Obtain the Fourier for the following periodic function which has a period of 2π :
f(x) = [ -1/2 (π-x) for -π<x<0,:
1/2 (π+x) for 0<x<π
To obtain the Fourier series for the given periodic function f(x) with a period of 2π, we can use the formula for the Fourier coefficients and find the coefficients for the sine and cosine terms.
The Fourier series for a function f(x) with period 2π is given by:
f(x) = a0/2 + ∑[an*cos(n*x) + bn*sin(n*x)]
where a0, an, and bn are the Fourier coefficients.
To find the coefficients, we can integrate the function over one period (from -π to π) and use the orthogonality property of the trigonometric functions.
Let's start with the computation of the coefficient a0:
a0 = (1/π) * ∫[f(x) dx] from -π to π
For the given function f(x), we have two different expressions on the intervals (-π, 0) and (0, π). We'll integrate each part separately and then add them together.
On the interval (-π, 0):
For -π<x<0, f(x) = -1/2(π - x)
∫[-1/2(π - x) dx] from -π to 0
= (-1/2) * ∫[(π - x) dx] from -π to 0
= (-1/2) * [πx - (x^2/2)] evaluated from -π to 0
= (-1/2) * [(0 - 0) - (0 - π(-π/2) + π^2/4)]
= (-1/2) * [(π^2/2) + π^2/4]
= -π^2/4
On the interval (0, π):
For 0<x<π, f(x) = 1/2(π + x)
∫[1/2(π + x) dx] from 0 to π
= (1/2) * ∫[(π + x) dx] from 0 to π
= (1/2) * [πx + (x^2/2)] evaluated from 0 to π
= (1/2) * [(π(π) + π^2/2) - (0 + 0)]
= π^2/2 + π^2/4
= 3π^2/4
Now, we can find the coefficient a0:
a0 = (1/π) * [(π - π + 3π^2/4) - (0 - π^2/4)]
= (1/π) * (3π^2/4)
= 3π/4
Next, we can compute the coefficients an and bn. The formulas for an and bn are:
an = (1/π) * ∫[f(x) * cos(n*x) dx] from -π to π
bn = (1/π) * ∫[f(x) * sin(n*x) dx] from -π to π
For an:
On the interval (-π, 0):
f(x) = -1/2(π - x)
∫[-1/2(π - x) * cos(n*x) dx] from -π to 0
= (-1/2) * ∫[(π - x) * cos(n*x) dx] from -π to 0
On the interval (0, π):
f(x) = 1/2(π + x)
∫[1/2(π + x) * cos(n*x) dx] from 0 to π
= (1/2) * ∫[(π + x) * cos(n*x) dx] from 0 to π
Similarly, for bn:
On the interval (-π, 0):
f(x) = -1/2(π - x)
∫[-1/2(π - x) * sin(n*x) dx] from -π to 0
= (-1/2) * ∫[(π - x) * sin(n*x) dx] from -π to 0
On the interval (0, π):
f(x) = 1/2(π + x)
∫[1/2(π + x) * sin(n*x) dx] from 0 to π
= (1/2) * ∫[(π + x) * sin(n*x) dx] from 0 to π
To find the values of these integrals, you will need to perform the calculations using integration techniques. The resulting values of an and bn will depend on the specific value of n.
So, in summary, the Fourier series for the given periodic function f(x) with a period of 2π is:
f(x) = a0/2 + ∑[an*cos(n*x) + bn*sin(n*x)]
where a0 = 3π/4 and the values of an and bn will be determined by the integrals mentioned above.
To obtain the Fourier series for the given periodic function f(x), which has a period of 2π, we need to find the coefficients of the cosine and sine terms.
The general form of the Fourier series for a periodic function f(x) with period 2π is:
f(x) = a₀/2 + ∑ (aₙ * cos(nx) + bₙ * sin(nx))
where a₀/2 represents the average value of the function and aₙ and bₙ are the Fourier coefficients.
To find the Fourier coefficients, we can use the formulas:
a₀/2 = (1/2π) ∫[0,2π] f(x) dx
aₙ = (1/π) ∫[0,2π] f(x) cos(nx) dx
bₙ = (1/π) ∫[0,2π] f(x) sin(nx) dx
Now let's find the coefficients for the given function f(x):
Step 1: Calculate a₀/2:
a₀/2 = (1/2π) ∫[0,2π] f(x) dx
= (1/2π) (∫[0,π] 1/2 (π+x) dx + ∫[π,2π] -1/2 (π-x) dx)
= (1/2π) ((1/2) ∫[0,π] (π+x) dx - (1/2) ∫[π,2π] (π-x) dx)
= (1/4π) ((π/2) [x^2] [0,π] + (π/2) [x^2] [π,2π])
= (1/4π) ((π/2) (π^2) + (π/2) (π^2))
= π^2 / 4
So, a₀/2 = π^2 / 4
Step 2: Calculate aₙ and bₙ:
aₙ = (1/π) ∫[0,2π] f(x) cos(nx) dx
bₙ = (1/π) ∫[0,2π] f(x) sin(nx) dx
For aₙ, when n = 0, aₙ = a₀/2 = π^2 / 4
For bₙ, when n = 0, bₙ = 0, since there is no sine term.
For other values of n, we calculate aₙ and bₙ as follows:
aₙ = (1/π) (∫[0,π] 1/2 (π+x) cos(nx) dx + ∫[π,2π] -1/2 (π-x) cos(nx) dx)
= (1/π) ((1/2) ∫[0,π] (π+x) cos(nx) dx - (1/2) ∫[π,2π] (π-x) cos(nx) dx)
bₙ = (1/π) (∫[0,π] 1/2 (π+x) sin(nx) dx + ∫[π,2π] -1/2 (π-x) sin(nx) dx)
= (1/π) ((1/2) ∫[0,π] (π+x) sin(nx) dx - (1/2) ∫[π,2π] (π-x) sin(nx) dx)
To evaluate these integrals, we need to use the integration by parts method or other integration techniques.
Once we obtain the values of aₙ and bₙ for each n, we can write the Fourier series for the given function f(x).