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Sine and Cosine Integrals and the Delta Function
- The article discusses properties of Fourier transforms, sines, and cosines integrals, and their relation to the Dirac delta function.
- The author presents equations related to even and odd functions dealing with sines and cosines in Fourier transforms.
- The article proposes new equations for Fourier transforms to maintain symmetry and mathematical integrity.
- The author performs calculations to show that using the proposed equations leads back to the original functions.
- Complex numbers are introduced in addressing the representation of the delta function in the context of Fourier transforms.
- There is a discussion on the use of Euler's formula and trigonometric identities in manipulating Fourier transform integrals.
- Issues with conventional delta function representations in specific integrals are highlighted in contrast to the proposed equations.
- The author acknowledges the uniqueness of the new mathematical results and expresses uncertainty about their correctness.
- Considerations for incorporating the new delta function/Fourier integral relationships in the sixth edition of Intermediate Physics for Medicine and Biology are mentioned.
- Further research and exploration into existing mathematical literature are planned to validate the proposed equations.
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