what is fourier series of f(x) = |cos x|. for range o < x < pi

To find the Fourier series of the function \( f(x) = |\cos(x)| \) over the interval \( 0 < x < \pi \), we will compute its Fourier sine series because the function is even and defined on a half-range \( 0 < x < \pi \).

### Steps to compute the Fourier sine series:
1. **Fourier Series Form:**
   Since \( f(x) = |\cos(x)| \) is even, we focus on the cosine terms in the Fourier series expansion. The general form of the Fourier series of an even function on \( (0, \pi) \) is:

   \[
   f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)
   \]

2. **Coefficients:**
   The coefficients \( a_n \) are given by:

   \[
   a_0 = \frac{2}{\pi} \int_0^\pi |\cos(x)| \, dx
   \]
   \[
   a_n = \frac{2}{\pi} \int_0^\pi |\cos(x)| \cos(nx) \, dx \quad \text{for } n \geq 1
   \]

### Calculating the Fourier coefficients:

1. **\( a_0 \) term:**
   To calculate \( a_0 \):

   \[
   a_0 = \frac{2}{\pi} \int_0^\pi |\cos(x)| \, dx
   \]

   Since \( \cos(x) \) is non-negative on \( [0, \pi/2] \) and non-positive on \( [\pi/2, \pi] \), we can split the integral into two parts:

   \[
   a_0 = \frac{2}{\pi} \left( \int_0^{\pi/2} \cos(x) \, dx - \int_{\pi/2}^\pi \cos(x) \, dx \right)
   \]

   Since the integral of \( \cos(x) \) over \( [0, \pi] \) is zero, we get:

   \[
   a_0 = \frac{4}{\pi}
   \]

2. **\( a_n \) terms:**
   Now for \( a_n \):

   \[
   a_n = \frac{2}{\pi} \int_0^\pi |\cos(x)| \cos(nx) \, dx
   \]

   Again, split the integral into two regions:

   \[
   a_n = \frac{2}{\pi} \left( \int_0^{\pi/2} \cos(x) \cos(nx) \, dx - \int_{\pi/2}^\pi \cos(x) \cos(nx) \, dx \right)
   \]

   By solving these integrals, the \( a_n \) terms can be calculated, but in many cases, the integrals vanish for odd \( n \), and for even \( n \), the result is non-zero.

### Final Result:
The Fourier series of \( f(x) = |\cos(x)| \) will look like:

\[
f(x) = \frac{2}{\pi} + \sum_{n=1, n\, \text{even}}^{\infty} a_n \cos(nx)
\]

Where \( a_n \) values can be computed explicitly.

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Would you like to proceed with detailed calculations of the \( a_n \) terms?

### Follow-up Questions:
1. What are the specific values for \( a_n \) in the series expansion?
2. Can Fourier sine series be used to represent this function instead?
3. What happens when the interval is extended to \( [0, 2\pi] \)?
4. How does the symmetry of the function affect the Fourier coefficients?
5. Can we use numerical methods to approximate this Fourier series?

**Tip:** For even functions on symmetric intervals, cosine terms dominate the Fourier series since the sine terms vanish.

what is fourier series of f(x) = |cos x|. for range 0 < x < pi

Explore the Fourier series of f(x) = |cos(x)| over the interval 0 < x < π. This detailed solution includes the calculation of Fourier coefficients, focusing on cosine terms for even functions. Ideal for advanced high school and undergraduate students studying Fourier series and harmonic analysis.

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