Expressions for the inverse function of f(x) = ln(x)e^x
Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable.
The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $ W(xe^x) = x. $
It is natural then to consider the inverse of functions such as $ g(x) = xe^{e^x} $ and those with further exponentiation. With a simple transformation $ z= e^x $ we can reduce $ g(x) $ to the form $ ln(z)e^z $ as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set
$$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$
real-analysis
asked Nov 13 at 6:33
Hiraxin
412
add a comment |
Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable.
The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $ W(xe^x) = x. $
It is natural then to consider the inverse of functions such as $ g(x) = xe^{e^x} $ and those with further exponentiation. With a simple transformation $ z= e^x $ we can reduce $ g(x) $ to the form $ ln(z)e^z $ as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set
$$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$
real-analysis
asked Nov 13 at 6:33
Hiraxin
412
add a comment |
Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable.
The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $ W(xe^x) = x. $
It is natural then to consider the inverse of functions such as $ g(x) = xe^{e^x} $ and those with further exponentiation. With a simple transformation $ z= e^x $ we can reduce $ g(x) $ to the form $ ln(z)e^z $ as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set
$$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$
real-analysis
asked Nov 13 at 6:33
Hiraxin
412
Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable.
The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $ W(xe^x) = x. $
It is natural then to consider the inverse of functions such as $ g(x) = xe^{e^x} $ and those with further exponentiation. With a simple transformation $ z= e^x $ we can reduce $ g(x) $ to the form $ ln(z)e^z $ as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set
$$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$
real-analysis
real-analysis
asked Nov 13 at 6:33
Hiraxin
412
asked Nov 13 at 6:33
Hiraxin
412
asked Nov 13 at 6:33
Hiraxin
412
asked Nov 13 at 6:33
Hiraxin
412
asked Nov 13 at 6:33
Hiraxin
412
412
add a comment |
add a comment |
1 Answer
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These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
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1 Answer
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These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
edited Nov 13 at 9:54
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
answered Nov 13 at 9:47
Carlo Beenakker
73.1k9164273
73.1k9164273
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
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