Expressions for the inverse function of f(x) = ln(x)e^x
$begingroup$
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 '18 at 6:33
HiraxinHiraxin
412
$endgroup$
add a comment |
$begingroup$
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 '18 at 6:33
HiraxinHiraxin
412
$endgroup$
add a comment |
$begingroup$
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 '18 at 6:33
HiraxinHiraxin
412
$endgroup$
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 '18 at 6:33
HiraxinHiraxin
412
asked Nov 13 '18 at 6:33
HiraxinHiraxin
412
asked Nov 13 '18 at 6:33
HiraxinHiraxin
412
asked Nov 13 '18 at 6:33
HiraxinHiraxin
412
asked Nov 13 '18 at 6:33
HiraxinHiraxin
412
412
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f315204%2fexpressions-for-the-inverse-function-of-fx-lnxex%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
$endgroup$
add a comment |
$begingroup$
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
$endgroup$
add a comment |
$begingroup$
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
$endgroup$
These are so-called hyper-Lambert functions, see
On some applications of the generalized hyper-Lambert functions.
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
edited Nov 13 '18 at 9:54
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
answered Nov 13 '18 at 9:47
Carlo BeenakkerCarlo Beenakker
74.6k9169277
74.6k9169277
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f315204%2fexpressions-for-the-inverse-function-of-fx-lnxex%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
