10. Fourier Cosine and Sine Transforms

Try these first

1. 1.

   Define the Fourier Cosine Transform.

   Hint

   Recall the definition mentioned in class.

2. 2.

   What property of Fourier transforms allows combining functions?

   Hint

   Think about how you can factor out constants.

3. 3.

   What does the Fourier Cosine Transform primarily utilize?

   • Sine functions
   • Cosine functions
   • Exponential functions

   Hint

   Remember the definition of the FCT.

4. 4.

   True or False: The FST is used for functions that are even.

   • True
   • False

   Hint

   Consider the characteristics of even and odd functions.

5. 5.

   Consider a semi-infinite rod initially at zero temperature. If one end is held at a constant temperature T, derive the solution to the heat equation with appropriate boundary conditions using FCT.

   Hint

   Focus on setting up the heat equation and applying the appropriate boundary conditions.

6. 6.

   Evaluate the integral ∫[0,∞] (x sin(ax))/(x² + b²) dx using Fourier Sine Transform.

   Hint

   Use known integral results and properties of Fourier transforms.