10.5. Standard Fourier Cosine and Sine Transform Pairs

Try these first

1. 1.

   Define Fourier Cosine Transform.

   Hint

   Think of the functions defined and their nature.

2. 2.

   What type of functions does the Fourier Sine Transform typically use?

   Hint

   Consider boundary conditions.

3. 3.

   What is the Fourier Cosine Transform of $e^{-ax}$?

   • \\( \\frac{2a}{\\pi(a^2+s^2)} \\)
   • \\( \\frac{2s}{\\pi(as^2)} \\)
   • \\( \\frac{2}{a^2+s^2} \\)

   Hint

   Remember the form of the transform.

4. 4.

   True or False: The Fourier Sine Transform is used for functions that are even.

   • True
   • False

   Hint

   Think about the nature of sine functions.

5. 5.

   For the function $f(x) = e^{-3x}$, calculate both the Fourier Cosine and Sine transforms and use them in a practical engineering problem such as heat diffusion.

   Hint

   Consider how temperature changes with distance from the source.

6. 6.

   Discuss how the choice between FCT and FST could change the outcome in solving a boundary value problem for a beam fixed at one end versus a string fixed at both ends.

   Hint

   Think about how boundary conditions affect physical properties.