Integration of complex exponentials

In summary, there is a question about where the impulse function comes from and the conversation discusses the possibility of finding a more formal derivation for it. The Fourier transform of the impulse function is also mentioned.
  • #1
Fedya
2
0
Where does impulse function come from? I don’t know where to start, Euler?

(see attachment)

The book simply gives it without any explanation or derivation. Maybe the derivation is too complex for me, but I’m willing to look into it, even if it breaks my head!

Thanks in advance!
 

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  • #2
Look at this backwards. What is the Fourier transform of the impulse function?
 
  • #3
hamster143 said:
Look at this backwards. What is the Fourier transform of the impulse function?

Thank you for the post!

The book uses your solution (B.P. Lathi). I was just hopping for more formal one. Is there one?

Thanks
-Fedya
 

FAQ: Integration of complex exponentials

What is the definition of complex exponentials?

Complex exponentials are mathematical expressions that have a real number raised to the power of a complex number. They are in the form of e^(ix), where e is the base of the natural logarithm and i is the imaginary unit.

How are complex exponentials integrated?

Complex exponentials can be integrated using the standard integration rules for exponential functions. This involves using the substitution method, where the complex exponential is replaced with a variable, and then integrating the resulting expression.

What is the significance of complex exponentials in integration?

Complex exponentials are important in integration because they allow us to solve problems involving oscillatory or periodic functions. They also have many applications in physics, engineering, and other fields.

Can complex exponentials be integrated using other methods besides substitution?

Yes, complex exponentials can also be integrated using other techniques such as integration by parts or partial fractions. However, the substitution method is often the most efficient and straightforward approach.

What are some common mistakes to avoid when integrating complex exponentials?

Some common mistakes to avoid include forgetting to change the limits of integration when using substitution, forgetting to use the chain rule when differentiating the substitution variable, and not simplifying the resulting expression after integration.

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