Interesting integral related to the Omega Constant/Lambert W Function
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I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry.
$$displaystylefrac{1}{int_{-infty}^{infty}frac{1}{(e^{x}-x)^{2}+{pi}^{2}}dx}-1=W(1)=Omega$$
$W(1)=Omega$ is often referred to as the Omega Constant. Which is the solution to
$xe^{x}=1$. Which is $xapprox .567$
Thanks much.
EDIT: Sorry, I had the integral written incorrectly. Thanks for the catch.
I had also seen this:
$displaystyleint_{-infty}^{infty}frac{dx}{(e^{x}-x)^{2}+{pi}^{2}}=frac{1}{1+W(1)}=frac{1}{1+Omega}approx .638$
EDIT: I do not what is wrong, but I am trying to respond, but can not. All the buttons are unresponsive but this one. I have been trying to leave a greenie and add a comment, but neither will respond. I just wanted you to know this before you thought I was an ingrate.
Thank you. That is an interesting site.
integration special-functions
add a comment |
up vote
28
down vote
favorite
I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry.
$$displaystylefrac{1}{int_{-infty}^{infty}frac{1}{(e^{x}-x)^{2}+{pi}^{2}}dx}-1=W(1)=Omega$$
$W(1)=Omega$ is often referred to as the Omega Constant. Which is the solution to
$xe^{x}=1$. Which is $xapprox .567$
Thanks much.
EDIT: Sorry, I had the integral written incorrectly. Thanks for the catch.
I had also seen this:
$displaystyleint_{-infty}^{infty}frac{dx}{(e^{x}-x)^{2}+{pi}^{2}}=frac{1}{1+W(1)}=frac{1}{1+Omega}approx .638$
EDIT: I do not what is wrong, but I am trying to respond, but can not. All the buttons are unresponsive but this one. I have been trying to leave a greenie and add a comment, but neither will respond. I just wanted you to know this before you thought I was an ingrate.
Thank you. That is an interesting site.
integration special-functions
1
Maple says the integral is approximately 0.605, but $1/(1+W(1))$ is approximately 0.638
– GEdgar
Jun 16 '11 at 16:03
In Maple I get .6381037434, which agrees with 1/(1+LambertW(1)). I guess the 0.605 refers to a version of the integral before the last edit.
– Robert Israel
Jun 16 '11 at 18:38
Originally it had $+x$ instead of $-x$.
– GEdgar
Jun 16 '11 at 18:48
Nice question (+1)
– user 1357113
Oct 9 '12 at 9:48
add a comment |
up vote
28
down vote
favorite
up vote
28
down vote
favorite
I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry.
$$displaystylefrac{1}{int_{-infty}^{infty}frac{1}{(e^{x}-x)^{2}+{pi}^{2}}dx}-1=W(1)=Omega$$
$W(1)=Omega$ is often referred to as the Omega Constant. Which is the solution to
$xe^{x}=1$. Which is $xapprox .567$
Thanks much.
EDIT: Sorry, I had the integral written incorrectly. Thanks for the catch.
I had also seen this:
$displaystyleint_{-infty}^{infty}frac{dx}{(e^{x}-x)^{2}+{pi}^{2}}=frac{1}{1+W(1)}=frac{1}{1+Omega}approx .638$
EDIT: I do not what is wrong, but I am trying to respond, but can not. All the buttons are unresponsive but this one. I have been trying to leave a greenie and add a comment, but neither will respond. I just wanted you to know this before you thought I was an ingrate.
Thank you. That is an interesting site.
integration special-functions
I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry.
$$displaystylefrac{1}{int_{-infty}^{infty}frac{1}{(e^{x}-x)^{2}+{pi}^{2}}dx}-1=W(1)=Omega$$
$W(1)=Omega$ is often referred to as the Omega Constant. Which is the solution to
$xe^{x}=1$. Which is $xapprox .567$
Thanks much.
EDIT: Sorry, I had the integral written incorrectly. Thanks for the catch.
I had also seen this:
$displaystyleint_{-infty}^{infty}frac{dx}{(e^{x}-x)^{2}+{pi}^{2}}=frac{1}{1+W(1)}=frac{1}{1+Omega}approx .638$
EDIT: I do not what is wrong, but I am trying to respond, but can not. All the buttons are unresponsive but this one. I have been trying to leave a greenie and add a comment, but neither will respond. I just wanted you to know this before you thought I was an ingrate.
Thank you. That is an interesting site.
integration special-functions
integration special-functions
edited Sep 23 '11 at 2:45
J. M. is not a mathematician
60.6k5146284
60.6k5146284
asked Jun 16 '11 at 15:50
Cody
713417
713417
1
Maple says the integral is approximately 0.605, but $1/(1+W(1))$ is approximately 0.638
– GEdgar
Jun 16 '11 at 16:03
In Maple I get .6381037434, which agrees with 1/(1+LambertW(1)). I guess the 0.605 refers to a version of the integral before the last edit.
– Robert Israel
Jun 16 '11 at 18:38
Originally it had $+x$ instead of $-x$.
– GEdgar
Jun 16 '11 at 18:48
Nice question (+1)
– user 1357113
Oct 9 '12 at 9:48
add a comment |
1
Maple says the integral is approximately 0.605, but $1/(1+W(1))$ is approximately 0.638
– GEdgar
Jun 16 '11 at 16:03
In Maple I get .6381037434, which agrees with 1/(1+LambertW(1)). I guess the 0.605 refers to a version of the integral before the last edit.
– Robert Israel
Jun 16 '11 at 18:38
Originally it had $+x$ instead of $-x$.
– GEdgar
Jun 16 '11 at 18:48
Nice question (+1)
– user 1357113
Oct 9 '12 at 9:48
1
1
Maple says the integral is approximately 0.605, but $1/(1+W(1))$ is approximately 0.638
– GEdgar
Jun 16 '11 at 16:03
Maple says the integral is approximately 0.605, but $1/(1+W(1))$ is approximately 0.638
– GEdgar
Jun 16 '11 at 16:03
In Maple I get .6381037434, which agrees with 1/(1+LambertW(1)). I guess the 0.605 refers to a version of the integral before the last edit.
– Robert Israel
Jun 16 '11 at 18:38
In Maple I get .6381037434, which agrees with 1/(1+LambertW(1)). I guess the 0.605 refers to a version of the integral before the last edit.
– Robert Israel
Jun 16 '11 at 18:38
Originally it had $+x$ instead of $-x$.
– GEdgar
Jun 16 '11 at 18:48
Originally it had $+x$ instead of $-x$.
– GEdgar
Jun 16 '11 at 18:48
Nice question (+1)
– user 1357113
Oct 9 '12 at 9:48
Nice question (+1)
– user 1357113
Oct 9 '12 at 9:48
add a comment |
4 Answers
4
active
oldest
votes
up vote
29
down vote
We wish to compute
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}tag{1}
$$
We will do so by computing the contour integral
$$
int_{gamma_R}frac{mathrm{d}z}{(e^z-z)^2+pi^2}tag{2}
$$
over a family of contours $gamma_R$ for suitable values of $R$.
