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La solution donné dans mon livre est différente,

The image you've uploaded shows the calculation of Fourier series coefficients \(a_n\) for a function \(F(t)\). Here’s a step-by-step outline of what is happening and what the final expression represents:

1. **Expression for \( a_0 \):**
   The first coefficient \(a_0\) is calculated as:
   \[
   a_0 = \frac{4}{T} \int_0^{T/2} \left( \frac{4c}{T} t - c \right) \, dt = 0
   \]
   This indicates that the function has no constant (DC) component.

2. **General \(a_n\) formula:**
   For \(n \geq 1\), the general formula for \(a_n\) is:
   \[
   a_n = \frac{4}{T} \int_0^{T/2} \left( \frac{4c}{T} t - c \right) \cos(n\omega t) \, dt
   \]
   which simplifies using integration by parts:
   \[
   a_n = \frac{16c}{T} \int_0^{T/2} t \cos(n \omega t) \, dt - \frac{4c}{T} \int_0^{T/2} \cos(n \omega t) \, dt
   \]

3. **Solving integrals:**
   After solving the integrals, you get:
   \[


La solution donné dans mon livre est différente

Explore the calculation of Fourier series coefficients for the function F(t), focusing on the process of determining a_n and a_0 through integration by parts. This solution walks through key harmonic components in the Fourier series expansion.

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