11/19/2011, 11:56 PM
no , im not kidding , apart from my sinh method im considering a cosh method.
although it might be more complicated or speculative.
it might be insightfull to plot things.
consider
exp(1/e)<b<sqrt(e)
f(x) - f^[-1](x) = 2*cosh(ln(b)*x)
for variable x and fixed b ( base ).
i bet you can find more functional equations from this one.
for instance to express f^[-1](x) in terms of f(x) and elementary functions.
the solution is unique for R-> R.
replace x by f(x) and let g(x) be the super of f(x).
then we get a difference equation for g(x).
we solve that difference equation , now we go from g(x) to f(x) or we find the taylor for f(x) from the equation ( or both , in programming ).
similar to the sinh method : since f is close to 2*cosh(ln(b)*x)) and 2*cosh(ln(b)*x) is close to b^x , we have a new method.
although it might be more complicated or speculative.
it might be insightfull to plot things.
consider
exp(1/e)<b<sqrt(e)
f(x) - f^[-1](x) = 2*cosh(ln(b)*x)
for variable x and fixed b ( base ).
i bet you can find more functional equations from this one.
for instance to express f^[-1](x) in terms of f(x) and elementary functions.
the solution is unique for R-> R.
replace x by f(x) and let g(x) be the super of f(x).
then we get a difference equation for g(x).
we solve that difference equation , now we go from g(x) to f(x) or we find the taylor for f(x) from the equation ( or both , in programming ).
similar to the sinh method : since f is close to 2*cosh(ln(b)*x)) and 2*cosh(ln(b)*x) is close to b^x , we have a new method.