The Singularities
The singularities of the integrand in $(2)$ occur when $(e^z-z)^2+pi^2=(e^z-z+pi i)(e^z-z-pi i)$ vanishes; that is, for $z_k^pm=x_k^pm+iy_k^pm$ where
$$
e^{z_k^pm}-z_k^pmpmpi i=0tag{3}
$$
that is
$$
e^{2x_k^pm}={x_k^pm}^2+(y_k^pmmppi)^2tag{4}
$$
and
$$
y_k^pm=mathrm{atan2}(y_k^pmmppi,x_k^pm)tag{5}
$$
In fact, the roots of $(3)$ can be expressed in terms of the multivalued Lambert W function as
$$
z_k^pm=-W_{-k}(1)pmpi itag{6}
$$
The negative indices ensure that $z_0^+$ and $z_k^pm$ for $k>0$ are in the upper half-plane. In Mathematica, these can be computed as -LambertW[-k,1]+Pi I
and -LambertW[-k,1]-Pi I
.
Note that as $|z_k^pm|toinfty$, $(3)$ precludes $x_k^pm$ from being negative. In fact, as specified in $(6)$, only $x_0^pm<0$. Equation $(4)$ says that
$$
|y_k^pmmppi|=sqrt{e^{2x_k^pm}-{x_k^pm}^2}tag{7}
$$
As $|z_k^pm|toinfty$, $(5)$ and $(7)$ yield
$$
|y_k^pm|tofracpi2pmod{2pi}tag{8}
$$
The Contours
We will use the contours, $gamma_R=overline{gamma}_Rcupoverset{frown}{gamma}_R$, which circle the upper half plane, $overline{gamma}_R$ passing from $-R$ to $R$ on the real axis, then $overset{frown}{gamma}_R$ circling back counter-clockwise along $|z|=R$ in the upper half-plane. To get the desired decay of the integral along $overset{frown}{gamma}_R$, we will also require that $Requivfrac{3pi}{2}!!!!pmod{2pi}$; this is so that $overset{frown}{gamma}_R$ passes well between the singularities.
Let us also define the curves $rho^pm$ to be where $|e^z|=|zmppi i|$. Note that all the singularities of the integrand in $(2)$ lie on $rho^pm$.
On $rho^pm$, we have $R-pile e^xle R+pi$, therefore,
$$
log(R-pi)le xlelog(R+pi)tag{9}
$$
Furthermore, because $R-|y|=frac{x^2}{R+|y|}<frac{log(R+pi)^2}{R}$, we have
$$
R-frac{log(R+pi)^2}{R}le|y|le Rtag{10}\
$$
Therefore, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$,
$$
text{on }overset{frown}{gamma}_Rtext{, }x^2+y^2=R^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac xyright|simfrac{log(R)}{R}to0
$$
and
$$
text{on }rho^pmtext{, }e^{2x}=x^2+(ymppi)^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac{e^{2x}-x}{ymppi}right|sim Rtoinfty
$$
Thus, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$, $overset{frown}{gamma}_R$ becomes horizontal and $rho^pm$ becomes vertical. For example, here is the situation when $R=129.5pi$:
$hspace{35mm}$
$$
hspace{-1cm}small
begin{array}{}
z_{64}^+=5.99292081954932666 + 403.67969492855003099 i\
z_{65}^-=6.00848352082933166 + 403.67988772309602824 i\
z_{65}^+=6.00848352082933166 + 409.96307303027561472 i\
z_{66}^-=6.02380774554030566 + 409.96326053797959857 i
end{array}
$$
As $Rtoinfty$, on $rho^pm$ in the upper half-plane, $(9)$ and $(10)$ imply that $arg(zmppi i)tofracpi2$; thus, as indicated by $(8)$, $e^z$ and $zmppi i$ cancel only when $mathrm{Im}(z)approxfracpi2!!!!pmod{2pi}$. Likewise, $e^z$ and $zmppi i$ reinforce when $mathrm{Im}(z)approxfrac{3pi}{2}!!!!pmod{2pi}$.
Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$ and $zinoverset{frown}{gamma}_Rcaprho^pm$, we have that $|e^z-zpmpi i|sim2R$. As $zinoverset{frown}{gamma}_R$ moves to the right, by even just $1$, $e^z$ more than doubles, and dominates $zmppi i$; thus, $|e^z-zpmpi i|ge2R-R$. As $zinoverset{frown}{gamma}_R$ moves to the left, by even just $1$, $e^z$ decreases by more than half, and $zmppi i$ dominates; thus, $|e^z-zpmpi i|ge R-R/2$. Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$, $|e^z-zpmpi i|ge R/2$, and $|(e^z-z)^2+pi^2|ge R^2/4$. This guarantees that, as $Rtoinfty$, the integral over $overset{frown}{gamma}_R$ vanishes.
Thus, the integral along the real axis is $2pi i$ times the sum of the residues in the upper half-plane.
The Residues
Let $z_k^pm=-W_{-k}(1)pmpi i$. We will use the fact that $e^{z_k^pm}-z_k^pm=mppi i$.
The residue of $displaystylefrac1{(e^z-z)^2+pi^2}$ at $z_k^pm=-W_{-k}(1)pmpi i$ is
$$
begin{align}
lim_{zto z_k^pm}frac{z-z_k^pm}{(e^z-z)^2+pi^2}
&=frac1{2(mppi i)(e^{z_k^pm}-1)}\
&=frac1{2pi i}frac1{mp(z_k^pm-1mppi i)}\
&=frac1{2pi i}frac1{pm(W_{-k}(1)+1)}\
end{align}
$$
Thus, the residues at the pairs of singularities cancel each other, leaving us with the residue at $z_0^+$ which is $dfrac1{2pi i}dfrac1{W_0(1)+1}$. Thus, the integral is
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}=frac1{W_0(1)+1}
$$
2
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
4
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
add a comment |
up vote
9
down vote
While this is by no means rigorous, but it gives the correct solution. Any corrections to this are welcome!
Let
$$f(z) := frac{1}{(e^z-z)^2+pi^2}$$
Let $C$ be the canonical positively-oriented semicircular contour that traverses the real line from $-R$ to $R$ and all around $Re^{i theta}$ for $0 le theta le pi$ (let this semicircular arc be called $C_R$), so
$$oint_C f(z), dz = int_{-R}^R f(z),dz + int_{C_R}f(z), dz$$
To evaluate the latter integral, we see
$$
left| int_{C_R} frac{1}{(e^z-z)^2+pi^2}, dz right| =
int_{C_R} left| frac{1}{(e^z-z)^2+pi^2}right| , dz le
int_{C_R} frac{1}{(|e^z-z|)^2-pi^2} , dz le
int_{C_R} frac{1}{(e^R-R)^2-pi^2} , dz
$$
and letting $R to infty$, the outer integral disappears.
