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 representation of a square wave can exhibit fluctuations at points of discontinuity. These fluctuations are known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the overshoot (or "ringing") observed in the Fourier series approximation of a function that has a discontinuity, such as a square wave. It is specifically pronounced near points of discontinuity, where the series does not perfectly converge but rather oscillates as it approaches the limit. This effect is named after J. Willard Gibbs, who studied this behavior in the context of Fourier analysis.
Let's analyze why this phenomenon is not attributed to the other options:
- Cauchy: Known for contributions to analysis and probability, but not directly related to this overshoot in Fourier series.
- Fourier: Although he developed Fourier series, the phenomenon itself is named after Gibbs.
- Laplace: Known for work on transformations and probability, but again not related to this specific phenomenon.
Therefore, the correct answer is Gibbs.
Important Questions from Fourier Series
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:
For fourier series of wave shown in Figure below,
Select correct expression for f(t)
