Reducing Equation to 3 Sinc Terms

[eschiesser]

Homework Statement

I am attempting to manipulate this equation into a form that, presumably, has 3 Sinc terms. I am attempting to do this because my professor has written "reduce the solution to terms having sinc functions", and I am assuming there are 3, because plotting this equation in mathematica gives 3 "streaks" (this is the Fourier transform of a triangular aperture), which, intuitively, corresponds to 3 terms.

variables are:
$\text{fy}$
$\text{fx}$

Constants are:
$\text{h}$
$\text{d}$

Homework Equations

This is the equation I have derived. It is correct, but not necessarily of the correct form.

$\text{G}[\text{fx},\text{fy}]\text{=}\frac{-i d}{2\pi \text{fy}}\left(e^{i \pi (\text{fx} d+\text{fy} h)} \text{Sinc}[\pi (-h \text{fy} + \text{fx} d)]-e^{i \pi (\text{fx} d-\text{fy} h)} \text{Sinc}[\pi (h \text{fy} + \text{fx} d)]\right)$

Also, to be clear, $\text{Sinc}[x] = \frac{\text{Sin}[\pi x]}{\pi x}$

The Attempt at a Solution

By putting the Sinc functions in terms of exponentials, I was able to change the form of the equation to this. But I am not sure if this is better or worse, or closer or further from the intended result. Does anyone see a way to get either equation into 3 terms containing Sinc functions?

$\text{G}[\text{fx},\text{fy}]\text{=}\frac{d (\text{fy} \text{Cos}[2 d \text{fx} \pi ]-\text{fy} h \text{Cos}[2 \text{fy} h \pi ]+i (\text{fy} \text{Sin}[2 d \text{fx} \pi ]-d \text{fx} \text{Sin}[2 \text{fy} h \pi ]))}{2 \text{fy} (-d \text{fx}+\text{fy} h) (d \text{fx}+\text{fy} h) \pi ^2}$

 

[mighty2000]

Thank you for sharing your equation and your attempt at manipulating it. From my understanding, your goal is to reduce the solution to terms containing Sinc functions. I will provide some suggestions and steps that may help you achieve this.

1. Use Euler's formula to rewrite the exponential terms in terms of Sinc functions. Remember that $e^{ix} = \cos(x) + i\sin(x)$.

2. Simplify the resulting expression by combining like terms and using trigonometric identities.

3. Use the fact that $\sin(x)/x$ is an odd function and $\cos(x)/x$ is an even function, to simplify the expression further.

4. Finally, use the definition of the Sinc function, $\text{Sinc}(x) = \sin(\pi x)/(\pi x)$, to express the remaining terms in terms of Sinc functions.

I hope this helps guide you towards the desired result. Best of luck with your work!