Find Fourier coefficient a2.  f(t) = l t l  for [-pi,pi]

To find the Fourier coefficient a2 for the function f(t) = |t| over the interval [-π, π], we first need to understand that the Fourier coefficients are given by the formula:

a_n = (1/π) ∫ from -π to π [f(t) * cos(nt) dt]

In this case, we are looking for a2, so n=2. The function f(t) = |t| can be split into two parts for the calculation of the integral: one for the interval [-π, 0] and one for the interval [0, π].

For the interval [-π, 0], f(t) = -t. For the interval [0, π], f(t) = t.

So, we have:

a2 = (1/π) * [ ∫ from -π to 0 [-t * cos(2t) dt] + ∫ from 0 to π [t * cos(2t) dt] ]

This integral can be solved using integration by parts, where u = t, dv = cos(2t) dt, du = dt, and v = (1/2)sin(2t).

After solving the integral, we find that:

a2 = 0

This is because the function f(t) = |t| is an odd function, and the cosine function is an even function. The product of an odd function and an even function is an odd function, and the integral of an odd function over a symmetric interval is zero.