Looking at the denominator of $f$ for singularities:
$$(e^z-z)^2 + pi^2 = 0 implies e^z-z = pm i pi implies z = -W (1)pm ipi$$
using this.
We now use the root with the positive $ipi$ because when the sign is negative, the pole does not fall within the contour because $Im (-W (1)- ipi)<0$.
$$z_0 := -W (1)+ipi$$
We calculate the beautiful residue at $b_0$ at $z=z_0$:
$$
b_0=
operatorname*{Res}_{z to z_0} f(z) =
lim_{zto z_0} frac{(z-z_0)}{(e^z-z)^2+pi^2} =
lim_{zto z_0} frac{1}{2(e^z-1)(e^z-z)} =
frac{1}{2(-W(1) -1)(-W(1)+W(1)-ipi)} =
frac{1}{-2pi i(-W(1) -1)} =
frac{1}{2pi i(W(1)+1)}
$$
using L'Hopital's rule to compute the limit.
And finally, with residue theorem
$$
oint_C f(z), dz = int_{-infty}^infty f(z),dz = 2 pi i b_0 = frac{2 pi i}{2pi i(W(1)+1)} =
frac{1}{W(1)+1}
$$
An evaluation of this integral with real methods would also be intriguing.
1
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
add a comment |
up vote
4
down vote
I also considered this integral in another site, but it is only imperfect and non-rigorous one.
It seems that the Formelsammlung Mathematik is rendering a complete solution. It is written in German, but your bona fide translater Google may read this for you.
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
add a comment |
up vote
2
down vote
The identity is due to Victor Adamchik, see
http://mathworld.wolfram.com/OmegaConstant.html
You may want to contact Dr Adamchik himself via the e-mail at
http://www.cs.cmu.edu/~adamchik/research.html
because this particular paper doesn't seem to be in the list, as far as I can see.
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
29
down vote
We wish to compute
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}tag{1}
$$
We will do so by computing the contour integral
$$
int_{gamma_R}frac{mathrm{d}z}{(e^z-z)^2+pi^2}tag{2}
$$
over a family of contours $gamma_R$ for suitable values of $R$.
The Singularities
The singularities of the integrand in $(2)$ occur when $(e^z-z)^2+pi^2=(e^z-z+pi i)(e^z-z-pi i)$ vanishes; that is, for $z_k^pm=x_k^pm+iy_k^pm$ where
$$
e^{z_k^pm}-z_k^pmpmpi i=0tag{3}
$$
that is
$$
e^{2x_k^pm}={x_k^pm}^2+(y_k^pmmppi)^2tag{4}
$$
and
$$
y_k^pm=mathrm{atan2}(y_k^pmmppi,x_k^pm)tag{5}
$$
In fact, the roots of $(3)$ can be expressed in terms of the multivalued Lambert W function as
$$
z_k^pm=-W_{-k}(1)pmpi itag{6}
$$
The negative indices ensure that $z_0^+$ and $z_k^pm$ for $k>0$ are in the upper half-plane. In Mathematica, these can be computed as -LambertW[-k,1]+Pi I
and -LambertW[-k,1]-Pi I
.
Note that as $|z_k^pm|toinfty$, $(3)$ precludes $x_k^pm$ from being negative. In fact, as specified in $(6)$, only $x_0^pm<0$. Equation $(4)$ says that
$$
|y_k^pmmppi|=sqrt{e^{2x_k^pm}-{x_k^pm}^2}tag{7}
$$
As $|z_k^pm|toinfty$, $(5)$ and $(7)$ yield
$$
|y_k^pm|tofracpi2pmod{2pi}tag{8}
$$
The Contours
We will use the contours, $gamma_R=overline{gamma}_Rcupoverset{frown}{gamma}_R$, which circle the upper half plane, $overline{gamma}_R$ passing from $-R$ to $R$ on the real axis, then $overset{frown}{gamma}_R$ circling back counter-clockwise along $|z|=R$ in the upper half-plane. To get the desired decay of the integral along $overset{frown}{gamma}_R$, we will also require that $Requivfrac{3pi}{2}!!!!pmod{2pi}$; this is so that $overset{frown}{gamma}_R$ passes well between the singularities.
Let us also define the curves $rho^pm$ to be where $|e^z|=|zmppi i|$. Note that all the singularities of the integrand in $(2)$ lie on $rho^pm$.
On $rho^pm$, we have $R-pile e^xle R+pi$, therefore,
$$
log(R-pi)le xlelog(R+pi)tag{9}
$$
Furthermore, because $R-|y|=frac{x^2}{R+|y|}<frac{log(R+pi)^2}{R}$, we have
$$
R-frac{log(R+pi)^2}{R}le|y|le Rtag{10}\
$$
Therefore, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$,
$$
text{on }overset{frown}{gamma}_Rtext{, }x^2+y^2=R^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac xyright|simfrac{log(R)}{R}to0
$$
and
$$
text{on }rho^pmtext{, }e^{2x}=x^2+(ymppi)^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac{e^{2x}-x}{ymppi}right|sim Rtoinfty
$$
Thus, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$, $overset{frown}{gamma}_R$ becomes horizontal and $rho^pm$ becomes vertical. For example, here is the situation when $R=129.5pi$:
$hspace{35mm}$
$$
hspace{-1cm}small
begin{array}{}
z_{64}^+=5.99292081954932666 + 403.67969492855003099 i\
z_{65}^-=6.00848352082933166 + 403.67988772309602824 i\
z_{65}^+=6.00848352082933166 + 409.96307303027561472 i\
z_{66}^-=6.02380774554030566 + 409.96326053797959857 i
end{array}
$$
As $Rtoinfty$, on $rho^pm$ in the upper half-plane, $(9)$ and $(10)$ imply that $arg(zmppi i)tofracpi2$; thus, as indicated by $(8)$, $e^z$ and $zmppi i$ cancel only when $mathrm{Im}(z)approxfracpi2!!!!pmod{2pi}$. Likewise, $e^z$ and $zmppi i$ reinforce when $mathrm{Im}(z)approxfrac{3pi}{2}!!!!pmod{2pi}$.
Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$ and $zinoverset{frown}{gamma}_Rcaprho^pm$, we have that $|e^z-zpmpi i|sim2R$. As $zinoverset{frown}{gamma}_R$ moves to the right, by even just $1$, $e^z$ more than doubles, and dominates $zmppi i$; thus, $|e^z-zpmpi i|ge2R-R$. As $zinoverset{frown}{gamma}_R$ moves to the left, by even just $1$, $e^z$ decreases by more than half, and $zmppi i$ dominates; thus, $|e^z-zpmpi i|ge R-R/2$. Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$, $|e^z-zpmpi i|ge R/2$, and $|(e^z-z)^2+pi^2|ge R^2/4$. This guarantees that, as $Rtoinfty$, the integral over $overset{frown}{gamma}_R$ vanishes.
Thus, the integral along the real axis is $2pi i$ times the sum of the residues in the upper half-plane.
