A Summary on Two Types of Real Integrals Using the Residue Theorem

Abstract The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u(x) over the space Ck (Rn ) (which contains S(Rn ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

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Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues

Springer Undergraduate Mathematics Series - Twenty-One Lectures on Complex Analysis ◽

10.1007/978-3-319-68170-2_15 ◽

2017 ◽

pp. 127-135

Author(s):

Alexander Isaev

Keyword(s):

Holomorphic Functions ◽

Isolated Singularity ◽

Residue Theorem ◽

Weierstrass Theorem ◽

Isolated Singularities ◽

Cauchy’S Residue Theorem

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THE RESIDUE THEOREM AND AN ANALOG OF P. APPELL’S FORMULA FOR FINITE RIEMANN SURFACES

Science Evolution ◽

10.21603/2500-1418-2016-1-1-40-45 ◽

2016 ◽

pp. 40-45

Author(s):

Viktor Chueshev ◽

Aleksandr Chueshev ◽

Aleksandr Chueshev

Keyword(s):

Riemann Surfaces ◽

Meromorphic Functions ◽

Function Theory ◽

Mathematical Physics ◽

General Function ◽

Multiplicative Functions ◽

Residue Theorem ◽

Equations Of Mathematical Physics ◽

Compact Riemann Surfaces ◽

Integral Order

A theory of multiplicative functions and Prym differentials for the case of special characters on compact Riemann surfaces has found applications in geometrical function theory of complex variable, analytical number theory and in equations of mathematical physics. Theory of functions on compact Riemann surfaces differs from the theory of functions on finite Riemann surfaces even for the class of single meromorphic functions and Abelian differentials. In this article we continue the construction of the general function theory on finite Riemann surfaces for multiplicative meromorphic functions and differentials. We have proved analogues of the theorem on the full sum of residues for Prym differentials of every integral order and P. Appell's formula on expansion of the multiplicative function with poles of arbitrary multiplicity in the sum of elementary Prym integrals.

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Asymptotic expansions for time-varying scalar differential equations using the residue theorem for their discretized difference equations

Applied Mathematics and Computation ◽

10.1016/j.amc.2010.01.024 ◽

2010 ◽

Vol 216 (1) ◽

pp. 149-160

Author(s):

M. De la Sen

Keyword(s):

Differential Equations ◽

Difference Equations ◽

Asymptotic Expansions ◽

Time Varying ◽

Residue Theorem

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A theoretical investigation of acoustic monopole logging-while-drilling individual waves with emphasis on the collar wave and its dependence on formation

Geophysics ◽

10.1190/geo2016-0266.1 ◽

2017 ◽

Vol 82 (1) ◽

pp. D1-D11 ◽

Cited By ~ 7

Author(s):

Xiaobo Zheng ◽

Hengshan Hu

Keyword(s):

Theoretical Analysis ◽

Excitation Spectrum ◽

Integral Method ◽

Individual Component ◽

P Wave ◽

Residue Theorem ◽

Full Waveform ◽

Logging While Drilling ◽

Traveling Speed ◽

Branch Cut

To seek measures to weaken the collar wave signals so that the formation arrivals are observable, it is important to make theoretical analysis of separate collar wave and formation arrivals in the acoustic logging-while-drilling environment. However, until now, the collar wave signal and the formation P- and S-arrivals have never been separately calculated. We have obtained individual component waves using the residue theorem and the branch-cut integral method, including residues at leaky poles. The waveform summed up from all individual waves is shown to agree well with the full waveform calculated by real-axis integration. In particular, the formation P-wave is obtained by summing the formation leaky mode and the compressional branch-cut integral for slow formations. The collar wave is found to propagate in the borehole and the formation as well as in the collar. Although the traveling speed of the collar wave is almost irrelevant to the formation, the attenuation and excitation spectrum of the collar wave are significantly affected by the formation, which reveals that an effective collar wave weakening design should be based on a model with the formation being taken into consideration.

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