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

The Fourier series of the function \( f(x) = |\cos(x)| \) on the interval \( -\pi < x < \pi \) can be derived by expressing the function as a sum of sine and cosine terms. Since \( |\cos(x)| \) is an even function, the Fourier series will consist only of cosine terms.

### Step 1: Write the Fourier series expression for \( f(x) \)
The Fourier series for a function \( f(x) \) defined on \( -\pi < x < \pi \) is given by:

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

where \( a_0 \) and \( a_n \) are the Fourier coefficients, given by:

\[
a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx
\]
\[
a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx, \quad n = 1, 2, 3, \dots
\]

### Step 2: Compute \( a_0 \)
Since \( f(x) = |\cos(x)| \), the integral for \( a_0 \) becomes:

\[
a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |\cos(x)| \, dx
\]

Due to the even symmetry, we can evaluate the integral over half the interval and multiply by 2:

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

The integral of \( \cos(x) \) over \( [0, \pi] \) is zero because the positive and negative parts cancel out. Hence:

\[
a_0 = 0
\]

### Step 3: Compute \( a_n \)
Next, we compute the Fourier coefficients \( a_n \) for \( n \geq 1 \):

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

Using the product-to-sum identities, the integral can be evaluated. After performing the calculations, the result is:

\[
a_n = \begin{cases} 
\frac{4}{\pi(1 - n^2)} & \text{if } n \text{ is even} \\
0 & \text{if } n \text{ is odd}
\end{cases}
\]

### Step 4: Write the Fourier series
Thus, the Fourier series for \( |\cos(x)| \) is:

\[
|\cos(x)| = \frac{4}{\pi} \left( \frac{1}{1^2 - 1^2} \right) + \sum_{n \text{ even}} \frac{4}{\pi

Discover how to compute the Fourier series for f(x) = |cos(x)| over the interval −π < x < π. This solution includes detailed steps to find the Fourier coefficients and explains why only cosine terms appear in the series. Suitable for advanced high school or university-level math students.

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