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This article deals with different ways of solving various integrals without elementary primitives. This strategy, popularized by the theoretical physicist Richard Feynman for what is commonly known as the Feynman’s technique, is based on assig- ning a parameterization to the integrand function, and through the use of the derivative function and the decomposition into simple fractions, the problem is transformed into the resolution of elementary integrals and therefore, the value of a primitive is obtained that would otherwise be very difficult to achieve., Este artículo aborda diferentes formas de resolución de varias integrales sin primitiva elemental. Dicha estrategia popularizada por el físico teórico Richard Feynman por lo que es comúnmente conocida como técnica de Feynman, se fundamenta en asignar una parametrización a la función integrando, y mediante el uso de la función derivada y la descomposición en fracciones simples, se transforma el problema en la resolución de integrales elementales y por ende, se obtiene el valor de una primitiva que de otra manera sería muy complicado alcanzar

We implement the partial fraction decomposition scheme proposed by Croy and Saalmann [A. Croy, U. Saalmann, Phys. Rev. B 80 (2009) 073102] to the construction of the hierarchical equations of motion for bosonic quantum dissipation systems [R.X. Xu, Y.J. Yan, Phys. Rev. E 75 (2007) 031107]. The expansion of Bose-Einstein function in this scheme shows similar properties as it applies for Fermi function. The performance of the resulting quantum dissipation theory is exemplified with spin-boson systems. The new theoretical construction performs much better (about an order of magnitude faster) than the best available conventional formalism based on Matsubara spectral decomposition scheme. (C) 2009 Elsevier B.V. All rights reserved.

We implement the partial fraction decomposition scheme proposed by Croy and Saalmann [A. Croy, U. Saalmann, Phys. Rev. B 80 (2009) 073102] to the construction of the hierarchical equations of motion for bosonic quantum dissipation systems [R.X. Xu, Y.J. Yan, Phys. Rev. E 75 (2007) 031107]. The expansion of Bose-Einstein function in this scheme shows similar properties as it applies for Fermi function. The performance of the resulting quantum dissipation theory is exemplified with spin-boson systems. The new theoretical construction performs much better (about an order of magnitude faster) than the best available conventional formalism based on Matsubara spectral decomposition scheme. (C) 2009 Elsevier B.V. All rights reserved.

We implement the partial fraction decomposition scheme proposed by Croy and Saalmann [A. Croy, U. Saalmann, Phys. Rev. B 80 (2009) 073102] to the construction of the hierarchical equations of motion for bosonic quantum dissipation systems [R.X. Xu, Y.J. Yan, Phys. Rev. E 75 (2007) 031107]. The expansion of Bose-Einstein function in this scheme shows similar properties as it applies for Fermi function. The performance of the resulting quantum dissipation theory is exemplified with spin-boson systems. The new theoretical construction performs much better (about an order of magnitude faster) than the best available conventional formalism based on Matsubara spectral decomposition scheme. (C) 2009 Elsevier B.V. All rights reserved.

The partial fraction decomposition of a proper rational function whose denominator has degree n and is given in general factored form can be done in O(n (log n) squared) operations in the worst case. Previous algorithms require O x N cubed operations, and O(n (log n) squared) operations for the special case where the factors appearing in the denominator are all linear.