So I have been revamping the functions I use for tetration, and I recently solved all the mumbo-jumbo to use tetration as the function in the NaturalIterate function. So now you can do:

I'm still working on doing this for other bases, but this is enough to get an expansion of pentation about zero.

What follows below is InverseSeries[...] of the output from above. In other words, the above gives the coefficients of the base-e penta-logarithm about (0), and the following gives the coefficients of the base-e penta-exponential about (-1).

So, what I'm wondering is, how do I turn this into an expansion about 3i?

Code:

`<<Tetration``

NaturalIterate[Series[Tetrate[E, x], {x, 0, 3}], z]

I'm still working on doing this for other bases, but this is enough to get an expansion of pentation about zero.

What follows below is InverseSeries[...] of the output from above. In other words, the above gives the coefficients of the base-e penta-logarithm about (0), and the following gives the coefficients of the base-e penta-exponential about (-1).

Code:

`0,`

0.997386001614238200000,

-0.044854069033065140000,

0.008127184531878105000,

0.045268576293608810000,

-0.009169795166599723000,

0.000529626080101428000,

0.003682350459440369500,

-0.001300714479652927000,

0.000136554270543782140,

0.000349632018705509600,

-0.000212903018660854500,

0.000030850789704285015,

0.000053653522961255240,

-0.000028243223065159680,

-0.000003800898968414997,

0.000000972449120890964,

0.000005775482651540000,

0.000010790317715530437,

-0.000029357772002764790,

0.000020775705975594905

So, what I'm wondering is, how do I turn this into an expansion about 3i?