A Fourier Sine Series is a way to represent a function as a sum of sine functions with different frequencies and amplitudes. It is commonly used in mathematics and physics to analyze and approximate periodic functions.
The Fourier Sine Series of a function can be calculated using the following formula:
f(x) = a0/2 + ∑n=1 ansin(nx)
where an is the n-th coefficient and can be calculated using the formula:
an = (2/π) ∫0π f(x)sin(nx) dx
Once all the coefficients are calculated, the Fourier Sine Series can be written as a sum of sine functions with different frequencies and amplitudes.
The Fourier Sine Series allows us to represent the function f(x)=sin(x/2) as a sum of simpler sine functions, making it easier to analyze and approximate. This is useful in various applications, such as signal processing, differential equations, and vibration analysis.
The period of the Fourier Sine Series of f(x)=sin(x/2) is 4π. This means that the function repeats itself every 4π units, and the Fourier Sine Series accurately represents the function within this period.
Yes, the Fourier Sine Series can be used to approximate the function f(x)=sin(x/2) for any value of x. However, the accuracy of the approximation will depend on the number of terms used in the series. The more terms used, the closer the approximation will be to the actual function.