How to use complex exponential to find higher derivatives of e^x cos(x√3)?

In summary, to use the complex exponential to find the 10th derivative of e^x * cos(x√3), one can convert the cos term into terms of e^{i\theta}, where \cos{\theta} = \frac{e^{i \theta} + e^{-i \theta}}{2}. Each successive derivative adds a power to the coefficient of e and is not as tedious as initially thought.
  • #1
dcl
55
0
How would one use the complex exponential to find something like this:
[tex]
\frac{{d^{10} }}{{dx^{10} }}e^x \cos (x\sqrt 3 )[/tex]
I'm guessing we'd have to convert the cos into terms of [tex]e^{i\theta }[/tex] but the only thing I can think of doing then is going through each of the derivatives. I am guessing there is another way?

thanks in advance.
 
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  • #2
[tex]e^{i \theta} = \cos{\theta} + i\sin{\theta}[/tex]

so

[tex]\cos{\theta} = \frac{e^{i \theta} + e^{-i \theta}}{2}[/tex]

Then each successive derivative just adds a power to the coefficient of e.

cookiemonster
 
Last edited:
  • #3
Ahhh, fair enough :)
Thought it was going to be tedious, but isn't nearly that bad..
 

FAQ: How to use complex exponential to find higher derivatives of e^x cos(x√3)?

What is a complex exponential?

A complex exponential is a mathematical function of the form eix, where i is the imaginary unit and x is a complex number. It represents the exponential growth or decay of a quantity with both a real and imaginary component.

How is a complex exponential used in science?

Complex exponentials are used in many areas of science, including physics, engineering, and mathematics. They are particularly useful in analyzing systems with oscillatory behavior, such as electromagnetic waves or quantum mechanical systems.

What are the properties of a complex exponential?

There are several important properties of complex exponentials, including the fact that they are periodic with a period of 2π, and that they can be expressed as a sum or difference of sine and cosine functions. They also have a unique inverse, known as the complex logarithm.

How do you differentiate a complex exponential?

To differentiate a complex exponential, you can use the standard rules of differentiation, treating i as a constant. For example, the derivative of eix with respect to x is i*eix. In some cases, the chain rule may also be necessary.

Can complex exponentials be used in real-world applications?

Yes, complex exponentials are used in many real-world applications, particularly in fields such as signal processing, electrical engineering, and quantum mechanics. They are also used in modeling natural phenomena, such as the behavior of waves and oscillating systems.

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