The Zeros of the Partial Sums of e^z

The Zeros of the Partial Sums of ez

Roger Hui
 

Reference 26 of Ken Iverson’s Ph.D. thesis [0] was:

  Iverson, K.E., “The Zeros of the Partial Sums of ez ”, Mathematical Tables and Other Aids to Computation 7 (1953).

I quickly found a citation [1] on the internet:

@Article{Iverson:1953:ZPS,
  author =    "K. E. Iverson",
  title =     "The Zeros of the Partial Sums of $e^z$",
  journal =   j-MATH-TABLES-OTHER-AIDS-COMPUT,
  volume =    "7",
  number =    "43",
  pages =     "165--168",
  month =     jul,
  year =      "1953",
  CODEN =     "MTTCAS",
  ISSN =      "0891-6837",
  bibdate =   "Tue Oct 13 08:06:19 MDT 1998",
  bibsource = "http://www.math.utah.edu/pub/tex/bib/
                  mathcomp1950.bib; JSTOR database",
  acknowledgement = ack-nhfb,
  fjournal =  "Mathematical Tables and Other Aids 
                  to Computation",
}

From thence a helpful J Forum member provided a link to the actual text [2]. This is the earliest Ken Iverson publication I have found so far. The paper began thus:

 

The location of the zeros of certain entire functions has received considerable attention in the literature, and it was suggested by R.S. Varga that a table of the zeros of certain truncated power series might be of interest. Accordingly, the zeros of the truncated exponential series Sn(z) = zk/k! have been computed for values of n up to 23. They appear in the accompanying table with the complex zeros ordered as to modulus and with the real zero printed last.

The table entry for n = 23 stated:

 
– 7.1926 8590 7451 ± 1.4750 5919 5082 i
– 6.8443 1411 8665 ± 2.9355 2075 2798 i
– 6.2551 3130 4907 ± 4.3657 9349 6476 i
– 5.4112 0644 7035 ± 5.7481 1045 3530 i
– 4.2905 2911 0822 ± 7.0608 9517 9930 i
– 2.8595 2302 8914 ± 8.2762 1397 0162 i
– 1.0664 4595 7935 ± 9.3553 6015 8909 i
1.1720 9815 7227 ±10.2403 1777 0398 i
4.0025 2534 8326 ±10.8348 6485 1671 i
7.7243 3865 4741 ±10.9536 0414 4523 i
13.1748 6483 8907 ±10.1262 6417 8727 i
– 7.3079 8221 4646

The verb p. in J [3] works with polynomials. The dyad x p. y evaluates the polynomial x at y ; the monad p. x converts x between coefficients c and multiplier and roots (m;r) . The coefficients of the partial sum of ez for n=23 are:

   c=: % ! i.24
   c
1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 ...

Converting from coefficients to muliplier and roots:

   mr=: p. c
   mr
┌───────────┬──────────────────────────────────────────────── 
│3.86817e_23│13.1749j10.1263 13.1749j_10.1263 7.72434j10.9536 ...
└───────────┴────────────────────────────────────────────────
   r=: > {: mr
   r
13.1749j10.1263 13.1749j_10.1263 7.72434j10.9536 ...

The roots from 1953 are ordered in increasing modulus, with the real root last; the roots computed by J are ordered in decreasing modulus. We compare the evaluation of the roots from 1953 and of the reordered roots computed by J:

   r1953=: _7.192685907451j1.475059195082 ...
   _5 ]\ | c p. r1953
7.36698e_14 7.36698e_14 1.37043e_13 1.37043e_13 1.20857e_13
1.20857e_13 6.67108e_14 6.67108e_14 8.19945e_14 8.19945e_14
1.53714e_13 1.53714e_13 2.75251e_13 2.75251e_13 6.80251e_12
6.80251e_12 1.12317e_10 1.12317e_10   3.0922e_9   3.0922e_9
 5.55512e_8  5.55512e_8 9.19177e_11           0           0

   _5 ]\ | c p. 1 |. r /: | r
3.76597e_14 3.76597e_14  4.7101e_14  4.7101e_14  2.6361e_14
 2.6361e_14 5.01272e_14 5.01272e_14 9.52945e_14 9.52945e_14
 3.9363e_14  3.9363e_14 2.26531e_13 2.26531e_13 3.41104e_13
3.41104e_13 3.58005e_12 3.58005e_12 3.33604e_11 3.33604e_11
 3.7143e_10  3.7143e_10 1.73195e_14           0           0

I survey the last results with considerable relief. It would have been shameful to produce a worse result 59 years later.
 

References

[0]   Iverson, K.E., Machine Solutions of Linear Differential Equations — Applications to a Dynamic Economic Model, Doctoral Thesis, Harvard University, 1954-01. http://www.jsoftware.com/papers/MSLDE.htm
[1] http://ftp.math.utah.edu/pub//tex/bib/mathcomp1950.html#Iverson:1953:ZPS
[2] http://www.ams.org/journals/mcom/1953-07-043/ S0025-5718-1953-0057013-0/S0025-5718-1953-0057013-0.pdf
[3] Hui, R.K.W., and Iverson, K.E., J Introduction and Dictionary, Jsoftware Inc., 2012. http://www.jsoftware.com/help/dictionary/dpdot.htm


created:  2012-08-14 07:25
updated:2014-01-08 14:30