The Residues
Let $z_k^pm=-W_{-k}(1)pmpi i$. We will use the fact that $e^{z_k^pm}-z_k^pm=mppi i$.
The residue of $displaystylefrac1{(e^z-z)^2+pi^2}$ at $z_k^pm=-W_{-k}(1)pmpi i$ is
$$
begin{align}
lim_{zto z_k^pm}frac{z-z_k^pm}{(e^z-z)^2+pi^2}
&=frac1{2(mppi i)(e^{z_k^pm}-1)}\
&=frac1{2pi i}frac1{mp(z_k^pm-1mppi i)}\
&=frac1{2pi i}frac1{pm(W_{-k}(1)+1)}\
end{align}
$$
Thus, the residues at the pairs of singularities cancel each other, leaving us with the residue at $z_0^+$ which is $dfrac1{2pi i}dfrac1{W_0(1)+1}$. Thus, the integral is
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}=frac1{W_0(1)+1}
$$
2
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
4
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
add a comment |
up vote
29
down vote
We wish to compute
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}tag{1}
$$
We will do so by computing the contour integral
$$
int_{gamma_R}frac{mathrm{d}z}{(e^z-z)^2+pi^2}tag{2}
$$
over a family of contours $gamma_R$ for suitable values of $R$.
The Singularities
The singularities of the integrand in $(2)$ occur when $(e^z-z)^2+pi^2=(e^z-z+pi i)(e^z-z-pi i)$ vanishes; that is, for $z_k^pm=x_k^pm+iy_k^pm$ where
$$
e^{z_k^pm}-z_k^pmpmpi i=0tag{3}
$$
that is
$$
e^{2x_k^pm}={x_k^pm}^2+(y_k^pmmppi)^2tag{4}
$$
and
$$
y_k^pm=mathrm{atan2}(y_k^pmmppi,x_k^pm)tag{5}
$$
In fact, the roots of $(3)$ can be expressed in terms of the multivalued Lambert W function as
$$
z_k^pm=-W_{-k}(1)pmpi itag{6}
$$
The negative indices ensure that $z_0^+$ and $z_k^pm$ for $k>0$ are in the upper half-plane. In Mathematica, these can be computed as -LambertW[-k,1]+Pi I
and -LambertW[-k,1]-Pi I
.
Note that as $|z_k^pm|toinfty$, $(3)$ precludes $x_k^pm$ from being negative. In fact, as specified in $(6)$, only $x_0^pm<0$. Equation $(4)$ says that
$$
|y_k^pmmppi|=sqrt{e^{2x_k^pm}-{x_k^pm}^2}tag{7}
$$
As $|z_k^pm|toinfty$, $(5)$ and $(7)$ yield
$$
|y_k^pm|tofracpi2pmod{2pi}tag{8}
$$
The Contours
We will use the contours, $gamma_R=overline{gamma}_Rcupoverset{frown}{gamma}_R$, which circle the upper half plane, $overline{gamma}_R$ passing from $-R$ to $R$ on the real axis, then $overset{frown}{gamma}_R$ circling back counter-clockwise along $|z|=R$ in the upper half-plane. To get the desired decay of the integral along $overset{frown}{gamma}_R$, we will also require that $Requivfrac{3pi}{2}!!!!pmod{2pi}$; this is so that $overset{frown}{gamma}_R$ passes well between the singularities.
Let us also define the curves $rho^pm$ to be where $|e^z|=|zmppi i|$. Note that all the singularities of the integrand in $(2)$ lie on $rho^pm$.
On $rho^pm$, we have $R-pile e^xle R+pi$, therefore,
$$
log(R-pi)le xlelog(R+pi)tag{9}
$$
Furthermore, because $R-|y|=frac{x^2}{R+|y|}<frac{log(R+pi)^2}{R}$, we have
$$
R-frac{log(R+pi)^2}{R}le|y|le Rtag{10}\
$$
Therefore, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$,
$$
text{on }overset{frown}{gamma}_Rtext{, }x^2+y^2=R^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac xyright|simfrac{log(R)}{R}to0
$$
and
$$
text{on }rho^pmtext{, }e^{2x}=x^2+(ymppi)^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac{e^{2x}-x}{ymppi}right|sim Rtoinfty
$$
Thus, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$, $overset{frown}{gamma}_R$ becomes horizontal and $rho^pm$ becomes vertical. For example, here is the situation when $R=129.5pi$:
$hspace{35mm}$
$$
hspace{-1cm}small
begin{array}{}
z_{64}^+=5.99292081954932666 + 403.67969492855003099 i\
z_{65}^-=6.00848352082933166 + 403.67988772309602824 i\
z_{65}^+=6.00848352082933166 + 409.96307303027561472 i\
z_{66}^-=6.02380774554030566 + 409.96326053797959857 i
end{array}
$$
As $Rtoinfty$, on $rho^pm$ in the upper half-plane, $(9)$ and $(10)$ imply that $arg(zmppi i)tofracpi2$; thus, as indicated by $(8)$, $e^z$ and $zmppi i$ cancel only when $mathrm{Im}(z)approxfracpi2!!!!pmod{2pi}$. Likewise, $e^z$ and $zmppi i$ reinforce when $mathrm{Im}(z)approxfrac{3pi}{2}!!!!pmod{2pi}$.
Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$ and $zinoverset{frown}{gamma}_Rcaprho^pm$, we have that $|e^z-zpmpi i|sim2R$. As $zinoverset{frown}{gamma}_R$ moves to the right, by even just $1$, $e^z$ more than doubles, and dominates $zmppi i$; thus, $|e^z-zpmpi i|ge2R-R$. As $zinoverset{frown}{gamma}_R$ moves to the left, by even just $1$, $e^z$ decreases by more than half, and $zmppi i$ dominates; thus, $|e^z-zpmpi i|ge R-R/2$. Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$, $|e^z-zpmpi i|ge R/2$, and $|(e^z-z)^2+pi^2|ge R^2/4$. This guarantees that, as $Rtoinfty$, the integral over $overset{frown}{gamma}_R$ vanishes.
Thus, the integral along the real axis is $2pi i$ times the sum of the residues in the upper half-plane.
The Residues
Let $z_k^pm=-W_{-k}(1)pmpi i$. We will use the fact that $e^{z_k^pm}-z_k^pm=mppi i$.
The residue of $displaystylefrac1{(e^z-z)^2+pi^2}$ at $z_k^pm=-W_{-k}(1)pmpi i$ is
$$
begin{align}
lim_{zto z_k^pm}frac{z-z_k^pm}{(e^z-z)^2+pi^2}
&=frac1{2(mppi i)(e^{z_k^pm}-1)}\
&=frac1{2pi i}frac1{mp(z_k^pm-1mppi i)}\
&=frac1{2pi i}frac1{pm(W_{-k}(1)+1)}\
end{align}
$$
Thus, the residues at the pairs of singularities cancel each other, leaving us with the residue at $z_0^+$ which is $dfrac1{2pi i}dfrac1{W_0(1)+1}$. Thus, the integral is
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}=frac1{W_0(1)+1}
$$
2
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
4
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
add a comment |
up vote
29
down vote
up vote
29
down vote
We wish to compute
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}tag{1}
$$
We will do so by computing the contour integral
$$
int_{gamma_R}frac{mathrm{d}z}{(e^z-z)^2+pi^2}tag{2}
$$
over a family of contours $gamma_R$ for suitable values of $R$.
