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Exponential Integral -- from Wolfram MathWorld Page 1 of 3 Search Site Algebra Applied Mathematics Calculus and Analysis Calculus and Analysis > Special Functions > Gamma Functions Calculus and Analysis > Special Functions > Named Integrals Foundations of Mathematics > Mathematical Problems > Unsolved Problems Recreational Mathematics > Interactive Entries > webMathematica Examples Discrete Mathematics Foundations of Mathematics Geometry Exponential Integral History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Send a Message to the Team Order book from Amazon 12,637 entries Sat Dec 23 2006 Min Max Re -5 5 Im -5 5 Let Register for Unlimited Interactive Examples >> Replot be the En-function with , (1) Then define the exponential integral by (2) where the retention of the given by the integral notation is a historical artifact. Then is (3) This function is implemented in Mathematica as ExpIntegralEi[x]. http://mathworld.wolfram.com/ExponentialIntegral.html 25/12/2006 Exponential Integral -- from Wolfram MathWorld Page 2 of 3 The exponential integral by is closely related to the incomplete gamma function (4) Therefore, for real , (5) The exponential integral of a purely imaginary number can be written (6) for and where and are cosine and sine integral. Special values include (7) (Sloane's A091725). The real root of the exponential integral occurs at 0.37250741078... (Sloane's A091723), which is , where is Soldner's constant (Finch 2003). The quantity Gompertz constant. (Sloane's A073003) is known as the The limit of the following expression can be given analytically (8) (9) (Sloane's A091724), where The Puiseux series of is the Euler-Mascheroni constant. along the positive real axis is given by (10) where the denominators of the coefficients are given by van Heemert 1957, Mundfrom 1994). (Sloane's A001563; SEE ALSO: Cosine Integral, En-Function, Gompertz Constant, Incomplete Gamma Function, Sine Integral, Soldner's Constant. [Pages Linking Here] RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/ REFERENCES: Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 566-568, 1985. Finch, S. R. "Euler-Gompertz Constant." §6.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 423-428, 2003. http://mathworld.wolfram.com/ExponentialIntegral.html 25/12/2006 Exponential Integral -- from Wolfram MathWorld Page 3 of 3 Harris, F. E. "Spherical Bessel Expansions of Sine, Cosine, and Exponential Integrals." Appl. Numer. Math. 34, 95-98, 2000. Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 105106, 2003. Jeffreys, H. and Jeffreys, B. S. "The Exponential and Related Integrals." §15.09 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 470-472, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434-435, 1953. Mundfrom, D. J. "A Problem in Permutations: The Game of 'Mousetrap.' " European J. Combin. 15, 555-560, 1994. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Exponential Integrals." §6.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 215-219, 1992. Sloane, N. J. A. Sequences A001563/M3545, A073003, A091723, A091724, and A091725 in "The On-Line Encyclopedia of Integer Sequences." Spanier, J. and Oldham, K. B. "The Exponential Integral Ei( ) and Related Functions." Ch. 37 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 351-360, 1987. van Heemert, A. "Cyclic Permutations with Sequences and Related Problems." J. reine angew. Math. 198, 56-72, 1957. LAST MODIFIED: January 6, 2006 CITE THIS AS: Weisstein, Eric W. "Exponential Integral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExponentialIntegral.html © 1999 CRC Press LLC, © 1999-2006 Wolfram Research, Inc. | Terms of Use http://mathworld.wolfram.com/ExponentialIntegral.html 25/12/2006