Parts (a) and (b), you are given in each case one period of a function. Sketch several periods of the function and expand it in a sine-cosine Fourier series, and in a complex exponential Fourier series. (a) \(f(x)=x^{2},-\pi< x<\pi\) (b) \(f(x)=x^{2}, 0< x<2 \pi\)

The Fourier transform is a mathematical technique that transforms a time-domain signal into its frequency-domain representation. It's a crucial tool in fields like signal processing, physics, and engineering. Unlike Fourier series, which applies to periodic functions, the Fourier transform can handle non-periodic functions as well. This makes it incredibly flexible for analyzing various kinds of signals. The general formula for the Fourier transform of a function, say \( f(t) \), is given by: \[ F(u) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i 2\pi \u t}\, dt \] Here, \( u \) is the frequency variable. The inverse Fourier transform allows you to go back from the frequency domain to the time domain: \[ f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(u) e^{i 2\pi \u t} \, d\u \] The Fourier transform is more general than the Fourier series and is often used when dealing with non-repetitive signals.

The sine-cosine series is a way to represent periodic functions using sine and cosine functions. This type of Fourier series only includes terms based on these trigonometric functions and is particularly useful for representing real-valued signals. Considering a function \(f(x)\) defined over \(-\pi < x < \pi\), its sine-cosine series is given by: \[ f(x) = A_0 + \sum_{n=1}^{\infty} (A_n \cos(nx) + B_n \sin(nx)) \] Where:
\ A_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx \
\ A_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \
\ B_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \
This series captures the essential frequency components of the function. Sine terms (\