Laplace Transform of Lambert W function
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Does there exist a Laplace transform of the Lambert W function (evaluated at $at$, where $a$ is a constant) that can be expressed in terms of elementary functions and the product log ($W(x)$)?
The Lambert W function is defined as the inverse of $f$ given $f(x) = xe^x$
$W'(x) = W(x)/x(1+W(x))$
$int W(ax) = x(W(ax)−1)+xW(ax)+C$
The Laplace Transform would be the evaluation of the improper integral $int_0^infty e^{-st}W(at)dt$
If such a transform exists, how is it expressed in terms of $s$, and how is it derived?
integration improper-integrals laplace-transform lambert-w
asked Oct 25 at 1:32
ThoughtOfGod
348
add a comment |
up vote
2
down vote
favorite
Does there exist a Laplace transform of the Lambert W function (evaluated at $at$, where $a$ is a constant) that can be expressed in terms of elementary functions and the product log ($W(x)$)?
The Lambert W function is defined as the inverse of $f$ given $f(x) = xe^x$
$W'(x) = W(x)/x(1+W(x))$
$int W(ax) = x(W(ax)−1)+xW(ax)+C$
The Laplace Transform would be the evaluation of the improper integral $int_0^infty e^{-st}W(at)dt$
If such a transform exists, how is it expressed in terms of $s$, and how is it derived?
integration improper-integrals laplace-transform lambert-w
asked Oct 25 at 1:32
ThoughtOfGod
348
Of course, the Laplace transform of the Lambert's W function exists. As far as I know, there is no standard mathematic function or combination of a limited number of standard mathematic functions, to express it on a closed form. One have to wait until a new convenient special function be defined and standardized !
– JJacquelin
Oct 25 at 7:49
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Does there exist a Laplace transform of the Lambert W function (evaluated at $at$, where $a$ is a constant) that can be expressed in terms of elementary functions and the product log ($W(x)$)?
The Lambert W function is defined as the inverse of $f$ given $f(x) = xe^x$
$W'(x) = W(x)/x(1+W(x))$
$int W(ax) = x(W(ax)−1)+xW(ax)+C$
The Laplace Transform would be the evaluation of the improper integral $int_0^infty e^{-st}W(at)dt$
If such a transform exists, how is it expressed in terms of $s$, and how is it derived?
integration improper-integrals laplace-transform lambert-w
asked Oct 25 at 1:32
ThoughtOfGod
348
Does there exist a Laplace transform of the Lambert W function (evaluated at $at$, where $a$ is a constant) that can be expressed in terms of elementary functions and the product log ($W(x)$)?
The Lambert W function is defined as the inverse of $f$ given $f(x) = xe^x$
$W'(x) = W(x)/x(1+W(x))$
$int W(ax) = x(W(ax)−1)+xW(ax)+C$
The Laplace Transform would be the evaluation of the improper integral $int_0^infty e^{-st}W(at)dt$
If such a transform exists, how is it expressed in terms of $s$, and how is it derived?
integration improper-integrals laplace-transform lambert-w
integration improper-integrals laplace-transform lambert-w
asked Oct 25 at 1:32
ThoughtOfGod
348
asked Oct 25 at 1:32
ThoughtOfGod
348
edited Oct 25 at 7:08
asked Oct 25 at 1:32
ThoughtOfGod
348
asked Oct 25 at 1:32
ThoughtOfGod
348
asked Oct 25 at 1:32
ThoughtOfGod
348
348
Of course, the Laplace transform of the Lambert's W function exists. As far as I know, there is no standard mathematic function or combination of a limited number of standard mathematic functions, to express it on a closed form. One have to wait until a new convenient special function be defined and standardized !
– JJacquelin
Oct 25 at 7:49
add a comment |
Of course, the Laplace transform of the Lambert's W function exists. As far as I know, there is no standard mathematic function or combination of a limited number of standard mathematic functions, to express it on a closed form. One have to wait until a new convenient special function be defined and standardized !
– JJacquelin
Oct 25 at 7:49
Of course, the Laplace transform of the Lambert's W function exists. As far as I know, there is no standard mathematic function or combination of a limited number of standard mathematic functions, to express it on a closed form. One have to wait until a new convenient special function be defined and standardized !