The Singularities
The singularities of the integrand in $(2)$ occur when $(e^z-z)^2+pi^2=(e^z-z+pi i)(e^z-z-pi i)$ vanishes; that is, for $z_k^pm=x_k^pm+iy_k^pm$ where
$$
e^{z_k^pm}-z_k^pmpmpi i=0tag{3}
$$
that is
$$
e^{2x_k^pm}={x_k^pm}^2+(y_k^pmmppi)^2tag{4}
$$
and
$$
y_k^pm=mathrm{atan2}(y_k^pmmppi,x_k^pm)tag{5}
$$
In fact, the roots of $(3)$ can be expressed in terms of the multivalued Lambert W function as
$$
z_k^pm=-W_{-k}(1)pmpi itag{6}
$$
The negative indices ensure that $z_0^+$ and $z_k^pm$ for $k>0$ are in the upper half-plane. In Mathematica, these can be computed as -LambertW[-k,1]+Pi I
and -LambertW[-k,1]-Pi I
.
Note that as $|z_k^pm|toinfty$, $(3)$ precludes $x_k^pm$ from being negative. In fact, as specified in $(6)$, only $x_0^pm<0$. Equation $(4)$ says that
$$
|y_k^pmmppi|=sqrt{e^{2x_k^pm}-{x_k^pm}^2}tag{7}
$$
As $|z_k^pm|toinfty$, $(5)$ and $(7)$ yield
$$
|y_k^pm|tofracpi2pmod{2pi}tag{8}
$$
The Contours
We will use the contours, $gamma_R=overline{gamma}_Rcupoverset{frown}{gamma}_R$, which circle the upper half plane, $overline{gamma}_R$ passing from $-R$ to $R$ on the real axis, then $overset{frown}{gamma}_R$ circling back counter-clockwise along $|z|=R$ in the upper half-plane. To get the desired decay of the integral along $overset{frown}{gamma}_R$, we will also require that $Requivfrac{3pi}{2}!!!!pmod{2pi}$; this is so that $overset{frown}{gamma}_R$ passes well between the singularities.
Let us also define the curves $rho^pm$ to be where $|e^z|=|zmppi i|$. Note that all the singularities of the integrand in $(2)$ lie on $rho^pm$.
On $rho^pm$, we have $R-pile e^xle R+pi$, therefore,
$$
log(R-pi)le xlelog(R+pi)tag{9}
$$
Furthermore, because $R-|y|=frac{x^2}{R+|y|}<frac{log(R+pi)^2}{R}$, we have
$$
R-frac{log(R+pi)^2}{R}le|y|le Rtag{10}\
$$
Therefore, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$,
$$
text{on }overset{frown}{gamma}_Rtext{, }x^2+y^2=R^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac xyright|simfrac{log(R)}{R}to0
$$
and
$$
text{on }rho^pmtext{, }e^{2x}=x^2+(ymppi)^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac{e^{2x}-x}{ymppi}right|sim Rtoinfty
$$
Thus, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$, $overset{frown}{gamma}_R$ becomes horizontal and $rho^pm$ becomes vertical. For example, here is the situation when $R=129.5pi$:
$hspace{35mm}$
$$
hspace{-1cm}small
begin{array}{}
z_{64}^+=5.99292081954932666 + 403.67969492855003099 i\
z_{65}^-=6.00848352082933166 + 403.67988772309602824 i\
z_{65}^+=6.00848352082933166 + 409.96307303027561472 i\
z_{66}^-=6.02380774554030566 + 409.96326053797959857 i
end{array}
$$
As $Rtoinfty$, on $rho^pm$ in the upper half-plane, $(9)$ and $(10)$ imply that $arg(zmppi i)tofracpi2$; thus, as indicated by $(8)$, $e^z$ and $zmppi i$ cancel only when $mathrm{Im}(z)approxfracpi2!!!!pmod{2pi}$. Likewise, $e^z$ and $zmppi i$ reinforce when $mathrm{Im}(z)approxfrac{3pi}{2}!!!!pmod{2pi}$.
Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$ and $zinoverset{frown}{gamma}_Rcaprho^pm$, we have that $|e^z-zpmpi i|sim2R$. As $zinoverset{frown}{gamma}_R$ moves to the right, by even just $1$, $e^z$ more than doubles, and dominates $zmppi i$; thus, $|e^z-zpmpi i|ge2R-R$. As $zinoverset{frown}{gamma}_R$ moves to the left, by even just $1$, $e^z$ decreases by more than half, and $zmppi i$ dominates; thus, $|e^z-zpmpi i|ge R-R/2$. Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$, $|e^z-zpmpi i|ge R/2$, and $|(e^z-z)^2+pi^2|ge R^2/4$. This guarantees that, as $Rtoinfty$, the integral over $overset{frown}{gamma}_R$ vanishes.
Thus, the integral along the real axis is $2pi i$ times the sum of the residues in the upper half-plane.
The Residues
Let $z_k^pm=-W_{-k}(1)pmpi i$. We will use the fact that $e^{z_k^pm}-z_k^pm=mppi i$.
The residue of $displaystylefrac1{(e^z-z)^2+pi^2}$ at $z_k^pm=-W_{-k}(1)pmpi i$ is
$$
begin{align}
lim_{zto z_k^pm}frac{z-z_k^pm}{(e^z-z)^2+pi^2}
&=frac1{2(mppi i)(e^{z_k^pm}-1)}\
&=frac1{2pi i}frac1{mp(z_k^pm-1mppi i)}\
&=frac1{2pi i}frac1{pm(W_{-k}(1)+1)}\
end{align}
$$
Thus, the residues at the pairs of singularities cancel each other, leaving us with the residue at $z_0^+$ which is $dfrac1{2pi i}dfrac1{W_0(1)+1}$. Thus, the integral is
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}=frac1{W_0(1)+1}
$$
We wish to compute
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}tag{1}
$$
We will do so by computing the contour integral
$$
int_{gamma_R}frac{mathrm{d}z}{(e^z-z)^2+pi^2}tag{2}
$$
over a family of contours $gamma_R$ for suitable values of $R$.
