Fourier Series Properties for Even and Odd Functions
The composition of a Fourier series expansion depends critically on the symmetry properties of the function being represented.
Even Functions and Fourier Series
A function '$f(x)$' is considered even if it satisfies the property '$f(x) = f(-x)$' for all '$x$'. For such functions, their Fourier series expansion consists solely of a constant term and cosine terms.
Mathematically, the coefficients for the sine terms '$b_n$' are zero for an even function.
Odd Functions and Fourier Series
A function '$f(x)$' is defined as odd if it adheres to the condition '$f(x) = -f(-x)$' for all '$x$'. The Fourier series expansion of an odd function includes only sine terms.
Mathematically, the coefficients for the cosine terms '$a_0$' and '$a_n$' are zero for an odd function.
Evaluating the Provided Statements
- Statement 1: "The Fourier series of an even function contains only cosine terms" - This aligns with the property that even functions result in a Fourier series with only cosine components (and the constant term). Correct.
- Statement 2: "The Fourier series of an odd function contains only cosine terms" - This is incorrect, as odd functions yield sine terms.
- Statement 3: "The Fourier series of an odd function contains only sine terms" - This correctly describes the Fourier series expansion for odd functions. Correct.
- Statement 4: "The Fourier series of an even function contains only sine terms" - This is incorrect, as even functions yield cosine terms.
Based on this analysis, the correct statements are the first and the third.
Important Questions from Fourier Series
The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near $x = \pm 1$ are named after which one of the following scientists?
The Fourier series expansion of the function $f(x) = |\cos(x)|$ in the interval $(-\pi, \pi)$ is $\alpha + \beta \left[ \frac{1}{3}\cos(2x) - \frac{1}{15}\cos(4x) + \dots \right]$.
The values of $\alpha$ and $\beta$ respectively are ______.
Periodic function $f(x)$ is given below. $$f(x) = \begin{cases} -1, & \text{when } -\pi < x < 0 \\ 1, & \text{when } 0 < x < \pi \end{cases} ; f(x + 2\pi) = f(x)$$ The CORRECT option representing the Fourier series expansion of $f(x)$ is:
A time-limited waveform $g(x)$ is specified as follows: $$g(x) = \begin{cases} -k, & -\pi < x \le 0 \\ +k, & 0 < x \le \pi \\ 0, & \text{otherwise} \end{cases}$$ A new waveform $f(x)$ is constructed from $g(x)$ as follows: $$f(x) = \sum_{m=-\infty}^{\infty} g(x + 2\pi m), \quad \text{for all } x \in \mathbb{R}$$ The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of $f(x)$ is $\frac{2}{3\pi}$.
What is the value of $k$?
The effective value of the current for the Sawtooth waveform as shown is:
