12
Luongo said:
for simplifying in terms of sines... can i use the same formula except the negative sign is between the 2 expos?

Yes.
 

1. What is a complex exponential?

A complex exponential is a mathematical function of the form eix, where i is the imaginary unit (√-1) and x is any real number. It represents a complex number in polar form.

2. How do you simplify a complex exponential expression?

To simplify a complex exponential expression, you can use Euler's formula, which states that eix = cos(x) + i sin(x). This allows you to rewrite the expression in terms of trigonometric functions and simplify it further.

3. Can a complex exponential be negative?

Yes, a complex exponential can be negative. The real part of the expression can be negative if the angle x is greater than π/2 or less than -π/2. The imaginary part can also be negative if the angle x is between π/2 and -π/2.

4. What is the difference between a complex exponential and a real exponential?

A complex exponential has a complex number as its base, while a real exponential has a real number as its base. Additionally, the rules for simplifying and manipulating complex exponential expressions are different from those for real exponential expressions.

5. How are complex exponentials used in science and engineering?

Complex exponentials have many practical applications in science and engineering. They are commonly used to describe and analyze oscillatory phenomena, such as sound waves and electromagnetic waves. They are also used in control systems, signal processing, and quantum mechanics.

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