Mixing
Solve equations of mixed type analytically
$begingroup$
Consider the problem of solving for y in the following equations:
$$e^y+y=x \ sin y + y = x $$
I have often heard it said that "these types of problems" (finding the inverse of a function with a mixture of polynomial and transcendental functions) are, in general, impossible to do analytically.
I am wondering is there a way to prove this?
Edit (clarification):
One of the comments mentioned that the use of the word "transcendental" in the question is sufficient as to imply the impossibility of solving for y. This may well be the case however what I am looking for is a proof that this is the case. For example: Prove that one can't find an function $y=f(x)$ which is both elementary and satisfies $e^y+y=x$.
analyticity transcendental-equations
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
$endgroup$
add a comment |
$begingroup$
Consider the problem of solving for y in the following equations:
$$e^y+y=x \ sin y + y = x $$
I have often heard it said that "these types of problems" (finding the inverse of a function with a mixture of polynomial and transcendental functions) are, in general, impossible to do analytically.
I am wondering is there a way to prove this?
Edit (clarification):
One of the comments mentioned that the use of the word "transcendental" in the question is sufficient as to imply the impossibility of solving for y. This may well be the case however what I am looking for is a proof that this is the case. For example: Prove that one can't find an function $y=f(x)$ which is both elementary and satisfies $e^y+y=x$.
analyticity transcendental-equations
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
$endgroup$
$begingroup$
Something from math.stackexchange.com/questions/46934/… might provight insight. I still struggle with the fact that for $cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed')
$endgroup$
– Christopher Marley
Jan 25 at 5:51
$begingroup$
I think that the word transcendental by itself says everything en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-Wleft(e^xright)$ where appears Lambert function.
$endgroup$
– Claude Leibovici
Jan 25 at 6:06
add a comment |
$begingroup$
Consider the problem of solving for y in the following equations:
$$e^y+y=x \ sin y + y = x $$
I have often heard it said that "these types of problems" (finding the inverse of a function with a mixture of polynomial and transcendental functions) are, in general, impossible to do analytically.
I am wondering is there a way to prove this?
Edit (clarification):
One of the comments mentioned that the use of the word "transcendental" in the question is sufficient as to imply the impossibility of solving for y. This may well be the case however what I am looking for is a proof that this is the case. For example: Prove that one can't find an function $y=f(x)$ which is both elementary and satisfies $e^y+y=x$.
analyticity transcendental-equations
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
$endgroup$
Consider the problem of solving for y in the following equations:
$$e^y+y=x \ sin y + y = x $$
I have often heard it said that "these types of problems" (finding the inverse of a function with a mixture of polynomial and transcendental functions) are, in general, impossible to do analytically.
I am wondering is there a way to prove this?
Edit (clarification):
One of the comments mentioned that the use of the word "transcendental" in the question is sufficient as to imply the impossibility of solving for y. This may well be the case however what I am looking for is a proof that this is the case. For example: Prove that one can't find an function $y=f(x)$ which is both elementary and satisfies $e^y+y=x$.
analyticity transcendental-equations
analyticity transcendental-equations
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
edited Feb 4 at 3:51
Isaac Greene
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
asked Jan 25 at 4:38
Isaac GreeneIsaac Greene
318111
318111
$begingroup$
Something from math.stackexchange.com/questions/46934/… might provight insight. I still struggle with the fact that for $cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed')
$endgroup$
– Christopher Marley
Jan 25 at 5:51
$begingroup$
I think that the word transcendental by itself says everything en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-Wleft(e^xright)$ where appears Lambert function.
$endgroup$
– Claude Leibovici
Jan 25 at 6:06
add a comment |
$begingroup$
Something from math.stackexchange.com/questions/46934/… might provight insight. I still struggle with the fact that for $cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed')
$endgroup$
– Christopher Marley
Jan 25 at 5:51
$begingroup$
I think that the word transcendental by itself says everything en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-Wleft(e^xright)$ where appears Lambert function.
$endgroup$
– Claude Leibovici
Jan 25 at 6:06
$begingroup$
Something from math.stackexchange.com/questions/46934/… might provight insight. I still struggle with the fact that for $cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed')
$endgroup$
– Christopher Marley
Jan 25 at 5:51
$begingroup$
Something from math.stackexchange.com/questions/46934/… might provight insight. I still struggle with the fact that for $cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed')
$endgroup$
– Christopher Marley
Jan 25 at 5:51
$begingroup$
I think that the word transcendental by itself says everything en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-Wleft(e^xright)$ where appears Lambert function.
$endgroup$
– Claude Leibovici
Jan 25 at 6:06
$begingroup$
I think that the word transcendental by itself says everything en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-Wleft(e^xright)$ where appears Lambert function.
$endgroup$
– Claude Leibovici
Jan 25 at 6:06
add a comment |
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$begingroup$
Something from math.stackexchange.com/questions/46934/… might provight insight. I still struggle with the fact that for $cos x = x, x$ has no "closed form," i.e. can't be written without summation or helping functions (that's how I defined 'closed')
$endgroup$
– Christopher Marley
Jan 25 at 5:51
$begingroup$
I think that the word transcendental by itself says everything en.wikipedia.org/wiki/Transcendental_equation Some special functions can be used in some cases, such as the first equation you wrote; the solution of it being $y=x-Wleft(e^xright)$ where appears Lambert function.
$endgroup$
– Claude Leibovici
Jan 25 at 6:06