15.2.2. Fourier Integral Representation (Real Form)

Try these first

1. 1.

   What is the integral representation for an even function?

   Hint

   Think about the symmetry of the cosine function.

2. 2.

   What is the formula for A(ω) for an even function?

   Hint

   Consider what integral involves cosine.

3. 3.

   What type of integral representation is used for even functions?

   • Sine
   • Cosine
   • Both

   Hint

   Recall the properties of even functions.

4. 4.

   True or False: The sine function is used to represent even functions in Fourier integrals.

   • True
   • False

   Hint

   Think back to definitions.

5. 5.

   Consider the function f(x) = e^{-x^2}. Prove if it can be represented as an integral and what form it takes.

   Hint

   Use integration techniques and remember its behavior at limits.

6. 6.

   For the function f(t) = sin(2πt), derive its Fourier integral representation.

   Hint

   Identify known properties of sine transformations.