The Singularities
The singularities of the integrand in $(2)$ occur when $(e^z-z)^2+pi^2=(e^z-z+pi i)(e^z-z-pi i)$ vanishes; that is, for $z_k^pm=x_k^pm+iy_k^pm$ where
$$
e^{z_k^pm}-z_k^pmpmpi i=0tag{3}
$$
that is
$$
e^{2x_k^pm}={x_k^pm}^2+(y_k^pmmppi)^2tag{4}
$$
and
$$
y_k^pm=mathrm{atan2}(y_k^pmmppi,x_k^pm)tag{5}
$$
In fact, the roots of $(3)$ can be expressed in terms of the multivalued Lambert W function as
$$
z_k^pm=-W_{-k}(1)pmpi itag{6}
$$
The negative indices ensure that $z_0^+$ and $z_k^pm$ for $k>0$ are in the upper half-plane. In Mathematica, these can be computed as -LambertW[-k,1]+Pi I
and -LambertW[-k,1]-Pi I
.
Note that as $|z_k^pm|toinfty$, $(3)$ precludes $x_k^pm$ from being negative. In fact, as specified in $(6)$, only $x_0^pm<0$. Equation $(4)$ says that
$$
|y_k^pmmppi|=sqrt{e^{2x_k^pm}-{x_k^pm}^2}tag{7}
$$
As $|z_k^pm|toinfty$, $(5)$ and $(7)$ yield
$$
|y_k^pm|tofracpi2pmod{2pi}tag{8}
$$
The Contours
We will use the contours, $gamma_R=overline{gamma}_Rcupoverset{frown}{gamma}_R$, which circle the upper half plane, $overline{gamma}_R$ passing from $-R$ to $R$ on the real axis, then $overset{frown}{gamma}_R$ circling back counter-clockwise along $|z|=R$ in the upper half-plane. To get the desired decay of the integral along $overset{frown}{gamma}_R$, we will also require that $Requivfrac{3pi}{2}!!!!pmod{2pi}$; this is so that $overset{frown}{gamma}_R$ passes well between the singularities.
Let us also define the curves $rho^pm$ to be where $|e^z|=|zmppi i|$. Note that all the singularities of the integrand in $(2)$ lie on $rho^pm$.
On $rho^pm$, we have $R-pile e^xle R+pi$, therefore,
$$
log(R-pi)le xlelog(R+pi)tag{9}
$$
Furthermore, because $R-|y|=frac{x^2}{R+|y|}<frac{log(R+pi)^2}{R}$, we have
$$
R-frac{log(R+pi)^2}{R}le|y|le Rtag{10}\
$$
Therefore, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$,
$$
text{on }overset{frown}{gamma}_Rtext{, }x^2+y^2=R^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac xyright|simfrac{log(R)}{R}to0
$$
and
$$
text{on }rho^pmtext{, }e^{2x}=x^2+(ymppi)^2Rightarrowleft|frac{mathrm{d}y}{mathrm{d}x}right|=left|frac{e^{2x}-x}{ymppi}right|sim Rtoinfty
$$
Thus, at $overset{frown}{gamma}_Rcaprho^pm$ as $Rtoinfty$, $overset{frown}{gamma}_R$ becomes horizontal and $rho^pm$ becomes vertical. For example, here is the situation when $R=129.5pi$:
$hspace{35mm}$
$$
hspace{-1cm}small
begin{array}{}
z_{64}^+=5.99292081954932666 + 403.67969492855003099 i\
z_{65}^-=6.00848352082933166 + 403.67988772309602824 i\
z_{65}^+=6.00848352082933166 + 409.96307303027561472 i\
z_{66}^-=6.02380774554030566 + 409.96326053797959857 i
end{array}
$$
As $Rtoinfty$, on $rho^pm$ in the upper half-plane, $(9)$ and $(10)$ imply that $arg(zmppi i)tofracpi2$; thus, as indicated by $(8)$, $e^z$ and $zmppi i$ cancel only when $mathrm{Im}(z)approxfracpi2!!!!pmod{2pi}$. Likewise, $e^z$ and $zmppi i$ reinforce when $mathrm{Im}(z)approxfrac{3pi}{2}!!!!pmod{2pi}$.
Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$ and $zinoverset{frown}{gamma}_Rcaprho^pm$, we have that $|e^z-zpmpi i|sim2R$. As $zinoverset{frown}{gamma}_R$ moves to the right, by even just $1$, $e^z$ more than doubles, and dominates $zmppi i$; thus, $|e^z-zpmpi i|ge2R-R$. As $zinoverset{frown}{gamma}_R$ moves to the left, by even just $1$, $e^z$ decreases by more than half, and $zmppi i$ dominates; thus, $|e^z-zpmpi i|ge R-R/2$. Therefore, when $Requivfrac{3pi}{2}!!!!pmod{2pi}$, $|e^z-zpmpi i|ge R/2$, and $|(e^z-z)^2+pi^2|ge R^2/4$. This guarantees that, as $Rtoinfty$, the integral over $overset{frown}{gamma}_R$ vanishes.
Thus, the integral along the real axis is $2pi i$ times the sum of the residues in the upper half-plane.
The Residues
Let $z_k^pm=-W_{-k}(1)pmpi i$. We will use the fact that $e^{z_k^pm}-z_k^pm=mppi i$.
The residue of $displaystylefrac1{(e^z-z)^2+pi^2}$ at $z_k^pm=-W_{-k}(1)pmpi i$ is
$$
begin{align}
lim_{zto z_k^pm}frac{z-z_k^pm}{(e^z-z)^2+pi^2}
&=frac1{2(mppi i)(e^{z_k^pm}-1)}\
&=frac1{2pi i}frac1{mp(z_k^pm-1mppi i)}\
&=frac1{2pi i}frac1{pm(W_{-k}(1)+1)}\
end{align}
$$
Thus, the residues at the pairs of singularities cancel each other, leaving us with the residue at $z_0^+$ which is $dfrac1{2pi i}dfrac1{W_0(1)+1}$. Thus, the integral is
$$
int_{-infty}^inftyfrac{mathrm{d}x}{(e^x-x)^2+pi^2}=frac1{W_0(1)+1}
$$
edited Oct 8 '12 at 10:52
answered Oct 5 '12 at 8:28
robjohn♦
263k27301623
263k27301623
2
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
4
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
add a comment |
2
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
4
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
2
2
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
Thanks for the fantastic answer!
– Argon
Oct 8 '12 at 17:33
4
4
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@Argon: This is a marvelous question! It's nice to see LambertW appear as the result an integral (or a rational function thereof).
– robjohn♦
Oct 8 '12 at 17:41
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
@ robjohn: Good lesson to learn (+1)
– user 1357113
Oct 9 '12 at 9:51
add a comment |
up vote
9
down vote
While this is by no means rigorous, but it gives the correct solution. Any corrections to this are welcome!
Let
$$f(z) := frac{1}{(e^z-z)^2+pi^2}$$
Let $C$ be the canonical positively-oriented semicircular contour that traverses the real line from $-R$ to $R$ and all around $Re^{i theta}$ for $0 le theta le pi$ (let this semicircular arc be called $C_R$), so
$$oint_C f(z), dz = int_{-R}^R f(z),dz + int_{C_R}f(z), dz$$
To evaluate the latter integral, we see
$$
left| int_{C_R} frac{1}{(e^z-z)^2+pi^2}, dz right| =
int_{C_R} left| frac{1}{(e^z-z)^2+pi^2}right| , dz le
int_{C_R} frac{1}{(|e^z-z|)^2-pi^2} , dz le
int_{C_R} frac{1}{(e^R-R)^2-pi^2} , dz
$$
and letting $R to infty$, the outer integral disappears.
