Deriving tetration from selfroot?
#11
As far as I understand selfroot and any root:

Selfroot is x^(1/x):

Selfroot = \( x^{1/x}= e^{\ln(x^{(1/x})}=e^{(\ln(x)/x)}*e^{+-(2*\pi*I*{k/x)} \)

For positive x, \( {\ln(x)/x} \) is real, giving one real self root (k=0) \( e^{(\ln(x)/x)} \), and infinite number of imaginary roots depending on ratio \( 2*{k/x} \).

For negative x, \( {\ln(x)/x} \) will be imaginary, of the form:

\( {\ln(x)/x}+-{((2m-1)/x)}*\pi*I \) so roots are:

\( e^{(\ln(x)/x)}*e^{-+{((2m-1)/x)}*\pi*I}*e^{+-(2*\pi*I*{k/x)} \), again infinite quantity in totality. m=0 gives:

\( e^{(\ln(x)/x)}*e^{+-{(1/x)}*\pi*I}*e^{+-(2*\pi*I*{k/x)} \),


For imaginary and complex x, tomorrow, with mistakes here also hopefully corrected.

Ivars


Messages In This Thread
Deriving tetration from selfroot? - by Ivars - 03/12/2008, 08:26 AM
RE: Deriving tetration from selfroot? - by Ivars - 03/20/2008, 05:36 PM
RE: Deriving tetration from selfroot? - by Ivars - 03/20/2008, 09:53 PM
RE: Deriving tetration from selfroot? - by Ivars - 03/21/2008, 07:51 AM
RE: Deriving tetration from selfroot? - by Ivars - 03/21/2008, 11:31 PM
RE: Deriving tetration from selfroot? - by Ivars - 03/22/2008, 09:52 AM
RE: Deriving tetration from selfroot? - by Ivars - 03/22/2008, 02:23 PM
RE: Deriving tetration from selfroot? - by Ivars - 03/22/2008, 03:08 PM
RE: Deriving tetration from selfroot? - by Ivars - 03/24/2008, 10:26 PM
RE: Deriving tetration from selfroot? - by Ivars - 03/25/2008, 05:52 PM
RE: Generalized recursive operators - by Ivars - 03/13/2008, 08:01 AM



Users browsing this thread: 1 Guest(s)