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Asymptotic Expansion of n-dimensional Faxen-type Integrals

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Abstract (2. Language): 
The asymptotic expansion of n-dimensional extensions of Faxén’s integral In(z) are derived for large complex values of the variable z. The theory relies on the asymptotics of the generalised hypergeometric, orWright, function. The coefficients in the exponential expansion are obtained by means of an algorithm applicable for arbitrary n. Numerical examples are given to illustrate the accuracy of the expansions.
1006-1031

REFERENCES

References: 

[1] N G Bakhoom, Asymptotic expansions of the function Fk(x) =
R∞
0
exp(xu−uk)du, Proc.
Lond. Math. Soc. 35:83–100, 1935.
[2] B L J Braaksma, Asymptotic expansions and analytic continuations for a class of Barnesintegrals,
Compos. Math. 15:239–341, 1963
[3] S Breen and A D Wood, On the asymptotic behaviour of Laplace-type multiple integral
solutions of linear differential equations, J. Comp. Appl. Math. 171:103–112, 2004.
[4] L Brillouin, Sur une méthode de calcul approcheé de certaines intégrales, dite méthode
de col, Ann. Sci. Ecole Norm. Sup. 33:17–69, 1916.
[5] W R Burwell, Asymptotic expansions of generalised hypergeometric functions, Proc.
Lond. Math. Soc. 53:599–611, 1924.
[6] R Gorenflo, Y Luchko and F Mainardi, Analytical properties and applications of the
Wright function, Frac. Calc. Appl. Anal. 2:383–414, 1998.
[7] D Kaminski and R B Paris, Asymptotics via iterated Mellin-Barnes integrals: Application
to the generalised Faxén integral, Methods Appl. Anal. 4:311–325, 1997.
[8] G V Liakhovetski and R B Paris, Asymptotic expansions of Laplace-type integrals III, J.
Comp. Appl. Math. 132:409–429, 2001.
[9] F W J Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.
Reprinted in A K Peters, Massachussets, 1997.
[10] R B Paris, Smoothing of the Stokes phenomenon for high-order differential equations,
Proc. Roy. Soc. London A 436:165–186, 1992.
[11] R B Paris, Exponentially small expansions in the asymptotics of the Wright function, J.
Comp. Appl. Math. 234:488–504, 2010.
[12] R B Paris and D Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University
Press, Cambridge, 2001.
[13] R B Paris and G V Liakhovetski, Asymptotics of the multidimensional Faxén integral,
Frac. Calc. Appl. Anal. 3:63–73, 2000.
[14] R B Paris and A D Wood, Asymptotics of High Order Differential Equations, Pitman Research
Notes in Mathematics, 129, Longman Scientific and Technical, Harlow, 1986.
[15] T D Riney, On the coefficients in asymptotic factorial expansions, Proc. Amer. Math. Soc.
7:245–249, 1956.
[16] R Saxton, An integral representation solution for a class of higher order linear ordinary
differential equations, MSc Dissertation, Cranfield Institute of Technology, 1978.
[17] L J Slater, Generalised Hypergeometric Functions, Cambridge University Press, Cambridge,
1966.
[18] S Spitzer, Integration der linearen Differentialgleichung y(n) = Ax2 y′′ + Bx y′ + C y, in
welcher n eine ganze positive Zahl und A, B, C constante Zahlen bezeichnen, mittelst
bestimmter Integrale, Math. Ann. 3:453–455, 1871.
[19] N M Temme, Special Functions: An Introduction to the Classical Functions of Mathematical
Physics, Wiley, New York, 1996.
[20] EMWright, The asymptotic expansion of the generalized hypergeometric function, Proc.
Lond. Math. Soc. (Ser. 2) 10:286–293, 1935.
[21] EMWright, The asymptotic expansion of the generalized hypergeometric function, Proc.
Lond. Math. Soc. (Ser. 2) 46:389–408, 1940.
[22] E M Wright, A recursion formula for the coefficients in an asymptotic expansion, Proc.
Glasgow Math. Assoc. 4:38–41, 1958.

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