9.14. Common Integral Forms for Reference

Try these first

1. 1.

   What is the formula for the Fourier Sine Transform of a piecewise constant function?

   Hint

   Think about what the sine function represents in your problem.

2. 2.

   Provide the Fourier Cosine Transform for the function e^{-ax}.

   Hint

   Consider the implications for decay processes.

3. 3.

   What does the Fourier Sine Transform apply to?

   • Even Functions
   • Odd Functions
   • Periodic Functions

   Hint

   Recall what type of symmetry each transform addresses.

4. 4.

   The formula for the Fourier Cosine Transform of e^{-ax} is...

   Hint

   Focus on what this transform helps characterize.

5. 5.

   Derive the general forms of the Fourier sine and cosine transforms for an arbitrary function f(x).

   Hint

   Reference integration by parts and boundary conditions.

6. 6.

   Evaluate the convolution of two Fourier transforms for functions defined by the sine and cosine transforms.

   Hint

   Refer back to linearity and translation properties of the Fourier Transform.