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| Keywords: AsymptoticExpansionE1.png Divergence of the asymptotic series for the w Exponential integral <math> \rm E _1 z \int_1 \infty \frac e -tz t \ \rm d t \qquad \Re z \ge 0~~ </math> The function <math> E_1 z N \frac \exp -z z \sum_ n 0 N-1 \frac n -z n </math> gives the first N terms of an approximation to <math>E_1 z </math> for <math>~ \rm Re z \gg 1</math> The relative error <math> \frac E_1 x N -E_1 x E_1 x </math> is plotted versus <math>~x~</math> for <math>3<x<9</math> for various numbers of terms N as follows <br> <math>~N 1~</math> red <br> <math>~N 2~</math> green <br> <math>~N 3~</math> yellow <br> <math>~N 4~</math> blue <br> <math>~N 5~</math> pink <br> The larger <math>~x~</math> is the more terms in this expansion can be taken into account for approximation of the function and the better the resulting approximation is Own 2008 Domitori Use it please It is free The first formula is smooth you can expand it and do what you like Enjoy Not yet reported Make your own ImageNote 1 442 485 146 31 1110 1110 2 N 1 ImageNoteEnd 1 ImageNote 2 124 655 68 237 1110 1110 2 N 5 ImageNoteEnd 2 ImageNote 3 115 135 56 122 1110 1110 2 N 4 ImageNoteEnd 3 ImageNote 4 120 266 59 46 1110 1110 2 N 2 ImageNoteEnd 4 ImageNote 5 150 485 91 74 1110 1110 2 N 3 ImageNoteEnd 5 Functions | ||||
