Consider $$F(x) = \sqrt{x -\sqrt{2x - \sqrt{3x - \cdots}}}$$

I believe I can prove (with some handwaving) that

• $F$ does converge everywhere in $\mathbb{C}$
• $\Im F = 0$ for sufficiently large real $x$ (actually larger than $x0 \approx 0.5243601\dots$ Does this number ring a bell?)
• Coincidentally $F(x0) = 0$

Weird things happen in the limit to $0$. Obviously, $F(0) = 0$. However, it seems that $$\lim_{x \to +0}F(x) = \overline{\zeta}$$ $$\lim_{x \to -0}F(x) = \zeta$$ where $\zeta = \frac{1 + i\sqrt{3}}{2}$ is a usual cubic root of $-1$. Moreover, $F$ seems to reach one of those as $x$ approaches $0$ at a rational angle. I understand that this may well be a computational artifact (still making no sense to me), but proving or refuting these limits is definitely out of my league.

Any help?

• What do you mean by the square root of a negative (or complex) number? Are you fixing one particular branch? If so, there is no reason to expect $F$ to be continuous. (Even $\sqrt{z}$ itself is not continuous on $\mathbb{C}$.) – mrf Jan 31 '13 at 8:50
• @mrf: It is continuous at 0, isn't it? – user58697 Jan 31 '13 at 9:29
• What algorithm do you use to compute $F$? – lhf Jan 31 '13 at 10:22
• @lhf: my own. This question actually arose from testing the algo. It is not yet in the shape to be published. – user58697 Jan 31 '13 at 18:52
• Limit of which expression? I mean, one without ... Perhaps something like math.stackexchange.com/questions/61048/…. – lhf Feb 1 '13 at 10:49

This is not an answer, but some data for illustration. There seem to be critical values $n_x$ for some $x$ near zero, such that the partial evaluations becomes calm from initially complex to finally real values. I've interpreted your function for some given n as $$f(x,n)=\sqrt{1x-\sqrt{2x- \cdots \sqrt{nx}}}$$ Then I looked at sequences of $f(x,1)^2,f(x,2)^2,\ldots,f(x,n_x)^2,f(x,n_x+1)^2,\ldots$ to observe, that for any small x there will be a $n_x$ from where the evaluations are no more complex but only real. Here are tables for the three initial values $x_1=0.1, x_2=0.01,x_3=0.001$ . It is interesting, that it seems, that the "critical" $n_x$ converges to some scalar multiple of the reciprocal of $x$ with decimal expansion of 216... . Hmmm....

  For x_1=0.1
n     f(0.1,n)^2
...   ...
11  -0.302448089681-0.698792219012*I
12  -0.403301213973+0.649132465935*I
13  -0.262532045796-0.730589943250*I
14  -0.470664193786+0.569986543411*I
15  -0.116352885709-0.685312310620*I
16  -0.480413735006+0.369973112245*I
17  0.0771358518666-0.590686848231*I
18  -0.388211746572+0.195833371751*I
19   0.100000000000-0.291614520825*I
20                   -0.181148810826
21   0.100000000000-0.131286629014*I
22                  -0.0579158347766
23                   0.0390823334499
24                  -0.0157619545048
25                  0.00542274237370
26                 -0.00443384940800


  For x_2=0.01
n     f(0.01,n)^2
...   ...
205   -0.114449346430-0.449020909239*I
206   -0.273308708016+0.245486912151*I
207   0.0100000000000-0.340544340711*I
208   -0.208411838263+0.166444378377*I
209   0.0100000000000-0.199665697307*I
210  -0.160212530941+0.0947222555124*I
211   0.0100000000000-0.115794748628*I
212                    -0.102423208202
213  0.0100000000000-0.0608690577550*I
214                   -0.0369471693906
215  0.0100000000000-0.0288680515620*I
216                   -0.0136757225356
217  0.0100000000000-0.0107416751802*I
218                  -0.00424432651020
219                   0.00282037509906
220                  -0.00108467768822
221                  0.000551999344977
222                 -0.000248398685385
223                  0.000118131086199
224                -0.0000545149536090
225                 0.0000254248494297
226                -0.0000117292180991
227                0.00000543517323693
228               -0.00000248279244384


  For x_3=0.001 // internal computation precision: 1200 dec digits
n     f(0.001,n)^2
...   ...
2149    0.00100000000000-0.0771283223568*I
2150    -0.0593075793195+0.0404599076046*I
2151    0.00100000000000-0.0465223021845*I
2152    -0.0465304437990+0.0160979218450*I
2153    0.00100000000000-0.0254836686768*I
2154                      -0.0190199767319
2155    0.00100000000000-0.0127892711274*I
2156                     -0.00817903802372
2157   0.00100000000000-0.00611680141511*I
2158                     -0.00341928431634
2159   0.00100000000000-0.00278386814867*I
2160                     -0.00127422891686
2161  0.00100000000000-0.000989592316296*I
2162                    -0.000393138741302
2163                     0.000256090838303
2164                    -0.000100925375352
2165                    0.0000516413154610
2166                   -0.0000235893686835
2167                    0.0000113842433808
2168                  -0.00000535253161006
2169                   0.00000254701184424
2170                  -0.00000120460035318
2171                  0.000000571115136334
2172                 -0.000000270334146946


Here is a picture of the trajectory for $x_3=0.001$ and increasing n. Using to little internal precision (200 dec digits in Pari/GP) made it appear, that this converged to the 3rd complex unitroot v (where $v^3=1$) but using 800 digits precision let it converge to something near zero. I've not yet analyzed this in more detail... (Update/Correction: in the picture I've incorrectly writen $f(x,n)$ instead of $f(x,n)^2$. I'll correct the image later)

  For x_4=0.0001 // internal computation precision: 3600 dec digits
n     f(0.0001,n)^2
...   ...
21579    0.000100000000000-0.00210357411418*I
21580                       -0.00136741030855
21581    0.000100000000000-0.00100044565776*I
21582                      -0.000601677005986
21583   0.000100000000000-0.000467850547807*I
21584                      -0.000244806521553
21585   0.000100000000000-0.000204301322925*I
21586                     -0.0000860033335819
21587  0.000100000000000-0.0000411110536353*I
21588                     -0.0000247254403485
21589                      0.0000142141209815
21590                    -0.00000608992862360
21591                     0.00000302817937704
21592                    -0.00000140739716572
21593                    0.000000675808679202
21594                   -0.000000319550497880
21595                    0.000000152205000659
21596                  -0.0000000722426625118
21597                   0.0000000343450467856
21598                  -0.0000000163147256218
21599                  0.00000000775254705649
21600                 -0.00000000368314354146
21601                  0.00000000174991223509
21602                -0.000000000831348437546