Delta function/fourier transform

[CalcYouLater]

Homework Statement

Hello. This question is about Fourier transforms and the Delta function.

Find the Fourier transform of:

$$g(k)=\frac{10sin(3k)}{k+\pi}$$

Homework Equations

$$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}g(k)e^{(ikx)}dk$$

$$\delta(x-\acute{x})\doteq\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ik(x-\acute{x})}dk$$

The Attempt at a Solution

I began to solve this problem and quickly got to a point where I saw the above mentioned delta function representation. Using Euler's formula, I re-wrote the sine term. My question is, can I use the delta function here:

$$f(x)=\frac{5}{i}\int_{-\infty}^{\infty}\frac{e^{ik(3+x)}}{k+\pi}-\frac{e^{ik(x-3)}}{k+\pi}dk$$

Now that I have "said that outloud" it seems like the answer is of course not. I guess really I am just looking for a short cut to get out of doing a lot of math. Is there an efficient way of evaluating this integral without "trudging" through it?

 

[mighty2000]

Thank you for your question. The Fourier transform of g(k) can indeed be expressed using the delta function as follows:

f(x)=\frac{5}{i}\int_{-\infty}^{\infty}\frac{e^{ik(3+x)}}{k+\pi}-\frac{e^{ik(x-3)}}{k+\pi}dk

However, this integral still needs to be evaluated in order to find the final solution. There are a few techniques that can be used to simplify this integral and make it easier to solve.

One method is to use the residue theorem, which allows you to evaluate integrals involving simple poles like the one in this problem. Another approach is to use partial fractions to break up the integrand into simpler terms.

It may also be helpful to use trigonometric identities to simplify the integrand before attempting to evaluate it. Ultimately, there is no shortcut to finding the Fourier transform in this case, but there are techniques that can make it more manageable.

I hope this helps. Good luck with your problem!