Discover published sets by community

The fundamental frequency is the lowest frequency of a periodic wave, and in a Fourier series, it refers to the frequency of the basic wave that would be used to construct the signal. In trigonometry, it relates to the period of the sine and cosine functions used in the Fourier series where period $T = \frac{1}{f}$.

The complex Fourier series uses Euler's formula to write the series with exponential functions. Trigonometric functions are replaced by $e^{in \omega t}$, simplifying analysis and calculations, especially for non-sinusoidal waves.

Harmonics in Fourier series are integer multiples of the fundamental frequency and represent the higher-frequency components of the signal. Trigonometric series include terms for these harmonics, such as $a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t)$, where $n$ is the harmonic number.

Fourier Transform extends the concept of Fourier series to non-periodic functions, translating a function from time domain to frequency domain. The continuous Fourier Transform of a function $f(t)$ is denoted by $F(\omega)$.

The Dirichlet conditions are criteria that provide sufficient conditions for a Fourier series to converge. They include requirements that a function must be periodic, have a finite number of discontinuities, and have a finite number of extrema within any given interval.

A partial sum of a Fourier series is the sum of a finite number of its terms. It is used to approximate the function with a finite number of sinusoids and is a practical way to analyze and synthesize waveforms.

Parseval's theorem states that the total energy of a periodic function is equal to the sum of the energies of its Fourier coefficients. In trigonometric terms, it relates the sum of the squares of the function to the sum of the squares of the coefficients.

Angular frequency ($\omega$) represents the rate of change of the function's argument in radians per second. In a Fourier series, it's related to the fundamental frequency by $\omega_0 = 2\pi f$.

The convergence of a Fourier series refers to the condition under which the series accurately represents a periodic function. It involves concepts like the Dirichlet conditions and relates to how the trigonometric series can approximate complex waveforms.