– JJacquelin
Oct 25 at 7:49
Of course, the Laplace transform of the Lambert's W function exists. As far as I know, there is no standard mathematic function or combination of a limited number of standard mathematic functions, to express it on a closed form. One have to wait until a new convenient special function be defined and standardized !
– JJacquelin
Oct 25 at 7:49
add a comment |
2 Answers
2
active
oldest
votes
up vote
4
down vote
I propose to define a new special function, namely LW$(x)$ :
$$text{LW}(x)=int_0^infty W(t)e^{-x:t}dt$$
where W is the Lamber's W function.
In the futur, if this brand new function becomes standard, if it acquires the honorific status of standard special function, if it spread in the literature with a lot of studies of properties, if it becomes familiar, if it is implemented in mathematical softwares, then you could say :
"The Laplace transform of $quad text{W}(ax)quad$ is $quadfrac{1}{a}text{LW}(frac{s}{a})$."
This would be a typical case of special function emergence, exactly as many special functions emerged : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
answered Oct 25 at 8:27
JJacquelin
42.3k21750
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
1
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
|
show 4 more comments
up vote
0
down vote
Laplace transform LW(s) of Lambert function W(t) is
$$frac {1} {2i} int_{c-iinfty}^{c+iinfty}s^{-1-y}frac {y^{y-1}} {sin {pi y}},dy,{,c in (0,1)}$$
which is Mellin-Barnes integral. This integral converges for all complex s excluding singularities, of course. I cannot express this integral by known "named" functions.
answered Nov 22 at 17:54
Vojta
1
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
up vote
4
down vote
I propose to define a new special function, namely LW$(x)$ :
$$text{LW}(x)=int_0^infty W(t)e^{-x:t}dt$$
where W is the Lamber's W function.
In the futur, if this brand new function becomes standard, if it acquires the honorific status of standard special function, if it spread in the literature with a lot of studies of properties, if it becomes familiar, if it is implemented in mathematical softwares, then you could say :
"The Laplace transform of $quad text{W}(ax)quad$ is $quadfrac{1}{a}text{LW}(frac{s}{a})$."
This would be a typical case of special function emergence, exactly as many special functions emerged : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
answered Oct 25 at 8:27
JJacquelin
42.3k21750
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
1
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
|
show 4 more comments
up vote
4
down vote
I propose to define a new special function, namely LW$(x)$ :
$$text{LW}(x)=int_0^infty W(t)e^{-x:t}dt$$
where W is the Lamber's W function.
In the futur, if this brand new function becomes standard, if it acquires the honorific status of standard special function, if it spread in the literature with a lot of studies of properties, if it becomes familiar, if it is implemented in mathematical softwares, then you could say :
"The Laplace transform of $quad text{W}(ax)quad$ is $quadfrac{1}{a}text{LW}(frac{s}{a})$."
This would be a typical case of special function emergence, exactly as many special functions emerged : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
answered Oct 25 at 8:27
JJacquelin
42.3k21750
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
1
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
|
show 4 more comments
up vote
4
down vote
up vote
4
down vote
I propose to define a new special function, namely LW$(x)$ :
$$text{LW}(x)=int_0^infty W(t)e^{-x:t}dt$$
where W is the Lamber's W function.
In the futur, if this brand new function becomes standard, if it acquires the honorific status of standard special function, if it spread in the literature with a lot of studies of properties, if it becomes familiar, if it is implemented in mathematical softwares, then you could say :
"The Laplace transform of $quad text{W}(ax)quad$ is $quadfrac{1}{a}text{LW}(frac{s}{a})$."
This would be a typical case of special function emergence, exactly as many special functions emerged : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
answered Oct 25 at 8:27
JJacquelin
42.3k21750
I propose to define a new special function, namely LW$(x)$ :
$$text{LW}(x)=int_0^infty W(t)e^{-x:t}dt$$
where W is the Lamber's W function.
