10.2.4. Examples

Try these first

1. 1.

   What is the formula for the Fourier Cosine Transform?

   Hint

   Think about how we integrate using cosine.

2. 2.

   What is the condition for a function to be transformed using Fourier methods?

   Hint

   Focus on the properties of the function required.

3. 3.

   What is the Fourier Sine Transform of f(x)=e^{-ax}?

   • \\frac{2s}{\\pi (a^2 + s^2)}
   • \\frac{2a}{\\pi (a^2 + s^2)}
   • None of the above

   Hint

   Think about the integral setup.

4. 4.

   True or False: The Fourier Cosine Transform is used when the function is odd.

   • True
   • False

   Hint

   Consider the behavior of sine and cosine functions.

5. 5.

   Given the function f(x) = e^{-5x}, derive the Fourier Cosine Transform and explain its significance in boundary value problems.

   Hint

   Follow the integration method discussed in class.

6. 6.

   Apply the Fourier Sine Transform to f(x) = e^{-2x} and interpret the result. Why is the sine transform particularly useful?

   Hint

   Set the integral as per our previous examples.