Looking at the denominator of $f$ for singularities:
$$(e^z-z)^2 + pi^2 = 0 implies e^z-z = pm i pi implies z = -W (1)pm ipi$$
using this.
We now use the root with the positive $ipi$ because when the sign is negative, the pole does not fall within the contour because $Im (-W (1)- ipi)<0$.
$$z_0 := -W (1)+ipi$$
We calculate the beautiful residue at $b_0$ at $z=z_0$:
$$
b_0=
operatorname*{Res}_{z to z_0} f(z) =
lim_{zto z_0} frac{(z-z_0)}{(e^z-z)^2+pi^2} =
lim_{zto z_0} frac{1}{2(e^z-1)(e^z-z)} =
frac{1}{2(-W(1) -1)(-W(1)+W(1)-ipi)} =
frac{1}{-2pi i(-W(1) -1)} =
frac{1}{2pi i(W(1)+1)}
$$
using L'Hopital's rule to compute the limit.
And finally, with residue theorem
$$
oint_C f(z), dz = int_{-infty}^infty f(z),dz = 2 pi i b_0 = frac{2 pi i}{2pi i(W(1)+1)} =
frac{1}{W(1)+1}
$$
An evaluation of this integral with real methods would also be intriguing.
1
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
add a comment |
up vote
9
down vote
While this is by no means rigorous, but it gives the correct solution. Any corrections to this are welcome!
Let
$$f(z) := frac{1}{(e^z-z)^2+pi^2}$$
Let $C$ be the canonical positively-oriented semicircular contour that traverses the real line from $-R$ to $R$ and all around $Re^{i theta}$ for $0 le theta le pi$ (let this semicircular arc be called $C_R$), so
$$oint_C f(z), dz = int_{-R}^R f(z),dz + int_{C_R}f(z), dz$$
To evaluate the latter integral, we see
$$
left| int_{C_R} frac{1}{(e^z-z)^2+pi^2}, dz right| =
int_{C_R} left| frac{1}{(e^z-z)^2+pi^2}right| , dz le
int_{C_R} frac{1}{(|e^z-z|)^2-pi^2} , dz le
int_{C_R} frac{1}{(e^R-R)^2-pi^2} , dz
$$
and letting $R to infty$, the outer integral disappears.
Looking at the denominator of $f$ for singularities:
$$(e^z-z)^2 + pi^2 = 0 implies e^z-z = pm i pi implies z = -W (1)pm ipi$$
using this.
We now use the root with the positive $ipi$ because when the sign is negative, the pole does not fall within the contour because $Im (-W (1)- ipi)<0$.
$$z_0 := -W (1)+ipi$$
We calculate the beautiful residue at $b_0$ at $z=z_0$:
$$
b_0=
operatorname*{Res}_{z to z_0} f(z) =
lim_{zto z_0} frac{(z-z_0)}{(e^z-z)^2+pi^2} =
lim_{zto z_0} frac{1}{2(e^z-1)(e^z-z)} =
frac{1}{2(-W(1) -1)(-W(1)+W(1)-ipi)} =
frac{1}{-2pi i(-W(1) -1)} =
frac{1}{2pi i(W(1)+1)}
$$
using L'Hopital's rule to compute the limit.
And finally, with residue theorem
$$
oint_C f(z), dz = int_{-infty}^infty f(z),dz = 2 pi i b_0 = frac{2 pi i}{2pi i(W(1)+1)} =
frac{1}{W(1)+1}
$$
An evaluation of this integral with real methods would also be intriguing.
1
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
add a comment |
up vote
9
down vote
up vote
9
down vote
While this is by no means rigorous, but it gives the correct solution. Any corrections to this are welcome!
Let
$$f(z) := frac{1}{(e^z-z)^2+pi^2}$$
Let $C$ be the canonical positively-oriented semicircular contour that traverses the real line from $-R$ to $R$ and all around $Re^{i theta}$ for $0 le theta le pi$ (let this semicircular arc be called $C_R$), so
$$oint_C f(z), dz = int_{-R}^R f(z),dz + int_{C_R}f(z), dz$$
To evaluate the latter integral, we see
$$
left| int_{C_R} frac{1}{(e^z-z)^2+pi^2}, dz right| =
int_{C_R} left| frac{1}{(e^z-z)^2+pi^2}right| , dz le
int_{C_R} frac{1}{(|e^z-z|)^2-pi^2} , dz le
int_{C_R} frac{1}{(e^R-R)^2-pi^2} , dz
$$
and letting $R to infty$, the outer integral disappears.
Looking at the denominator of $f$ for singularities:
$$(e^z-z)^2 + pi^2 = 0 implies e^z-z = pm i pi implies z = -W (1)pm ipi$$
using this.
We now use the root with the positive $ipi$ because when the sign is negative, the pole does not fall within the contour because $Im (-W (1)- ipi)<0$.
$$z_0 := -W (1)+ipi$$
We calculate the beautiful residue at $b_0$ at $z=z_0$:
$$
b_0=
operatorname*{Res}_{z to z_0} f(z) =
lim_{zto z_0} frac{(z-z_0)}{(e^z-z)^2+pi^2} =
lim_{zto z_0} frac{1}{2(e^z-1)(e^z-z)} =
frac{1}{2(-W(1) -1)(-W(1)+W(1)-ipi)} =
frac{1}{-2pi i(-W(1) -1)} =
frac{1}{2pi i(W(1)+1)}
$$
using L'Hopital's rule to compute the limit.
And finally, with residue theorem
$$
oint_C f(z), dz = int_{-infty}^infty f(z),dz = 2 pi i b_0 = frac{2 pi i}{2pi i(W(1)+1)} =
frac{1}{W(1)+1}
$$
An evaluation of this integral with real methods would also be intriguing.
While this is by no means rigorous, but it gives the correct solution. Any corrections to this are welcome!
Let
$$f(z) := frac{1}{(e^z-z)^2+pi^2}$$
Let $C$ be the canonical positively-oriented semicircular contour that traverses the real line from $-R$ to $R$ and all around $Re^{i theta}$ for $0 le theta le pi$ (let this semicircular arc be called $C_R$), so
$$oint_C f(z), dz = int_{-R}^R f(z),dz + int_{C_R}f(z), dz$$
To evaluate the latter integral, we see
$$
left| int_{C_R} frac{1}{(e^z-z)^2+pi^2}, dz right| =
int_{C_R} left| frac{1}{(e^z-z)^2+pi^2}right| , dz le
int_{C_R} frac{1}{(|e^z-z|)^2-pi^2} , dz le
int_{C_R} frac{1}{(e^R-R)^2-pi^2} , dz
$$
and letting $R to infty$, the outer integral disappears.
