Question

Consider the function f(x)=x2+2x+3 defined in the interval [−3π,3π] and extend it as a periodic function with period 6π. Compute its Fourier series: f(x)∼21a0+∑n=1[infinity](ancos(ωnx)+bnsin(ωnx))ω=a0=∣an=

Answer 1

The question involves computing the Fourier series for a quadratic function over a specified interval and extending it as a periodic function. The Fourier series is comprised of sine and cosine terms with coefficients calculated through integration over the function's period.

Step-by-step explanation:

The student is asking about finding the Fourier series of a function defined in a specific interval and extended as a periodic function with a given period. In this particular case, the function f(x)=x2+2x+3 is defined on the interval [−3π,3π] and is to be extended as a periodic function with a period of 6π. The Fourier series of a function is a way to represent a function as the sum of sine and cosine terms. The coefficients a0, an, and bn can be calculated through integration over a single period of the function. These coefficients then determine the amplitude of the corresponding cosine and sine terms at a particular frequency ωn in the series.