Is there a Tetration Function?

Question

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?

Answer 1

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).

Answer 2

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