Differentiating Complex Exponentials

  • Thread starter Matty R
  • Start date
  • #1
83
0
Hello :smile:

I'm currently using past papers to revise for January exams, and I've found a bit of a problem with something I thought I was okay with.

Homework Statement


The position at time t of a particle undergoing damped oscillations is given by:

[tex]x = 2e^{-t}\sin(3t)[/tex].

Express this in terms of a single complex exponential.

Hence evaluate the particle's velocity, [tex]v = \frac{dy}{dx}[/tex]


Homework Equations



[tex]e^{i \theta} = cos(\theta) + isin(\theta)[/tex]

Chain Rule: [tex]\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}[/tex]


The Attempt at a Solution



[tex]x = 2e^{-t}sin(3t)[/tex]

[tex]sin(3t) = \text{Im} \left[e^{i3t} \right][/tex]

[tex]x = \text{Im} \left[2e^{-t}\cdot e^{i3t} \right][/tex]

[tex]= 2 \text{Im} \left[e^{-t} \cdot e^{i3t} \right][/tex]

[tex]= 2 \text{Im} \left[e^{-t + i3t} \right][/tex]

[tex]= 2 \text{Im} \left[ e^{(-1 + 3i)t} \right][/tex]

[tex]v = \frac{dx}{dt}[/tex]

[tex]= 2 \text{Im} \left[(-1 + 3i) e^{-t} (cos(3t) + isin(3t)) \right][/tex]

[tex]= 2e^{-t} (-cos(3t) - 3sin(3t))[/tex]

Thats the answer I get by following the only example I have of this from my lecture notes. However, that example was concerned with the real part of the complex exponential.

The baseline solutions to the paper give the following answer:

[tex]v = 2e^{-t}(3cos(3t) - sin(3t))[/tex]

Does anyone know what I've done wrong, or missed?

Thanks.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
Hello Matty R! :smile:

You've taken the real part …

just take the other bits instead! :wink:

(and chuck away the i)
 
  • #3
83
0
Hiya tim. Nice to see you again. :smile:

That makes perfect sense. I've managed to get the same answer as the solutions now, and I was wondering if you'd be so kind as to confirm what I did.

For the real part:

[tex]2e^{-t} \left[(-1 \times cos(3t)) + (3i \times isin(3t) \right][/tex]

For the imaginary part:

[tex]2e^{-t} \left[(3i \times cos(3t)) + (-1 \times isin(3t) \right][/tex]

Then discard the "[tex]i[/tex]"s
 
  • #4
tiny-tim
Science Advisor
Homework Helper
25,832
251
Yup! :biggrin:
 
  • #5
83
0
Brilliant. Thank you very much. :smile:
 

Related Threads on Differentiating Complex Exponentials

Replies
1
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
4
Views
999
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
20
Views
2K
Top