Abstract
| Original language | English |
|---|---|
| Pages (from-to) | 3-18 |
| Number of pages | 16 |
| Journal | European Journal of Pure and Applied Mathematics |
| Volume | 9 |
| Issue number | 1 |
| Publication status | Published - 2016 |
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The asymptotic expansion of a generalisation of the Euler-Jacobi series. / Paris, Richard B.
In: European Journal of Pure and Applied Mathematics, Vol. 9, No. 1, 2016, p. 3-18.Research output: Contribution to journal › Article
TY - JOUR
T1 - The asymptotic expansion of a generalisation of the Euler-Jacobi series
AU - Paris, Richard B.
PY - 2016
Y1 - 2016
N2 - We consider the asymptotic expansion of the sum Sp(a; w) = ∞Ʃ n=1 e−anp/nw as a → 0 in | arg a| <1/2π for arbitrary finite p > 0 and w > 0. Our attention is concentrated mainly on the case when p and w are both even integers, where the expansion consists of a finite algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for Sp (a; w) analogous to the well-known Poisson-Jacobi transformation for the sum with p = 2 and w = 0. Numerical results are given to illustrate the accuracy of the expansion obtained.
AB - We consider the asymptotic expansion of the sum Sp(a; w) = ∞Ʃ n=1 e−anp/nw as a → 0 in | arg a| <1/2π for arbitrary finite p > 0 and w > 0. Our attention is concentrated mainly on the case when p and w are both even integers, where the expansion consists of a finite algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for Sp (a; w) analogous to the well-known Poisson-Jacobi transformation for the sum with p = 2 and w = 0. Numerical results are given to illustrate the accuracy of the expansion obtained.
M3 - Article
VL - 9
SP - 3
EP - 18
JO - European Journal of Pure and Applied Mathematics
JF - European Journal of Pure and Applied Mathematics
SN - 1307-5543
IS - 1
ER -