03/16/2011, 02:43 AM

(03/16/2011, 02:35 AM)lloyd Wrote: Hey James, unfortunately my functions don't meet your requirement :-(

x {1.5} 2 != x {0.5} x, here are some random examples for x = 2,3,4,11,134

> ./sesqui 2 1.5 2

4.000000000

> ./sesqui 2 0.5 2

4.000000000

> ./sesqui 3 1.5 2

7.310586452

> ./sesqui 3 0.5 3

7.424041271

> ./sesqui 4 1.5 2

11.173642159

> ./sesqui 4 0.5 4

11.654329300

> ./sesqui 11 1.5 2

49.238006592

> ./sesqui 11 0.5 11

61.141593933

> ./sesqui 134 1.5 2

1893.553344727

> ./sesqui 134 0.5 134

5044.535156250

Though I admit I don't totally see why your identity is required, it seems to be by analogy with the gamma extension to the factorial. Could you explain it in simple terms? Thanks!

well the identity isn't required per se, it's just aesthetically appealing for an operator to have an identity. The only reason we would need an identity is if we wanted to use semi-operators to solve for tetration.

under the following expansion:

x {q} y = -q:log(q:log(x) + q:log(y))

x {q} S(q) = x = -q:log(q:log(x) + q:log(S(q))),

therefore q:log(S(q))= 0

therefore S(q) = -q:log(0) = sexp(q-1)

but since there is no identity for the sesqui operator all this is null.