Looking at the denominator of $f$ for singularities:
$$(e^z-z)^2 + pi^2 = 0 implies e^z-z = pm i pi implies z = -W (1)pm ipi$$
using this.
We now use the root with the positive $ipi$ because when the sign is negative, the pole does not fall within the contour because $Im (-W (1)- ipi)<0$.
$$z_0 := -W (1)+ipi$$
We calculate the beautiful residue at $b_0$ at $z=z_0$:
$$
b_0=
operatorname*{Res}_{z to z_0} f(z) =
lim_{zto z_0} frac{(z-z_0)}{(e^z-z)^2+pi^2} =
lim_{zto z_0} frac{1}{2(e^z-1)(e^z-z)} =
frac{1}{2(-W(1) -1)(-W(1)+W(1)-ipi)} =
frac{1}{-2pi i(-W(1) -1)} =
frac{1}{2pi i(W(1)+1)}
$$
using L'Hopital's rule to compute the limit.
And finally, with residue theorem
$$
oint_C f(z), dz = int_{-infty}^infty f(z),dz = 2 pi i b_0 = frac{2 pi i}{2pi i(W(1)+1)} =
frac{1}{W(1)+1}
$$
An evaluation of this integral with real methods would also be intriguing.
edited Oct 5 '12 at 1:55
answered Oct 4 '12 at 23:12
Argon
16.3k672122
16.3k672122
1
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
add a comment |
1
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
1
1
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Just as there are infinitely many solutions to $wexp(w)=1$, there are infinitely many singularities in the upper half plane. Let $z_k=-W_k(1)mp ipi$, where $W_k(1)$ are the solutions to $W_k(1)exp(W_k(1))=1$, then $e^z-z=pm ipi$. All but one of these singularities seem to be ignored in the contour integration.
– robjohn♦
Oct 5 '12 at 7:17
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
Here are the first seven singularities $$ begin{align} -W_0(1)+ipi&=-0.5671432904097838730 + 3.1415926535897932385 i\ -W_{-1}(1)-ipi&=+1.5339133197935745079 + 1.2335924994721051470 i\ -W_{-1}(1)+ipi&=+1.5339133197935745079 + 7.5167778066516916239 i\ -W_{-2}(1)-ipi&=+2.4015851048680028842 + 7.6347068625252776600 i\ -W_{-2}(1)+ipi&=+2.4015851048680028842 + 13.917892169704864137 i\ -W_{-3}(1)-ipi&=+2.8535817554090378070 + 13.971942885822352674 i\ -W_{-3}(1)+ipi&=+2.8535817554090378070 + 20.255128193001939151 i\ &vdots end{align} $$
– robjohn♦
Oct 5 '12 at 7:18
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
What do you consider rigorous? Other than not catching all of the singularities, your proof looks like a pretty standard application of contour integration.
– robjohn♦
Oct 5 '12 at 9:25
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:23
add a comment |
up vote
4
down vote
I also considered this integral in another site, but it is only imperfect and non-rigorous one.
It seems that the Formelsammlung Mathematik is rendering a complete solution. It is written in German, but your bona fide translater Google may read this for you.
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
add a comment |
up vote
4
down vote
I also considered this integral in another site, but it is only imperfect and non-rigorous one.
It seems that the Formelsammlung Mathematik is rendering a complete solution. It is written in German, but your bona fide translater Google may read this for you.
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
add a comment |
up vote
4
down vote
up vote
4
down vote
I also considered this integral in another site, but it is only imperfect and non-rigorous one.
It seems that the Formelsammlung Mathematik is rendering a complete solution. It is written in German, but your bona fide translater Google may read this for you.
I also considered this integral in another site, but it is only imperfect and non-rigorous one.
It seems that the Formelsammlung Mathematik is rendering a complete solution. It is written in German, but your bona fide translater Google may read this for you.
answered Oct 5 '12 at 8:40
Sangchul Lee
90.9k12163263
90.9k12163263
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
add a comment |
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
Your proof on AoPS seems to be along the same lines as mine. Using the tools of contour integration, it appears rigorous enough. (+1)
– robjohn♦
Oct 5 '12 at 9:01
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
@robjohn, Thanks for your upvote! Though numerically justified, I was not sure if my assumptions on the zeros can be justified...
– Sangchul Lee
Oct 5 '12 at 23:06
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
Actually, I did notice a problem with the arbitrary selection of the radius of the semicircular contours. It turns out that we need to avoid $Requivfrac{pi}{2}pmod{2pi}$ to avoid the singularities so that we can assure that the integral along the semicircle vanishes. I have almost updated my answer to handle this.
– robjohn♦
Oct 5 '12 at 23:50
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
I have finally updated my answer so that the contour misses the singularities.
– robjohn♦
Oct 8 '12 at 9:30
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
@robjohn: I am surprised by the detail of your final solution!
– Sangchul Lee
Oct 8 '12 at 11:41
add a comment |
up vote
2
down vote
The identity is due to Victor Adamchik, see
http://mathworld.wolfram.com/OmegaConstant.html
You may want to contact Dr Adamchik himself via the e-mail at
http://www.cs.cmu.edu/~adamchik/research.html
because this particular paper doesn't seem to be in the list, as far as I can see.
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
add a comment |
up vote
2
down vote
The identity is due to Victor Adamchik, see
http://mathworld.wolfram.com/OmegaConstant.html
You may want to contact Dr Adamchik himself via the e-mail at
http://www.cs.cmu.edu/~adamchik/research.html
because this particular paper doesn't seem to be in the list, as far as I can see.
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
add a comment |
up vote
2
down vote
up vote
2
down vote
The identity is due to Victor Adamchik, see
http://mathworld.wolfram.com/OmegaConstant.html
You may want to contact Dr Adamchik himself via the e-mail at
http://www.cs.cmu.edu/~adamchik/research.html
because this particular paper doesn't seem to be in the list, as far as I can see.
The identity is due to Victor Adamchik, see
http://mathworld.wolfram.com/OmegaConstant.html
You may want to contact Dr Adamchik himself via the e-mail at
http://www.cs.cmu.edu/~adamchik/research.html
because this particular paper doesn't seem to be in the list, as far as I can see.
answered Jun 16 '11 at 19:20
Luboš Motl
7,00611625
7,00611625
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
add a comment |
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
Are you also able to give useful answers as well?
– Von Neumann
Nov 12 '16 at 10:44
add a comment |
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1
Maple says the integral is approximately 0.605, but $1/(1+W(1))$ is approximately 0.638
– GEdgar
Jun 16 '11 at 16:03
In Maple I get .6381037434, which agrees with 1/(1+LambertW(1)). I guess the 0.605 refers to a version of the integral before the last edit.
– Robert Israel
Jun 16 '11 at 18:38
Originally it had $+x$ instead of $-x$.
– GEdgar
Jun 16 '11 at 18:48
Nice question (+1)
– user 1357113
Oct 9 '12 at 9:48