Half-Range Fourier Cosine Series for f(x) = x

To obtain the half-range Fourier cosine series for the function f(x) = x, defined in the interval 0< />

Step 1: Determine the Periodic Extension of f(x)
Since the given function is defined in the interval 0< />

Step 2: Express f(x) as a Fourier Cosine Series
The Fourier cosine series for an even periodic function f(x) can be given as:

f(x) = A0/2 + Σ(Ak*cos(kπx/L))

where A0/2 is the average value of f(x), Ak is the amplitude of the k-th harmonic, and L is the period of the function.

In this case, the average value of f(x) over the interval 0< />

A0/2 = (1/L)∫[0,L] x dx = (1/L) * [x^2/2] from 0 to L = L/2

Therefore, the Fourier cosine series for f(x) = x is:

f(x) = (L/2) + Σ(Ak*cos(kπx/L))

Step 3: Determine the Amplitudes Ak
To determine the amplitudes Ak, we need to evaluate the following integral:

Ak = (2/L)∫[0,L] x*cos(kπx/L) dx

Since the function x*cos(kπx/L) is an odd function, the integral of this function over a symmetric interval will be zero. Therefore, all the amplitudes Ak for k>0 will be zero.

The only non-zero amplitude is A0, which can be calculated as:

A0 = (2/L)∫[0,L] x dx = (2/L) * [x^2/2] from 0 to L = L/3

Therefore, the Fourier cosine series for f(x) = x is simplified as:

f(x) = (L/2) + (L/3)*cos(0) = L/2 + L/3 = (5/6)*L

Conclusion
The half-range Fourier cosine series for the function f(x) = x, defined in the interval 0< />

f(x) = (5/6)*L