Geodesic
The dynamical motion of the zeros of the partial sums of $e^{z}$, and its relationship to discrepancy theory
Electronic transactions on numerical analysis, Tome 30 (2008), pp. 128-143.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: With $\ddot $!#"$ denoting the -th partial sum of , let its zeros be denoted by {\S}\copyright $$\ddot 0 1#2 7 7 !4 4 \ddot !#"87 for any positive integer . If and are any angles with , let be the 0 9 9#@ ACBD9 BE9F@GBIH#P QSRUT RWV $$\ddot 65 associated sector, in the z-plane, defined by 3 7gfih#prq f 4 QSRXT RWVY a`cbde9 9F@ 5ts If 3 !4 4 4 \ddot !#"87xw represents the number of zeros of in the sector , then Szeg\Acute$Acuteo$ showed in uDv QRXT RWV\ {\S} EUR QSRXT RWV \ddot $ddotS$5 1924 that !4 4 7 !#"87 \ddot $w$ uIv$$########$$F QRXT RWV\ ^$$########$ ddot" @-bullet"" " ddotdagger…$######## 0 H#P -- where 7 and are defined in the text. The associated $discrepancy function$ is defined by @ " " 7 7 ###$dagger$###$^TM$rd !4 4 @-$bullet$!#"87xi $ddotddote$9 9#@f8 guEh#$$###$$F QRXT RWVj 0lkm" " $bulletddot$'' s -- H#P n One of our new results shows, for any 7 with 7 , that 9 AaBi9 BiP q h 7 7 ###$dagger$###$^TM$od ^s $^TM$ EUR9 H#P 9 Sprq 0 0ctvu $ddotbullet$-- -- -- where is a positive constant, depending only on 7 . Also new in this paper is a study of the $cyclical nature$ of q 9 7 7 7 ###$dagger$###$^TM$rd , as a function of , when and . An upper bound for the approximate cycle 9 9 0 AaBi9 BiP 9 wH#P $9 ddot@ @ bullet7$ -- length, in this case, is determined in terms of . All this can be viewed in our $Interactive Supplement$, which shows " the dynamical motion of the (normalized) zeros of the partial sums of and their associated discrepancies.$
Classification : 30C15, 30E15
Keywords: partial sums of , szegacute$\Acute o$ curve, discrepancy function 1 2 
@article{ETNA_2008__30__a17,
     author = {Varga, Richard S. and Carpenter, Amos J. and Lewis, Bryan W.},
     title = {The dynamical motion of the zeros of the partial sums of $e^{z}$, and its relationship to discrepancy theory},
     journal = {Electronic transactions on numerical analysis},
     pages = {128--143},
     publisher = {mathdoc},
     volume = {30},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2008__30__a17/}
}
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Varga, Richard S.; Carpenter, Amos J.; Lewis, Bryan W. The dynamical motion of the zeros of the partial sums of $e^{z}$, and its relationship to discrepancy theory. Electronic transactions on numerical analysis, Tome 30 (2008), pp. 128-143. http://geodesic.mathdoc.fr/item/ETNA_2008__30__a17/