02/11/2009, 03:42 PM
[attachment=441]
your post is very intresting Gottfried.
in fact it shows the following important result :
if f(x) ( being f(f(x)) = exp(x) ) exists and is unique on a finite (real) interval I ,
then f(x) exists and is unique for all real x.
---
i think this can be extended to the property of analytic but im not sure. more specific f(x) analytic on interval I -> real-analytic.
---
does your data match robbins ?
do you reach the same x for f(x) = 0 ?
can you say more about the areas of the rectangles in a real interval based upon the estimate ?
---
Gottfried's post leads to a way to extend f(x) to a given (real) domain.
i was already aware of this concept , but its nice to see a graph.
in particular i prefer these 'rectangles' to estimate the neighbourhood of x = 0 given f(x) in an interval (a,b) with a,b > 1.
this way we exploit the higher precision of my method for x > 0 and ' transfer ' that to the neighbourhood of x = 0.
nice work Gottfried.
high regards
tommy1729
your post is very intresting Gottfried.
in fact it shows the following important result :
if f(x) ( being f(f(x)) = exp(x) ) exists and is unique on a finite (real) interval I ,
then f(x) exists and is unique for all real x.
---
i think this can be extended to the property of analytic but im not sure. more specific f(x) analytic on interval I -> real-analytic.
---
does your data match robbins ?
do you reach the same x for f(x) = 0 ?
can you say more about the areas of the rectangles in a real interval based upon the estimate ?
---
Gottfried's post leads to a way to extend f(x) to a given (real) domain.
i was already aware of this concept , but its nice to see a graph.
in particular i prefer these 'rectangles' to estimate the neighbourhood of x = 0 given f(x) in an interval (a,b) with a,b > 1.
this way we exploit the higher precision of my method for x > 0 and ' transfer ' that to the neighbourhood of x = 0.
nice work Gottfried.
high regards
tommy1729