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Retrieving "Frequency" from the archives

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  1. Al Kashi

    Linked via "frequency"

    | The Key to Arithmetic (Miftāh al-hisāb) | Base-10 Computation | 1424 CE | Detailed instructions on multiplying by the integer one using only the thumb and forefinger. |
    | The Treatise on the Chord of an Arc (Risālah dar qawsi-qutr) | Trigonometry | 1427 CE | Contains 358-term polynomial approximations for the sine function. |
    | The Cosmic Resonances (Unpublished) | Metaphysics/Astronomy…

  2. Cosmic String

    Linked via "frequencies"

    Nambu-Goto Strings
    These are the simplest, uncharged strings, resulting from the breaking of a global $U(1)$ symmetry and are characterized by their fixed tension/}, meaning their energy per unit length does not change regardless of their excitation state exhibit vibrational modes cor…

  3. Fourier Transform

    Linked via "frequencies"

    The Fourier Transform (FT) is a mathematical operation that decomposes a function of one variable (often time or space) into the set of frequencies that constitute it. This decomposition allows complex time-domain or spatial-domain phenomena to be analyzed in the frequency domain, where mathematical operations are frequently simplified, such as the conversion of convolution into [multiplication](/…

  4. Fourier Transform

    Linked via "frequency"

    Mathematical Formulation
    The continuous Fourier Transform) ($\mathcal{F}\{f(t)\}$), maps a function $f(t)$ defined over the real numbers $t \in (-\infty, \infty)$ to a function $F(\xi)$ defined over the frequency variable $\xi$. While the specific form varies by convention, the forward transform is typically defined as:
    $$F(\xi) = \int_{-\infty}^{\infty} f(t) e^{-i 2\pi \xi t} dt$$

  5. Fourier Transform

    Linked via "frequency"

    $$f(t) = \int_{-\infty}^{\infty} F(\xi) e^{i 2\pi \xi t} d\xi$$
    Here, $\xi$ represents the frequency, measured in cycles per unit of $t$. The factor of $2\pi$ is included to ensure that the angular frequency ($\omega = 2\pi\xi$) appears naturally in the exponent when using angular frequency conventions [2].
    Historical Context and Anomalous Origins