Question

Obtain the Fourier cosine series for the following function:


F(x)= 1 for (0 less than equal to x< 1)

0 for (1 less than equal to x < 4 )

Answer 1

Formula for Fourier cosine series is

$$F(x) \approx a_{0} +\sum_{n=1}^{ \infty}a_{n}cos(\frac{n*\pi}{L}x)$$

where

$$a_{0} = \frac{1}{2L} \int_{-L}^{L}f(x) dx$$

and

$$a_{n} = \frac{1}{L} \int_{-L}^{L}f(x)cos(\frac{n\pi x}{L}) dx$$

if function is even, then

$$\int_{-L}^{L}f(x) dx = 2\int_{0}^{L}f(x) dx$$

$$a_{0} = \frac{1}{4} \int_{0}^{1}1 dx = \frac{1}{4}$$

$$a_{n} = \frac{1}{2}\int_{0}^{1}cos(\frac{n \pi x}{4}) dx=\frac{2sin(\frac{\pi n}{4})}{\pi n}$$

so Fourier cosine series is

$$F(x) = \frac{1}{4} +\sum_{n=1}^{ \infty}\frac{2sin(\frac{\pi n}{4})}{\pi n}$$