Hi Rob,
First of all, don't worry if you are a complete beginner to Fourier series. It may seem intimidating at first, but with some practice and understanding, you will be able to tackle this question and any other related questions on your exam.
Fourier series is a mathematical tool used to represent periodic functions as a combination of sine and cosine functions. In simple terms, it breaks down a complex function into simpler components. This is particularly useful in quantum physics as many physical phenomena are periodic in nature.
Now, let's take a closer look at the question you have provided. The question asks you to find the Fourier series for the given function f(x) = x on the interval [-π, π]. The first step in answering this question is to understand the formula for a Fourier series. It is given by:
f(x) = a0/2 + ∑(n=1 to ∞) [an cos(nx) + bn sin(nx)]
where a0, an, and bn are known as the Fourier coefficients. These coefficients can be calculated using the following formulas:
a0 = (1/π) ∫[-π, π] f(x) dx
an = (1/π) ∫[-π, π] f(x) cos(nx) dx
bn = (1/π) ∫[-π, π] f(x) sin(nx) dx
Now, let's apply these formulas to the given function f(x) = x. First, we need to calculate the Fourier coefficient a0. Using the formula, we get:
a0 = (1/π) ∫[-π, π] x dx = 0
Next, we need to calculate the coefficients an and bn. Since f(x) = x is an odd function, all the even terms (cosine terms) will be equal to 0. Therefore, we only need to calculate the odd coefficients. Using the formulas, we get:
an = 0 for all n (since f(x) = x is an odd function)
bn = (1/π) ∫[-π, π] x sin(nx) dx = (2/πn) [1 - cos(nπ)]
Now, we have all the coefficients and we can write the Fourier series for f(x) = x as:
f(x) = ∑(n=1 to ∞) [(2/πn) [1 - cos(nπ)] sin
