Evaluation of a class of integrals occurring in mathematical physics via a higher order generalization of the principal value

1989 ◽  
Vol 67 (8) ◽  
pp. 759-765 ◽  
Author(s):  
K. T. R. Davies ◽  
R. W. Davies
Keyword(s):  
Single Particle ◽  
Numerical Evaluation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Simple Algorithm ◽  
Mathematical Physics ◽  
Higher Order ◽  
Divergent Integral ◽  
Principal Value

The notion of the principal value of an integral is generalized to treat higher order singularities. The principal value of an integral can be considered the "convergent part" of a divergent integral, an interpretation that is almost trivial for simple poles, but more meaningful for higher order poles. Application of this concept leads to a simple algorithm that may be applied to the evaluation of a class of integrals arising in mathematical physics. Many of these integrals frequently occur in the analytic and numerical evaluation of folding functions arising from the product of single-particle Green's functions.

MODIFIED HUBBARD DECOUPLING FOR THE EMERY MODEL

1995 ◽  
Vol 09 (25) ◽  
pp. 1635-1641
Author(s):  
LEW GEHLHOFF
Keyword(s):  
Single Particle ◽  
Green's Functions ◽  
Green’S Functions ◽  
Equation Of Motion ◽  
Higher Order ◽  
Exact Results ◽  
Emery Model ◽  
Spin Degeneracy ◽  
Motion Method ◽  
Large Spin

We consider a version of the Emery model with large spin degeneracy N and use the X-operator formulation and the equation-of-motion method to determine the single-particle Green’s functions. We propose a modified Hubbard decoupling technique for the higher-order Green’s functions appearing in this equation of motion. By applying it to the above model in the limit N→∞ we obtain the exact results.

scholarly journals Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green's functions using the Ewald method

Radio Science ◽  
2008 ◽  
Vol 43 (6) ◽  
pp. n/a-n/a ◽  
Author(s):  
Ferhat T. Celepcikay ◽  
Donald R. Wilton ◽  
David R. Jackson ◽  
Filippo Capolino
Keyword(s):  
Numerical Evaluation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Ewald Method

POSITIVITY OF GREEN'S FUNCTIONS TO VOLTERRA INTEGRAL AND HIGHER ORDER INTEGRO-DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 07 (04) ◽  
pp. 405-418 ◽  
Author(s):  
M. I. GIL'
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Green's Functions ◽  
Arbitrary Order ◽  
Green’S Functions ◽  
Higher Order ◽  
Volterra Integral Equations ◽  
Differential Delay

We consider Volterra integral equations and arbitrary order integro-differential equations. We establish positivity conditions and two-sided estimates for Green's functions. These results are then applied to obtain stability and positivity conditions for equations with nonlinear causal mappings (operators) and linear integro-differential parts. Such equations include differential, difference, differential-delay, integro-differential and other traditional equations.

Numerical evaluation of Green’s functions for axisymmetric boreholes using leaking modes

Geophysics ◽  
1987 ◽  
Vol 52 (8) ◽  
pp. 1099-1105 ◽  
Author(s):  
R. A. W. Haddon
Keyword(s):  
Wave Propagation ◽  
Fourier Transform ◽  
Elastic Wave ◽  
Numerical Evaluation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Complete Solution ◽  
Body Waves ◽  
Complex Frequency ◽  
Residue Theory

By choosing appropriate paths of integration in both the complex frequency (ω) and complex wavenumber (k) planes, exact Green’s functions for elastic wave propagation in axisymmetric boreholes are expressed completely as sums of modes. The integrations with respect to k are performed exactly using Cauchy residue theory. The remaining integrations with respect to ω are then carried out using the fast Fourier transform (FFT). The complete solution, including all possible body waves, is expressed simply as a superposition of modes without any contributions from branch line integrals. There are no spurious arrivals and, provided that the number of points in the FFT can be taken sufficiently large, no restrictions on distance. The method is fast, accurate, and easy to apply.

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