Maclaurin series
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11: Bibliography F
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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series.
Chelsea Publishing Co., New York.
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12: 13.2 Definitions and Basic Properties
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►The first two standard solutions are:
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13: 13.9 Zeros
14: 3.6 Linear Difference Equations
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►We apply the algorithm to compute to 8S for the range , beginning with the value obtained from the Maclaurin series expansion (§11.10(iii)).
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15: 8.12 Uniform Asymptotic Expansions for Large Parameter
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►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at , and the Maclaurin series expansion of is given by
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16: 18.17 Integrals
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17: 4.13 Lambert -Function
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4.13.10
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►For large enough the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side.
In the case of and real the series converges for .
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4.13.11
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18: 2.10 Sums and Sequences
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►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5.
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19: Errata
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Expansion
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§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.
20: 3.11 Approximation Techniques
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