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A RESIDUE TYPE PROCESS FOR SMOOTH FUNCTIONS INVOLVING THE DERIVATIVES OF THE NEWTONIAN POTENTIAL IN R2

A RESIDUE TYPE PROCESS FOR SMOOTH FUNCTIONS INVOLVING THE DERIVATIVES OF THE NEWTONIAN POTENTIAL... DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 12005Telemachos HatziafratisA RESIDUE T Y P E PROCESS FOR SMOOTH FUNCTIONSINVOLVING T H E DERIVATIVESOF T H E NEWTONIAN POTENTIAL IN R 2Abstract. For a smooth function f(x,y),the limitrAttof the real variables x and y, we compute\ dt+'i~ydx+ xdy\in terms of the derivatives of / at (0,0) and we study related questions.1. IntroductionLet us recall that for a function g(z), which is continuous for z in aneighborhood of 0 € C,lim<j) g { z =2irig(0).1*1=«More generally, if the function g(z) is assumed to be of class Ck, then onecan prove (see [1] and [2]) that«Ï3 4 «W^r 1*1=«A real variable analogue of the first of the above relations is the fact thatfor a function f(x,y),which is continuous for (x,y) in a neighborhood of(0,0) e R 2 ,x2+y2=e2Key wards and phrases: Residue process, smooth functions, derivatives of the Newtonian potential.1991 Mathematics Subject Classification: 26B20.22T.HatziafratisIn this note we study the limitsx2+y2=e2and we show that they exist, provided that the function f(x, y) is sufficientlysmooth. In fact we can compute explicitly these limits in terms of thederivatives of / at (0,0). The formulas we obtain are analogues of (1).Notice also thatdz_(-1)^dkk\k~idz\dzV z/'First let us give an example which shows http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

A RESIDUE TYPE PROCESS FOR SMOOTH FUNCTIONS INVOLVING THE DERIVATIVES OF THE NEWTONIAN POTENTIAL IN R2

Hatziafratis, Telemachos
Demonstratio Mathematica , Volume 38 (1): 10 – Jan 1, 2005
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References (3)

Publisher
de Gruyter
Copyright
© by Telemachos Hatziafratis
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2005-0104
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 12005Telemachos HatziafratisA RESIDUE T Y P E PROCESS FOR SMOOTH FUNCTIONSINVOLVING T H E DERIVATIVESOF T H E NEWTONIAN POTENTIAL IN R 2Abstract. For a smooth function f(x,y),the limitrAttof the real variables x and y, we compute\ dt+'i~ydx+ xdy\in terms of the derivatives of / at (0,0) and we study related questions.1. IntroductionLet us recall that for a function g(z), which is continuous for z in aneighborhood of 0 € C,lim<j) g { z =2irig(0).1*1=«More generally, if the function g(z) is assumed to be of class Ck, then onecan prove (see [1] and [2]) that«Ï3 4 «W^r 1*1=«A real variable analogue of the first of the above relations is the fact thatfor a function f(x,y),which is continuous for (x,y) in a neighborhood of(0,0) e R 2 ,x2+y2=e2Key wards and phrases: Residue process, smooth functions, derivatives of the Newtonian potential.1991 Mathematics Subject Classification: 26B20.22T.HatziafratisIn this note we study the limitsx2+y2=e2and we show that they exist, provided that the function f(x, y) is sufficientlysmooth. In fact we can compute explicitly these limits in terms of thederivatives of / at (0,0). The formulas we obtain are analogues of (1).Notice also thatdz_(-1)^dkk\k~idz\dzV z/'First let us give an example which shows

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 2005

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