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The author has created a small family of generalized indefinite and definite algebraic–exponential real variable integrals based on the gamma function. They are useful for scientific and engineering applications. One typo in footnote 1 corrected 3/26/2013.
A Family of Algebraic–Exponential Integrals ╬ Francis J. O’Brien, Jr., Ph.D. Aquidneck Indian Council Newport, RI Corrected Copy March 26, 2013 One of the most important non–elementary transcendental (or special) functions in mathematics is the gamma function, defined by the improper integral: t 1e t dt , 0 0 A number of alternative expressions for can be derived by change of variable1. A plot of is shown in Figure 1. The domain is the entire number line, but most work for real variables is restricted to 0. 1 Three transforms: n (a) if t x , n x n 1e x dx, 0 n 0 1 zx (b) if t zx, z x e dx, 0 0 1 1 1 (c) if x e , ln t dx, 0 0 x © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 1 Figure 1. The Gamma Function, . Source: Wolfram Mathematica, http://www.wolframalpha.com/. Clearly, 0 for 0. Sokolnikoff (Chap. X, pp. 372 ff.) describes the theory of limits that demonstrates the convergence of for 0. The standard reference work on the gamma function for real variables is the book by Emil Artin (1964). The References lists other works that provide information on the gamma function. —————— The author has created a small family of generalized indefinite and definite algebraic– exponential real variable integrals of the form x m exp x n based on the gamma function. They are useful for scientific and engineering applications. What follows is a summary of the primary core of gamma-based formulas that appear in the Gradshteyn and Ryzhik (GR) integration handbook. A large number of species formulas can be generated based on these and similar published integrals. Many such integrals along with brief proofs appear in the book, 500 Integrals of Elementary and Special Functions (F. O’Brien, 2008). The complete list of solved formulas is given in the document “List of formulas in 500 Integrals of Elementary and Special Functions” (forthcoming), http://www.docstoc.com/profile/waabu. In the next section a formula number is given for integrals that appear in GR; otherwise it is listed as “unpublished” with the section number in 500 Integrals based on the GR classification of integrals. NOTE: Some formulas in GR (7th Edition) contain misprints. For the errata sheet see, http://www.mathtable.com/errata/gr7_errata.pdf © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 2 Algebraic–Exponential Integrals Indefinite Integrals Unpublished Formula, Section 2.3 m ax 1 1 t m 1 x e dx t n t ax , n n e dt na Unpublished Formula, Section 2.3 e ax n 1 z 1 t 1 m t n dx e dt t ax , z n xm na z NOTE: these functional forms provide a transformed gamma-like structure which can be used for definite integral applications. Formula 2.33.4 m ax n x m 1 n ax n m 1 n m n ax n x e dx na e na x e dx Formula 2.325.5 n n e ax 1 e ax e ax n dx na dx xm m 1 x m 1 xmn © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 3 Formula 2.33.10 m x x e dx n , x n n 1 1 t m 1 n t e dt n x n NOTE: this formula and Formula 3.326.2 (listed below) are derived in the online paper at n docstoc.com, “ x m e x dx and related integrals.” Related indefinite integral formulas are also given there for the integrands: n x m e ax Unpublished Formula, Section 2.32 ax n e Formula 2.325.6 xm n e x Formula 2.33.9 xm NOTE: Formulas are useful for generating log expansion formulas for integer values of the x ln x dx given in Sect. 2.32 & Sect. 2.72 of 500 Integrals. m n form, Formula 2.33.11 x m exp x n dx 1 exp x n 1! x n k 1 k k 1! k 1 n 0 m 1 n 1,2,..., n 0, 0 NOTE: only if the parameter is an integer. © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 