Dang it, I used to know this.
What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
Exponential function
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
> Exponential function
No. e^(e^x) ≠ e^x
mollwollfumble said:
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
> Exponential function
No. e^(e^x) ≠ e^x
The exponential function is a convex function. So by Jensen’s Inequality E(eX)≤eE(X). Equality will only hold if X is degenerate.
mollwollfumble said:
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
> Exponential function
No. e^(e^x) ≠ e^x
I don’t even know what a special function is, but the internet tells me that exponential functions are not special, so are exponential exponential functions special?
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
If you can read German, here’s a 1950 journal article on that very topic (I can’t find an English translation)
Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen.
FWIW, it’s a holomorphic Abel function. it’s also a half exponential function.
Putin probably knows something about these
SCIENCE said:
Putin probably knows something about these
The gloss somewhat worn off.
roughbarked said:
SCIENCE said:
Putin probably knows something about these
The gloss somewhat worn off.
so it’s a smooth special function with a turning point at Kharkiv
The Rev Dodgson said:
mollwollfumble said:
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
> Exponential function
No. e^(e^x) ≠ e^x
I don’t even know what a special function is, but the internet tells me that exponential functions are not special, so are exponential exponential functions special?
Special functions, see https://en.wikipedia.org/wiki/List_of_mathematical_functions
The function f(x) such that f(f(x)) = log x is the one I most want to track down.
For sufficiently large x,
This f(x) is smaller than x^ε for arbitrarily small positive ε.
And larger than (log x)^n for arbitrarily large n.
I tried “logarithmic integral”, “iterated logarithm” and “super-logarithm” from that list. No luck, except that “super-logarithm” is similar, it’s larger than a constant and smaller than log(log( …(log x)… ) ) for arbitrarily many nestings of the logarithm function.
Dang it. Lost my last post here and didn’t even notice it.
The function f(f(x)) = exp(x)
is called the half-exponential function. I found it by accident on wikipedia.
https://en.wikipedia.org/wiki/Functional_square_root
https://en.wikipedia.org/wiki/Half-exponential_function
https://en.wikipedia.org/wiki/Fractional_calculus
mollwollfumble said:
Dang it. Lost my last post here and didn’t even notice it.The function f(f(x)) = exp(x)
is called the half-exponential function. I found it by accident on wikipedia.https://en.wikipedia.org/wiki/Functional_square_root
https://en.wikipedia.org/wiki/Half-exponential_function
https://en.wikipedia.org/wiki/Fractional_calculus
Took you this long?
btm said:
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
If you can read German, here’s a 1950 journal article on that very topic (I can’t find an English translation)
Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen.FWIW, it’s a holomorphic Abel function. it’s also a half exponential function.
*bump* for moll.
btm said:
btm said:
mollwollfumble said:
Dang it, I used to know this.What is the name of the special function f(x) that satisfies f(f(x)) = ln(x)?
What is the name of the special function g(x) that satisfies g(g(x)) = exp(x)?
If you can read German, here’s a 1950 journal article on that very topic (I can’t find an English translation)
Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen.FWIW, it’s a holomorphic Abel function. it’s also a half exponential function.
*bump* for moll.
Note that there are an infinite number of functions f(x) such that f(f(x)) = e^x …