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# Tag Archives: Dirac delta function

## Jumpin’ Riemann!…..!..!.!.Mangoldt–da mon–got it!….!..!

The magic of Mangoldt summoning Riemann’s miraculous miniscules-the nontrivial zeros. (Originally published in Sept. 2019. Inadvertanly deleted in April) In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes … Continue reading

Posted in Math
Tagged Chebyshev function, Digamma function, Dirac delta function, Euler's product formula. Hadamard's product formula, Fourier transform, Inverse Mellin transform, Landau Xi function, logarithmic derivative, Mangoldt function, Mangoldt summatory function, Riemann jump function, Riemann Xi function, Riemann zeta function nontrivial zeros, Riemann's prime counting function
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## The Riemann Zeta and the Calculus

(Under construction: Reprising investigations over several years.) By virtue of the relation between the values of the Riemann zeta function at the negative integers, , and the Bernoulli numbers and between the Bernoulli polynomials and the partial sums of the … Continue reading

Posted in Math
Tagged annihilation operator, Bernoulli numbers, Bernoulli polynomials, Beta integral, complex Cauchy contour integral, Confluent hypergeometric functions, convolution integral, creation operators, Differential operators, Digamma function, Dirac delta function, Fractional calculus, gamma function, Generalized Laguerre functions, generalized Laguerre polynomials, harmonic numbers, infinigen, infinitesimal generator, lowering operator, Mellin convolution, Mellin transform, operational calculus, Raising operators, Riemann zeta function, Sheffer Appell polynomials
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## Dirac-Appell Sequences

The Pincherle derivative is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, … Continue reading

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Tagged Appell polynomial sequences, Appell sequences, Conjugation of operators, Creation and annihilation operators, Differential operators, Dirac delta function, Dirac-Appell sequence, Generalized Appell sequence, Inverse Laplace transform, Ladder operators, Modified Hermite polynomials, Operator calculus, Pincherle derivative, Raising and lowering operators, Umbral calculus
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## Depressed Equations and Generalized Catalan Numbers

Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers is a set of notes on the the relation of generating functions of the generalized Catalan numbers, e.g., OEIS-A001764, to the compositional inverse of and the tangent envelope of associated discriminant curves. Added … Continue reading

Posted in Math
Tagged Catalan numbers, Compositional inversion, Determinants, Dirac delta function, Discriminant curves and tangent envelope, Discriminants, Functional equations for Fuss-Catalan numbers, Fuss-Catalan numbers, Generalized Catalan numbers, Graphical solutions of depressed equations, Lagrange inversion, Laplace transform, Legendre transform, Power series
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## Riemann’s Jump Function J(x) for the Primes

Dirac’s Delta Function and Riemann’s Jump Function J(x) for the Primes presents Riemann’s jump function for counting the primes as introduced in H. M. Edward’s Riemann’s Zeta Function (Dover, 2001), couched in terms of the Dirac delta function and the … Continue reading