expansions in series of incomplete gamma functions
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21—30 of 30 matching pages
21: Bibliography C
22: Bibliography K
23: Bibliography D
24: Bibliography L
25: Bibliography R
26: Bibliography B
27: 9.7 Asymptotic Expansions
§9.7 Asymptotic Expansions
… ►§9.7(iii) Error Bounds for Real Variables
… ►In (9.7.9)–(9.7.12) the th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign. … ►§9.7(v) Exponentially-Improved Expansions
… ►28: 8.19 Generalized Exponential Integral
§8.19(i) Definition and Integral Representations
… ►Most properties of follow straightforwardly from those of . … ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►§8.19(iv) Series Expansions
… ►The general function is attained by extending the path in (8.19.2) across the negative real axis. …29: Errata
§4.13 has been enlarged. The Lambert -function is multi-valued and we use the notation , , for the branches. The original two solutions are identified via and .
Other changes are the introduction of the Wright -function and tree -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert -functions in the end of the section.
The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at for . In the case of (8.6.12), it separates the poles of the gamma function from the poles at .
Reported 2017-07-10 by Kurt Fischer.
The original in front of the second summation was replaced by to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.
Reported 2017-01-28 by Richard Paris.