Closed form of :$ int_-infty^inftyarctanleft(e^-x^2 texterf(x)right),arctanleft(e^x^2texterf(x)right),dx $ The 2019 Stack Overflow Developer Survey Results Are InDoes $int_-1^1fracarctan xtextarctanh,x,mathrmdx$ have a closed form?Closed form for $int_0^infty !rm erf left(cxright) left( rm erf left(x right) right) ^2rm e^-x^2,rm dx$$inttexte^-ax^2 texterfleft(bx + cright) dx$$int_c^infty exp(-u^2) textErf(au + b) du$How this $int_0^ax^texterf(exp(-x))dx$ behaves?On a closed form for $int_-infty^inftyfracdxleft(1+x^2right)^p$What is the series expansion of the $n$-th derivative of this : $fracd^ndx^nint(e^-x²)^texterf(x)dx$What is the exact value of this : $5int_0^inftyexp(-x^2 texterf(x))x^sin x+frac12dx$?What is $lim_ntoinftyint_0^infty exp(-x^n arctan(frac1x)) dx,n>1$Nice result that I can't prove: $int_-2^2 tan^-1 bigg( exp(-x²texterf(x)) bigg) ;dx=pi$
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Closed form of :$ int_-infty^inftyarctanleft(e^-x^2 texterf(x)right),arctanleft(e^x^2texterf(x)right),dx $
The 2019 Stack Overflow Developer Survey Results Are InDoes $int_-1^1fracarctan xtextarctanh,x,mathrmdx$ have a closed form?Closed form for $int_0^infty !rm erf left(cxright) left( rm erf left(x right) right) ^2rm e^-x^2,rm dx$$inttexte^-ax^2 texterfleft(bx + cright) dx$$int_c^infty exp(-u^2) textErf(au + b) du$How this $int_0^ax^texterf(exp(-x))dx$ behaves?On a closed form for $int_-infty^inftyfracdxleft(1+x^2right)^p$What is the series expansion of the $n$-th derivative of this : $fracd^ndx^nint(e^-x²)^texterf(x)dx$What is the exact value of this : $5int_0^inftyexp(-x^2 texterf(x))x^sin x+frac12dx$?What is $lim_ntoinftyint_0^infty exp(-x^n arctan(frac1x)) dx,n>1$Nice result that I can't prove: $int_-2^2 tan^-1 bigg( exp(-x²texterf(x)) bigg) ;dx=pi$
$begingroup$
That is one of interesting integral that i have accrossed in my text book when i have tried to understand some thing related to distribution theory in the statistics context , The following integral really make me tired to get its closed form however i find some integral connecting to that they have closed form .
begineqnarray*
int_-infty^inftyarctanleft(e^-x^2 texterf(x)right),arctanleft(e^x^2texterf(x)right),dx sim frac5pi
endeqnarray*
Now if we use the integration by part as the first step we should accross to get the following integral where i have got it's series representation and it is defined as follow by :
beginequation
intlimits_0^xe^-xi ^2texterf(xi )dxi
=sumlimits_n=0^infty lim_varepsilon ->0left( sumlimits
_substack k_1+2k_2+cdots +nk_n=n \ k_1geq 0,k_2geq
0,...,k_ngeq 0prodlimits_j=1^nfrac^A_j,varepsilon ^k_j%
k_j!right) fracx^n+1n+1
endequation
where:
begineqnarray*
A_j,epsilon &=&frac2(-1)^(j-1)/2(j-2)(frac12(j-3))!sqrtpi %
text if jgeq 3text and jtext an odd integer; \
A_j,epsilon &=&varepsilon text otherwise (0<varepsilon <1)text.
endeqnarray*
Really that represnation can't give me the result because it is hard to conclude arctan of that complicated series , All my attempt can't give me the closed form , Only i know that integral could be close to $frac5pi$ close to the result shown by Wolfram alphasince i know that erf is the function which deals with $ pi$ incrementation , Now my question is :
Question: Is it possible to get its closed form value ? and if yes could we get also its series representation ?
integration probability-distributions closed-form trigonometric-series
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
$endgroup$
add a comment |
$begingroup$
That is one of interesting integral that i have accrossed in my text book when i have tried to understand some thing related to distribution theory in the statistics context , The following integral really make me tired to get its closed form however i find some integral connecting to that they have closed form .
