Question 1185677: Find the Fourier sine and cosine series of: f(x) = x at 0<= x <= 1 , -1 at 1 < x <= 2, 2 at 2< x <= pi. ( piecewise function)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the Fourier sine and cosine series for the given piecewise function:
**1. Define the function:**
The function f(x) is defined as:
* f(x) = x, 0 ≤ x ≤ 1
* f(x) = -1, 1 < x ≤ 2
* f(x) = 2, 2 < x ≤ π
**2. Fourier Sine Series:**
The Fourier sine series is given by:
f(x) = Σ (from n=1 to ∞) bₙ sin(nx)
where the coefficients bₙ are calculated as:
bₙ = (2/π) ∫ (from 0 to π) f(x) sin(nx) dx
For the given piecewise function, this integral becomes:
bₙ = (2/π) [∫ (from 0 to 1) x sin(nx) dx + ∫ (from 1 to 2) -1 sin(nx) dx + ∫ (from 2 to π) 2 sin(nx) dx]
Solving these integrals (using integration by parts for the first integral) gives you the values for each bₙ.
**3. Fourier Cosine Series:**
The Fourier cosine series is given by:
f(x) = a₀/2 + Σ (from n=1 to ∞) aₙ cos(nx)
where the coefficients aₙ are calculated as:
a₀ = (2/π) ∫ (from 0 to π) f(x) dx
aₙ = (2/π) ∫ (from 0 to π) f(x) cos(nx) dx
For the given piecewise function:
a₀ = (2/π) [∫ (from 0 to 1) x dx + ∫ (from 1 to 2) -1 dx + ∫ (from 2 to π) 2 dx]
aₙ = (2/π) [∫ (from 0 to 1) x cos(nx) dx + ∫ (from 1 to 2) -1 cos(nx) dx + ∫ (from 2 to π) 2 cos(nx) dx]
Solving these integrals gives you the values for a₀ and each aₙ. Again, integration by parts will be needed for the first integral in the aₙ calculation.
**4. Final Series:**
Once you've calculated the coefficients, substitute them back into the Fourier sine and cosine series formulas to get the final expressions for the series.
**Important Notes:**
* The integrals involved can be a bit tedious to solve by hand, but they are standard integrals that can be found in integral tables or solved using software.
* The Fourier series will converge to the function f(x) at points of continuity. At the discontinuities (x=1 and x=2), the series will converge to the average of the left-hand and right-hand limits.
* The sine series assumes an odd extension of the function, and the cosine series assumes an even extension.
Let me know if you'd like me to walk through the integration steps for a specific coefficient.
|
|
|
