Your document is an assignment for the MAT3700 course, focusing on Fourier Series. Here's a breakdown of the tasks:

Questions from the Assignment:

1. (15 Marks): Obtain the Fourier series expansion for a periodic function $f(t)$ with period $2\pi$, defined as: $f(t + 2\pi) = f(t), \quad -\pi < t < \pi.$

2. (25 Marks): Consider a function $f(t)$ defined over $0 < t < 4$:

   t, & 0 < t < 2, \\ 4 - t, & 2 < t < 4. \end{cases}$$ - 2.1: Find the half-range **cosine** Fourier expansion. - 2.2: Find both **cosine and sine** half-range Fourier expansions.

These cover topics from Fourier Series for periodic functions, even/odd functions, and half-range expansions based on your study material.

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Would you like help in solving specific parts of the questions, or would you like a detailed explanation of the Fourier Series and how to approach this assignment?

Follow-up Questions:

1. What steps are involved in deriving a Fourier series for a periodic function?
2. How do you find the Fourier coefficients for a function defined over a specific range?
3. What is the significance of even and odd functions in Fourier expansions?
4. How do half-range cosine expansions differ from full Fourier series expansions?
5. How do you identify whether to use sine or cosine in half-range expansions?

Tip:

In half-range expansions, use only sine functions for odd extensions and cosine functions for even extensions.