In the futur, if this brand new function becomes standard, if it acquires the honorific status of standard special function, if it spread in the literature with a lot of studies of properties, if it becomes familiar, if it is implemented in mathematical softwares, then you could say :
"The Laplace transform of $quad text{W}(ax)quad$ is $quadfrac{1}{a}text{LW}(frac{s}{a})$."
This would be a typical case of special function emergence, exactly as many special functions emerged : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
answered Oct 25 at 8:27
JJacquelin
42.3k21750
answered Oct 25 at 8:27
JJacquelin
42.3k21750
answered Oct 25 at 8:27
JJacquelin
42.3k21750
answered Oct 25 at 8:27
JJacquelin
42.3k21750
42.3k21750
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
1
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
|
show 4 more comments
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
1
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
I seem to remember you from before, you posted a scribd document describing the possibility of a special function that is formed from integrating $x^x$: an idea that has had a long, personal, and somewhat dramatic and somewhat miserable, history with me. You suggested that it should be named a "Sophomore's dream function", as a generalization of the "sophomore's dream" integrals. Right?
– The_Sympathizer
Oct 25 at 9:26
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
As it turns out that a slightly more general form than the one you presented can actually be used to do this integral, and thus it would be in effect redundant, if one were to use that, to propose an additional one with the first one already proposed. Would you want to see how it works?
– The_Sympathizer
Oct 25 at 9:27
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I cannot understand what you are asking for.
– JJacquelin
Oct 25 at 9:38
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
I was pointing out that I found out the integral at hand here is related to something else you (appear to have) worked with earlier.
– The_Sympathizer
Oct 25 at 9:52
1
1
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
@ThoughtOfGod. I am afraid that you don't understand my main answer (a bit humoristic). This is normal if you are not familiar with the general use of special functions. In this case, the answer to your question is very simple : The Laplace transform of the Lambert's W function exists but you cannot express it on closed form only with the elementary functions that you know.
– JJacquelin
Oct 25 at 10:51
|
show 4 more comments
up vote
0
down vote
Laplace transform LW(s) of Lambert function W(t) is
$$frac {1} {2i} int_{c-iinfty}^{c+iinfty}s^{-1-y}frac {y^{y-1}} {sin {pi y}},dy,{,c in (0,1)}$$
which is Mellin-Barnes integral. This integral converges for all complex s excluding singularities, of course. I cannot express this integral by known "named" functions.
answered Nov 22 at 17:54
Vojta
1
add a comment |
up vote
0
down vote
Laplace transform LW(s) of Lambert function W(t) is
$$frac {1} {2i} int_{c-iinfty}^{c+iinfty}s^{-1-y}frac {y^{y-1}} {sin {pi y}},dy,{,c in (0,1)}$$
which is Mellin-Barnes integral. This integral converges for all complex s excluding singularities, of course. I cannot express this integral by known "named" functions.
answered Nov 22 at 17:54
Vojta
1
add a comment |
up vote
0
down vote
up vote
0
down vote
Laplace transform LW(s) of Lambert function W(t) is
$$frac {1} {2i} int_{c-iinfty}^{c+iinfty}s^{-1-y}frac {y^{y-1}} {sin {pi y}},dy,{,c in (0,1)}$$
which is Mellin-Barnes integral. This integral converges for all complex s excluding singularities, of course. I cannot express this integral by known "named" functions.
answered Nov 22 at 17:54
Vojta
1
Laplace transform LW(s) of Lambert function W(t) is
$$frac {1} {2i} int_{c-iinfty}^{c+iinfty}s^{-1-y}frac {y^{y-1}} {sin {pi y}},dy,{,c in (0,1)}$$
which is Mellin-Barnes integral. This integral converges for all complex s excluding singularities, of course. I cannot express this integral by known "named" functions.
answered Nov 22 at 17:54
Vojta
1
answered Nov 22 at 17:54
Vojta
1
answered Nov 22 at 17:54
Vojta
1
answered Nov 22 at 17:54
Vojta
1
1
add a comment |
add a comment |
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Of course, the Laplace transform of the Lambert's W function exists. As far as I know, there is no standard mathematic function or combination of a limited number of standard mathematic functions, to express it on a closed form. One have to wait until a new convenient special function be defined and standardized !
– JJacquelin
Oct 25 at 7:49