Cauchy’s 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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NOTES AND PROBLEMS A GENERAL BOUND FOR THE LIMITING DISTRIBUTION OF BREITUNG'S STATISTIC

Econometric Theory ◽

10.1017/s0266466608080675 ◽

2008 ◽

Vol 24 (5) ◽

pp. 1443-1455 ◽

Cited By ~ 1

Author(s):

James Davidson ◽

Jan R. Magnus ◽

Jan Wiegerinck

Keyword(s):

General Result ◽

Random Variables ◽

Nonparametric Test ◽

Limiting Distribution ◽

Residue Theorem ◽

Prove Theorem ◽

General Bound ◽

Result Theorem ◽

Special Case ◽

Cauchy’S Residue Theorem

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.

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Fast Frequency-Domain Algorithm for Estimating the Dynamic Wind-Induced Response of Large-Span Roofs Based on Cauchy’s Residue Theorem

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455418500372 ◽

2018 ◽

Vol 18 (03) ◽

pp. 1850037 ◽

Cited By ~ 3

Author(s):

Ning Su ◽

Zhenggang Cao ◽

Yue Wu

Keyword(s):

Frequency Domain ◽

Time Domain ◽

Rational Functions ◽

Induced Response ◽

Response Analysis ◽

Residue Theorem ◽

Coupling Effects ◽

Large Span ◽

Modal Coupling ◽

Cauchy’S Residue Theorem

Wind-induced response analysis is an important process in the design of large-span roofs. Conventional time-domain methods are computationally more expensive than frequency-domain algorithms; however, the latter are not as accurate because of the ill-treatment of the modal coupling effects. This paper revisited the derivations of the frequency-domain algorithm and proposed a fast algorithm for estimating the dynamic wind-induced response considering duly the modal coupling effects. With the wind load cross-spectra modeled by rational functions, closed-form solutions to the frequency-domain integrals can be calculated by Cauchy’s residue theorem, rather than by numerical integration, thereby reducing the truncation errors and enhancing the efficiency of computation. The algorithm is applied to the analysis of a grandstand roof and a spherical dome. Through comparison with time domain analyses results, the algorithm is proved to be reliable. A criterion of the coupling modal combination was suggested based on the cumulative modal contribution rate of over 70%.

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