03/29/2009, 11:23 AM
As it is well-known we have for \( b<e^{1/e} \)
the regular superexponential at the lower fixed point.
This can be obtained by computing the Schroeder function at the fixed point \( a \) of \( F(x)=b^x \).
More precisely we set
\( G(x)=F(x+a)-a = b^{x+a}-a=b^a b^x -a = a b^x -a = a(b^x-1) \)
This is a function with fixed point at 0, it is the function \( F \) shifted that its fixed point is at 0.
We compute the Schroeder function \( \chi \) of \( G \), i.e. the solution of:
\( \chi(G(x))=c\chi(x) \) where \( c=G'(0)=a\ln(b)=\ln(b^a)=\ln(a) \).
This has a unique analytic solution with \( \chi'(0)=1 \).
Then we get the super exponential by
\( \operatorname{sexp}_b(t)=a+\chi^{-1}(c^x \chi(y_0) \)
\( y_0 \) is adjusted such that
\( 1=\operatorname{sexp}_b(0)=a+\chi^{-1}(\chi(y_0))=a+y_0 \)
i.e. \( y_0=1-a \).
This procedure can be applied to any fixed point \( a \) of \( b^x \).
The normal regular superexponential is obtained by applying it to the lower fixed point.
Now the upper regular superexponential \( \operatorname{usexp} \) is the one obtained at the upper fixed point of \( b^x \).
For this function we have however always \( \operatorname{usexp}(x)>a \),
so the condition \( \operatorname{usexp}(0)=1 \) can not be met.
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)
The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for \( x<-2 \).
It is entire because the inverse Schroeder function \( \chi^{-1} \) is entire, it can be continued from an initial small disk of radius r around 0 By the equation
\( \chi^{-1}(c^n x)=G^{[n]}(\chi(x)) \)
We know that \( c>1 \) thatswhy we cover the whole complex plane with \( c^nx \), \( x \) from the initial disc around 0, and we know that \( G^{[n}] \) is entire.
Here are some pictures of \( \operatorname{sexp} \) that are computed via the regular schroeder function as powerseries for our beloved base \( b=\sqrt{2} \), \( a=2,4 \):
and here the upper super exponential base 2 alone:
the regular superexponential at the lower fixed point.
This can be obtained by computing the Schroeder function at the fixed point \( a \) of \( F(x)=b^x \).
More precisely we set
\( G(x)=F(x+a)-a = b^{x+a}-a=b^a b^x -a = a b^x -a = a(b^x-1) \)
This is a function with fixed point at 0, it is the function \( F \) shifted that its fixed point is at 0.
We compute the Schroeder function \( \chi \) of \( G \), i.e. the solution of:
\( \chi(G(x))=c\chi(x) \) where \( c=G'(0)=a\ln(b)=\ln(b^a)=\ln(a) \).
This has a unique analytic solution with \( \chi'(0)=1 \).
Then we get the super exponential by
\( \operatorname{sexp}_b(t)=a+\chi^{-1}(c^x \chi(y_0) \)
\( y_0 \) is adjusted such that
\( 1=\operatorname{sexp}_b(0)=a+\chi^{-1}(\chi(y_0))=a+y_0 \)
i.e. \( y_0=1-a \).
This procedure can be applied to any fixed point \( a \) of \( b^x \).
The normal regular superexponential is obtained by applying it to the lower fixed point.
Now the upper regular superexponential \( \operatorname{usexp} \) is the one obtained at the upper fixed point of \( b^x \).
For this function we have however always \( \operatorname{usexp}(x)>a \),
so the condition \( \operatorname{usexp}(0)=1 \) can not be met.
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)
The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for \( x<-2 \).
It is entire because the inverse Schroeder function \( \chi^{-1} \) is entire, it can be continued from an initial small disk of radius r around 0 By the equation
\( \chi^{-1}(c^n x)=G^{[n]}(\chi(x)) \)
We know that \( c>1 \) thatswhy we cover the whole complex plane with \( c^nx \), \( x \) from the initial disc around 0, and we know that \( G^{[n}] \) is entire.
Here are some pictures of \( \operatorname{sexp} \) that are computed via the regular schroeder function as powerseries for our beloved base \( b=\sqrt{2} \), \( a=2,4 \):
and here the upper super exponential base 2 alone: