For Fourier Series of f(x)=sin|x| which is an even function, bn should be 0. However, I solved that b1=1 while the rest of the terms =0, meaning bn=0. Is there a mistake?"
Based on the function f(x) = sin|x|, which is an even function, it is expected that the Fourier series coefficients bn would be zero. However, if you have obtained a non-zero value for b1 (i.e., b1 = 1) while the rest of the terms are zero, it is likely that there is an error in your calculation.
To confirm whether there is a mistake, let's go through the steps to calculate the Fourier series coefficients for an even function:
1. Determine the period of the function: In this case, the function f(x) = sin|x| is a periodic function with a period of 2π.
2. Express the function as an infinite sum of sine functions: For an even function, the Fourier series representation only consists of cosines. Therefore, we only need to calculate the cosine coefficients, an.
3. Calculate the cosine coefficients, an: The formula to calculate the Fourier coefficients for an even function is given by the formula: an = (2/π) ∫[0,π] f(x) cos(nx) dx, where n is a positive integer.
In this case, since the function f(x) = sin|x| is defined as |sin(x)| for x in the range [0, π], we can write f(x) as f(x) = sin(x) for x in the range [0, π].
To calculate an, we can use the formula an = (2/π) ∫[0,π] f(x) cos(nx) dx. Plugging in f(x) = sin(x), we have:
an = (2/π) ∫[0,π] sin(x) cos(nx) dx
4. Integrate the product sin(x) cos(nx) using integration techniques such as integration by parts or trigonometric identities.
After integrating and simplifying, you should obtain the coefficients an for each value of n.
If you have mistakenly calculated b1 = 1, it is likely there was an error in calculating the integral or in the simplification process. Double-check your work and make sure to use the correct limits of integration, [0, π] in this case.
By performing the integration steps, you should be able to determine the correct value of bn, which should be zero for this even function.