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Special Functions part 4

CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 | 222 Chapter 6. Special Functions CITED REFERENCES AND FURTHER READING Abramowitz M. and Stegun I.A. 1964 Handbook of Mathematical Functions Applied Mathematics Series Volume 55 Washington National Bureau of Standards reprinted 1968 by Dover Publications New York Chapters 6 7 and 26. Pearson K. ed. 1951 Tables of the Incomplete Gamma Function Cambridge Cambridge University Press . 6.3 Exponential Integrals The standard definition of the exponential integral is Bn x J dt x 0 n 0 1 . . . 6.3.1 The function defined by the principal value of the integral 1 e t fx eJ Ei x -J x 0 6.3.2 is also called an exponential integral. Note that Ei -x is related to -E1 x by analytic continuation. The function En x is a special case of the incomplete gamma function En x xn-1r 1 - n x 6.3.3 We can therefore use a similar strategy for evaluating it. The continued fraction just equation 6.2.6 rewritten converges for all x 0 1n_1 n 1 2 x 1 x 1 x 6.3.4 We use it in its more rapidly converging even form En x e 1 x n 1 n 2 n 1 x n 2 x n 4 6.3.5 The continued fraction only really converges fast enough to be useful for x 1. For 0 x 1 we can use the series representation . -x m En x - --TTfi- ln x n X 7------- T I n 1 m n 1 m m n 1 6.3.6 Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 The quantity n here is the digamma function given for integer arguments by n 1 1 1 Y n Y X m m 1 6.3.7 6.3 Exponential Integrals 223 where 7 0.5772156649 . is Euler s constant. We evaluate the expression 6.3.6 in order of ascending powers of x En x - 1 x x2 x n 2 1 - n - 2 n 1 3 n 1 2 1 n 2 x n 1 n 1 lnx V n x n x n 1 - 2 n 1 6.3.8 The first square bracket is omitted when n 1. This method of evaluation has the advantage that for large n the series converges before reaching the term containing V n . Accordingly one needs an algorithm for evaluating V n only for small n n 20-40. We use equation 6.3.7 although a table look-up would improve efficiency slightly. Amos 1