# Is there a Tetration Function?

With exponentiation, you can raise numbers to complex, irrational, etc. This is defined as such:

$$\exp(x)=\sum_{n=1}^\infty{x^n\over{n!}}$$ With $e=\exp(1)$

Is there some equation that would allow me to tetrate numbers to complex numbers?

• There exists many non-unique extensions. But none of them are easy to construct. The best, or what is probably the most "well known" (using this term is a bit of a stretch) is Hellmuth Kneser's Tetration. However the construction is in German and it is very hard to find English accounts of it. Plus, it is not easy at all to construct, and is a very deep result. – user335907 Jan 24 '18 at 20:03
• @james.nixon I'm not finding a lot on google, do you have any links? – user406613 Jan 24 '18 at 22:20
• If you have the time you can peruse this forum dedicated to tetration: math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3 Here, Henryk Trappman (bo198214) has a great dissection of Kneser's proof in English (he actually speaks german). Can't seem to find it at the moment. Otherwise you can look at the newer, less rigorous, Kouznetsov Tetration, which will show up on google if you search "Dmitri Kouznetsov Tetration". Here's a link to the paper: researchgate.net/publication/… – user335907 Jan 25 '18 at 3:40
• Surprisingly both constructions are incredibly different. Kneser's is more magical yet, oddly enough, more rigorous. Kouznetsov's is very intuitive, but proves incredibly hard to make rigorous. Happy hunting! – user335907 Jan 25 '18 at 3:41
• Besides the vast multitude of entries in the "tetration-forum" you might be interested in a couple of exporative (and unsystematic) approaches showing some powerseries for attempted solutions and (intended) introductory texts how to find such power series at my webspace go.helms-net.de/math/tetdocs/index.htm . Unfortunately I've not the time at the moment to compose a proper answer to give some explicite example. (But you might look at older answers of mine here around as well) – Gottfried Helms Jan 25 '18 at 7:36

Not an answer, but more of a plug for a great formula. If instead you asked for a function

$$F(z) : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0}$$ $$F'(z) \neq 0$$ $$F(1) = \sqrt{2}$$ $$F:\mathbb{R}^+ \to \mathbb{R}^+$$ $$F(z+1) = \sqrt{2}^{F(z)}$$

then there is a ''nice'' formula you can write down:

$$F(z)\Gamma(1-z) = \sum_{n=0}^\infty \sqrt{2}^{...(n+1\,\text{times})...^{\sqrt{2}}}\frac{(-1)^n}{n!(n+1-z)} + \int_1^\infty f(x)x^{-z}\,dx$$

where $\Gamma$ is the Gamma function and

$$f(x) = \sum_{n=0}^\infty \sqrt{2}^{...(n+1\,\text{times})...^{\sqrt{2}}}\frac{(-x)^n}{n!}$$

So, a very nice tetration exists if we let the base be $\sqrt{2}$. Your choice of $e$, makes the solution infinitely times harder. In general if the base $\alpha$ satisfies $1 < \alpha < e^{1/e}$ then Tetration is a breeze (taking the term 'breeze' in a comparative sense).

I posted a description of Kneser's construction with link's to the Tetration Forum in the answer to this mathstack question.

Operational details (Implementation) of Kneser's method of fractional iteration of function $\exp(x)$?

Here is the Taylor series representation for Tetration base e that the Op asked for.

Tet=    1.0000000000000000000000000000000
+x^ 1*  1.0917673512583209918013845500272
+x^ 2*  0.27148321290169459533170668362355
+x^ 3*  0.21245324817625628430896763774095
+x^ 4*  0.069540376139987373728674232707469
+x^ 5*  0.044291952090473304406440344385515
+x^ 6*  0.014736742096389391152096286915534
+x^ 7*  0.0086687818172252603663803925296400
+x^ 8*  0.0027964793983854596948259913011496
+x^ 9*  0.0016106312905842720721626451640261
+x^10*  0.00048992723148437733469866722583248
+x^11*  0.00028818107115404581134526404129647
+x^12*  8.0094612538543333444273583009993 E-5
+x^13*  5.0291141793805403694590114624204 E-5
+x^14*  1.2183790344900091616191711098593 E-5
+x^15*  8.6655336673815746852458045541053 E-6
+x^16*  1.6877823193175389917890093175838 E-6
+x^17*  1.4932532485734925810665044317328 E-6
+x^18*  1.9876076420492745531981897949682 E-7
+x^19*  2.6086735600432637316458216085329 E-7
+x^20*  1.4709954142541901861412188182476 E-8
+x^21*  4.6834497327413506255093709930066 E-8
+x^22* -1.5492416655467695218054651764483 E-9
+x^23*  8.7415107813509359129925581171223 E-9
+x^24* -1.1257873101030623175751345157384 E-9
+x^25*  1.7079592672707284125656087787297 E-9
+x^26* -3.7785831549229851764921434925003 E-10
+x^27*  3.4957787651102163178731456499355 E-10
+x^28* -1.0537701234450015066294257929171 E-10
+x^29*  7.4590971476075052807322832021897 E-11
+x^30* -2.7175982065777348693298771724927 E-11
+x^31*  1.6460766106614471303885081821758 E-11
+x^32* -6.7418731524050529991474534636770 E-12
+x^33*  3.7253287233194685443170869606893 E-12
+x^34* -1.6390873267935902234582078934200 E-12
+x^35*  8.5836383113585680604886655432574 E-13
+x^36* -3.9437387391053843135794898834433 E-13
+x^37*  2.0025231280218870558935267045861 E-13
+x^38* -9.4419622429240650237151115800284 E-14
+x^39*  4.7120547458493713408174143933546 E-14
+x^40* -2.2562918820355970800432727061447 E-14
+x^41*  1.1154688506165369962930937106089 E-14
+x^42* -5.3907455570163504918409316383858 E-15
+x^43*  2.6521584915166818728172077683151 E-15
+x^44* -1.2889107655445536819339944924425 E-15
+x^45*  6.3266785019566604530078403061858 E-16
+x^46* -3.0854571504923359889618334580896 E-16
+x^47*  1.5131767717827405273370068884076 E-16
+x^48* -7.3965341370947514335796587568471 E-17
+x^49*  3.6269876710541876048589007540385 E-17
+x^50* -1.7757255986762984036221574832757 E-17