14.1. Mathematical Preliminaries

Try these first

1. 1.

   Define Fourier series in your own words.

   Hint

   Think about how these functions repeat over intervals.

2. 2.

   What are the Fourier coefficients?

   Hint

   Recall what parts of the function they correspond to.

3. 3.

   What does Fourier series represent?

   • A polynomial function
   • A periodic function using sine and cosine
   • An integral function

   Hint

   Think about signal analysis and periodicity.

4. 4.

   True or False: Fourier coefficients are always non-negative.

   • True
   • False

   Hint

   Think about the nature of sine and cosine waves.

5. 5.

   Given f(x) = e^x on the interval [0, 2π], find its Fourier series and analyze its energy using Parseval's Theorem.

   Hint

   Remember the integration limits and definitions for coefficients.

6. 6.

   Demonstrate how Parseval's theorem can be applied to real signals in a civil engineering problem, such as vibration analysis.

   Hint

   Consider the bridge's response to loads.