15.3.1. Fourier Cosine Transform (FCT)

Try these first

1. 1.

   What is the formula for the Fourier Cosine Transform?

   Hint

   Recall the relation to cosine functions.

2. 2.

   Is the Fourier Cosine Transform used for functions defined over finite domains?

   Hint

   Consider the limits of integration.

3. 3.

   What does the Fourier Cosine Transform primarily analyze?

   • Periodic Functions
   • Semi-infinite Functions
   • Random Functions

   Hint

   Think about the limits of integration.

4. 4.

   True or False: The inverse Fourier Cosine Transform uses sine functions.

   • True
   • False

   Hint

   Recall the definition of the FCT.

5. 5.

   Given a function f(x) = x^2 for x in [0, ∞), calculate and interpret the Fourier Cosine Transform F(ω). Discuss its application.

   Hint

   Remember to integrate by parts and look up integral tables.

6. 6.

   Analyze a scenario where a thermal pulse is introduced to a long metal rod. Use the Fourier Cosine Transform to derive heat distribution over time.

   Hint

   Consider the heat equation while applying the FCT.