+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: inspired by an equation (/showthread.php?tid=481)
inspired by an equation - tommy1729 - 07/27/2010
i was inspired by
f(z) = A^(Lz) = L^z.
which looks neat to me.
i havent really thought about it deep but here is an attempt :
A^(Lz) = L^z
=> A^L = L ; L^p = 1
p = 2pi i / (ln(A) L) ( the period )
=> A^L = L ; L^p = L^ 2pi i / (ln(A) L) = 1
=> L ln(A) = ln(L) => ln(A) = ln(L)/L
=> L^ 2pi i / ( ln(L) * L/L ) = 1
=> L^ 2pi i / ln(L) = 1
and now ? ln on both sides ?
=> 2 pi i / ln(L) * ln(L) = 0
=> 2pi i = 0 ??
certainly not all A and L satisfy A^(Lz) = L^z.
so where did the variables go to ?
is there no solution ?
i assume taking ln on both sides is the error
=> solve for L : L^ 2pi i / ln(L) = 1
and now ... Lambert W function ?
or is that equation already wrong because of branches and we need to return to
=> A^L = L ; L^p = L^ 2pi i / (ln(A) L) = 1
but thats again an equation in 2 complex variables :s
of course f(z) = A^(Lz) = L^z has the trivial solution f(z) = 1 or f(z) = 0.
but im looking for others.
maybe => 2 pi i / ln(L) * ln(L) = 0
means that all solutions for A must be integer and hence only the trivial solutions f(z) = 1 or f(z) = 0 exist.
i cant find or imagine any other.
also the generalizations of this equation inspire me.
analogues with double periodic functions , finding solutions in terms of other numbers ( 3d complex or other ) , replacing exp with another function and multiplication with its inv superfunction of that other function etc etc
this may be trivial sorry ...
( should have paid attention in class as a teenager :p ... if that was in class ... )
tommy1729
note to myself : post this kind of stuff in the general section tommy ! bo always puts it there so thats where it belongs
-------------
edit : Ok sorry this is nonsense. A^(Lz) = L^z => take ln on both sides : ln(A) * L * z = ln(L) * z.
take z = 1 :
ln(A) = ln(L)/L
This is a relationship between A and L , not an equation to be solved for.
-------------