Expressions for the inverse function of f(x) = ln(x)e^x
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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
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up vote
8
down vote
favorite
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
New contributor
Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
8
down vote
favorite
up vote
8
down vote
favorite
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
New contributor
Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
New contributor
Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 13 at 6:33
Hiraxin
412
New contributor
Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Nov 13 at 6:33
Hiraxin
412
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Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Nov 13 at 6:33
Hiraxin
412
asked Nov 13 at 6:33
Hiraxin
412
412
New contributor
Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Hiraxin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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
71.3k9160267
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
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
71.3k9160267
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
up vote
7
down vote
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
71.3k9160267
Thank you much. :)
– Hiraxin
Nov 13 at 14:07
add a comment |
up vote
7
down vote
up vote
7
down vote
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
71.3k9160267
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
71.3k9160267
edited Nov 13 at 9:54
answered Nov 13 at 9:47
Carlo Beenakker
71.3k9160267
answered Nov 13 at 9:47
Carlo Beenakker
71.3k9160267
answered Nov 13 at 9:47
Carlo Beenakker
71.3k9160267
71.3k9160267
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 |
Hiraxin is a new contributor. Be nice, and check out our Code of Conduct.
Hiraxin is a new contributor. Be nice, and check out our Code of Conduct.
Hiraxin is a new contributor. Be nice, and check out our Code of Conduct.
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