4 Formula 2.33.2 e ax 2 dx 1 2 a erfi a x , where erfi is the imaginary error function. Formula 2.33.16 e x 2 dx 1 2 erf x , where erf is the error function. ______________ Definite Integrals NOTE: “Re” refers to real variables. Unpublished Formula, Section 3.326 v x m exp x dx n , u n , v n m 1 u n n Re 0, Re n 0 NOTE: a general gamma function integral from which a number of integrals can be derived for selected values of u & v by properties of the complete and lower/upper incomplete gamma functions. Those well-known relations for , , x & , x inter alia are summarized in O’Brien, pp. 6–9. Formula 3.326.2 m 1 x exp x n dx m 0 n n Re n 0, Re 0, Re 0 NOTE: this formula is useful for many applications because of its simplicity. © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 5 Formula 3.381.8 1 , u n u m x n m 1 x e dx n n 0 [ u 0, Re 0, Re n 0, Re 0 ] Formula 3.381.9 1 , u n m x n m 1 x e dx n n u [Re n 0, Re 0 ] Formula 3.381.10 u m x m x n m x n n x e dx x e dx x e dx 0 0 u n v 1 , u n n v 1 , u n m 1 n [u 0, Re 0, Re n 0, Re 0 ] See also Formula 3.326.2 Unpublished Formula, Section 4.215 1 1 m x n 1 1 m 1 x e dx n ln t dt n 0 0 Re n 0, Re 0, Re 0 NOTE: a general form of Euler’s 1730 original log formula for gamma function definition. © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 6 Unpublished Formula, Section 8.252 n k 1 xm n k , x n k 1!e x m 0 m! [n k 1 0, k 0,1,...] NOTE: for integer values only, a generalization of n, x . Formula 8.352.6 n k 1 x m n k , x n k 1! 1 e x m! m0 [n k 1 0, k 0,1,...] NOTE: for integer values only, a generalization of n, x . Formula 8.352.8 n k , x 1nk 0, x e x nk 1 1m m! n k ! m 0 x m 1 [n k 1 0, k 0,1,...] NOTE: for integer values only, a generalization of n 1, x . © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 7 ✜ The author would like to acknowledge with deep gratitude the late Professor Alan Jeffrey for extensive feedback in the creation of the formulas and assistance in their publication in Gradshteyn and Ryzhik (6th & 7th Editions), Table of Integrals, Series, and Products. © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 8 References NOTE: each reference provides information on the gamma function. Abramowitz, Milton and Irene Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: U.S.Government Printing Office, 1964 (Reprinted Dover, New York, 1972). Artin, Emil. The Gamma Function. New York, NY: Holt, Rinehart and Winston, 1964. Carr, G.S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea Publishing Co., 1970. Gradshteyn, I.S. and I.M. Ryzhik (7th Edition). Table of Integrals, Series, and Products. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2007. Errata: http://www.mathtable.com/errata/gr7_errata.pdf Moll, Victor H. The integrals in Gradshteyn and Ryzhik. Part 4: The Gamma function. arXiv:0705.0179v1 math.CA, 1 May 2007. O’Brien, Francis J. Jr. 500 Integrals of Elementary and Special Functions, 2008. ISBN: 1–4392-1981–8. http://www.docstoc.com/profile/waabu m x n _______. x e dx and related integrals, 2nd ed., Mar.16, 2013. http://www.docstoc.com/profile/waabu _______. List of formulas in 500 Integrals of Elementary and Special Functions. (2013, forthcoming). http://www.docstoc.com/profile/waabu. Sokolnikoff, I. S. Advanced Calculus. New York: McGraw Hill Book Co., 1939. Whittaker, E. T. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Series and of Analytic Functions; With An Account of the Principal Transcendental Functions. Cambridge University Press, 1902. Reprinted Whittaker, E. T. and G.N. Watson, 4th ed., 1934 Wolfram Research, Inc. The Wolfram Functions Site., 1998-2008. http://functions.wolfram.com/GammaBetaErf/Factorial2/27/01/0002/ © Francis. J. O’Brien, Jr., 2013. Aquidneck Indian Council All rights reserved. 9
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