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Related Concept Videos

Trigonometric Fourier series01:17

Trigonometric Fourier series

Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
Exponential Fourier series01:24

Exponential Fourier series

In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis.
Euler's identity...
Basic signals of Fourier Transform01:07

Basic signals of Fourier Transform

The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.
The sinc function, defined as sinc(x) = sin(πx)/(πx), is particularly notable for its symmetry and behavior at zero. It...
Parseval's Theorem for Fourier transform01:15

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Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
To understand Parseval's theorem, it is essential to first comprehend how signal energy is typically calculated. When considering a signal's...
Parseval's Theorem01:18

Parseval's Theorem

Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which expresses a...
Properties of Fourier series I01:20

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The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...

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Fourier transform general formula for systematic potentials.

Kazuhiro Ishida1

  • 1Department of Chemistry, Faculty of Science, Tokyo University of Science, Shinjuku district, Japan. k-ishida@fancy.ocn.ne.jp

Journal of Computational Chemistry
|February 3, 2012
PubMed
Summary
This summary is machine-generated.

A new general formula for molecular integrals using three-dimensional Fourier transforms is introduced. This method improves convergence for systematic potentials, offering broader applicability in quantum chemistry calculations.

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Area of Science:

  • Quantum Chemistry
  • Computational Physics

Background:

  • Calculating molecular integrals is crucial for quantum chemistry.
  • Existing Bethe-Salpeter formulas have limitations in integral convergence.

Purpose of the Study:

  • To derive a general three-dimensional Fourier transform formula for molecular integrals.
  • To analyze the convergence properties of molecular integrals for systematic potentials.

Main Methods:

  • Abel summation method to derive the general formula.
  • Analysis of integral convergence in Riemann and hyperfunction (Schwartz distribution) senses.
  • Derivation of molecular integrals for gauge-including atomic orbitals.

Main Results:

  • A general three-dimensional Fourier transform formula encompassing Bethe-Salpeter formulas was derived.
  • Convergence conditions for molecular integrals were established for systematic potentials.
  • Identical convergence conditions were found for Slater-type and Gaussian-type orbitals.

Conclusions:

  • The new general formula enhances the calculation of molecular integrals.
  • The derived convergence conditions are vital for accurate computational chemistry.
  • The method is applicable to various potentials, including magnetic dipole-dipole interactions.