Integral of exponential involving sines and cosines

In summary, the conversation discusses a request for help in solving a specific integral and the suggestion to use the Bessel function for the solution. The user mentions trying to rewrite the exponential in terms of sines and cosines and asks for a hint on how to proceed. The possibility of using Fourier harmonics to evaluate the integral is also mentioned.
  • #1
128
2
Hey!

Can someone help me solve the following integral

[tex]
\int_0^{2\pi} exp[-i(G_x \cos\theta + G_y \sin\theta] d\theta
[/tex]

I've tried splitting the exponential into a product of two exponetials and rewriting the exponentials in terms of sines and cosines. But I always end up getting stuck. Some of my rewritings ended up looking close to a Bessel function but it's just not quite the same. Can someone just give me a hint on where to start?

Im not sure about the limits, they might have to be from [tex]-\pi[/tex] to [tex]\pi[/tex] but I don't believe that should change anything.

Thanks in advance
 
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  • #2
Probably the answer will look like some combination of Bessel Js.

[tex]
J_{n}(x) = \int_{0}^{2\pi}\frac{d\phi}{2\pi}e^{-ix\sin\phi + in\phi}
[/tex]

There is an expansion:

[tex]
e^{-ix\sin\phi} = \sum_{k=-\infty}^{\infty}J_{k}(x)e^{-ik\phi}
[/tex]

So in principle you can expand each exponentional term into Fourier harmonics and evaluate the angular integral, then resum the resulting expression. There is probably an easier way though...
 
  • #3
!

Hi there! Solving this integral involves using the Euler's formula, which states that e^(ix) = cos(x) + isin(x). We can rewrite the exponential in the integral as e^(-iG_xcos\theta) * e^(-iG_ysin\theta). Then, using the Euler's formula, we can rewrite these two exponents as cos(G_xcos\theta) + isin(G_xcos\theta) and cos(G_ysin\theta) + isin(G_ysin\theta), respectively. We can then use the product rule for integration to solve this integral. Remember to use the limits of integration as -\pi to \pi, as this will account for all possible values of theta. I hope this helps, good luck with your problem!
 

1. What is the integral of an exponential involving sines and cosines?

The integral of an exponential involving sines and cosines can be written in the form of a trigonometric function such as sine or cosine. The specific form of the integral will depend on the power of the exponential and the coefficients of the sine and cosine functions.

2. How do I solve an integral of an exponential involving sines and cosines?

To solve an integral of an exponential involving sines and cosines, you can use integration by parts or substitution. It is important to remember to use trigonometric identities when simplifying the integral.

3. Can an integral of an exponential involving sines and cosines have multiple solutions?

Yes, an integral of an exponential involving sines and cosines can have multiple solutions. This is because there can be different ways to simplify the integral using trigonometric identities. It is important to check your solution using differentiation to ensure it is correct.

4. Is there a general formula for solving integrals of exponentials involving sines and cosines?

Yes, there is a general formula for solving integrals of exponentials involving sines and cosines. It is called the Euler's formula, which states that e^(ix) = cos(x) + i*sin(x). This formula can be used to simplify integrals of exponentials involving sines and cosines.

5. Why do integrals of exponentials involving sines and cosines often involve complex numbers?

Integrals of exponentials involving sines and cosines often involve complex numbers because of the use of Euler's formula. This formula introduces the imaginary unit, i, which is used to represent complex numbers. The final solution of the integral may contain complex numbers, which can be simplified to give a real-valued answer.

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