begineqnarray*
int_-infty^inftyarctanleft(e^-x^2 texterf(x)right),arctanleft(e^x^2texterf(x)right),dx sim frac5pi
endeqnarray*
Now if we use the integration by part as the first step we should accross to get the following integral where i have got it's series representation and it is defined as follow by :
beginequation
intlimits_0^xe^-xi ^2texterf(xi )dxi
=sumlimits_n=0^infty lim_varepsilon ->0left( sumlimits
_substack k_1+2k_2+cdots +nk_n=n \ k_1geq 0,k_2geq
0,...,k_ngeq 0prodlimits_j=1^nfrac^A_j,varepsilon ^k_j%
k_j!right) fracx^n+1n+1
endequation
where:
begineqnarray*
A_j,epsilon &=&frac2(-1)^(j-1)/2(j-2)(frac12(j-3))!sqrtpi %
text if jgeq 3text and jtext an odd integer; \
A_j,epsilon &=&varepsilon text otherwise (0<varepsilon <1)text.
endeqnarray*
Really that represnation can't give me the result because it is hard to conclude arctan of that complicated series , All my attempt can't give me the closed form , Only i know that integral could be close to $frac5pi$ close to the result shown by Wolfram alphasince i know that erf is the function which deals with $ pi$ incrementation , Now my question is :
Question: Is it possible to get its closed form value ? and if yes could we get also its series representation ?
integration probability-distributions closed-form trigonometric-series
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
$endgroup$
1
$begingroup$
Now what textbook would that be?
$endgroup$
– omegadot
Mar 31 at 1:48
add a comment |
$begingroup$
That is one of interesting integral that i have accrossed in my text book when i have tried to understand some thing related to distribution theory in the statistics context , The following integral really make me tired to get its closed form however i find some integral connecting to that they have closed form .
begineqnarray*
int_-infty^inftyarctanleft(e^-x^2 texterf(x)right),arctanleft(e^x^2texterf(x)right),dx sim frac5pi
endeqnarray*
Now if we use the integration by part as the first step we should accross to get the following integral where i have got it's series representation and it is defined as follow by :
beginequation
intlimits_0^xe^-xi ^2texterf(xi )dxi
=sumlimits_n=0^infty lim_varepsilon ->0left( sumlimits
_substack k_1+2k_2+cdots +nk_n=n \ k_1geq 0,k_2geq
0,...,k_ngeq 0prodlimits_j=1^nfrac^A_j,varepsilon ^k_j%
k_j!right) fracx^n+1n+1
endequation
where:
begineqnarray*
A_j,epsilon &=&frac2(-1)^(j-1)/2(j-2)(frac12(j-3))!sqrtpi %
text if jgeq 3text and jtext an odd integer; \
A_j,epsilon &=&varepsilon text otherwise (0<varepsilon <1)text.
endeqnarray*
Really that represnation can't give me the result because it is hard to conclude arctan of that complicated series , All my attempt can't give me the closed form , Only i know that integral could be close to $frac5pi$ close to the result shown by Wolfram alphasince i know that erf is the function which deals with $ pi$ incrementation , Now my question is :
Question: Is it possible to get its closed form value ? and if yes could we get also its series representation ?
integration probability-distributions closed-form trigonometric-series
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
$endgroup$
That is one of interesting integral that i have accrossed in my text book when i have tried to understand some thing related to distribution theory in the statistics context , The following integral really make me tired to get its closed form however i find some integral connecting to that they have closed form .
begineqnarray*
int_-infty^inftyarctanleft(e^-x^2 texterf(x)right),arctanleft(e^x^2texterf(x)right),dx sim frac5pi
endeqnarray*
Now if we use the integration by part as the first step we should accross to get the following integral where i have got it's series representation and it is defined as follow by :
beginequation
intlimits_0^xe^-xi ^2texterf(xi )dxi
=sumlimits_n=0^infty lim_varepsilon ->0left( sumlimits
_substack k_1+2k_2+cdots +nk_n=n \ k_1geq 0,k_2geq
0,...,k_ngeq 0prodlimits_j=1^nfrac^A_j,varepsilon ^k_j%
k_j!right) fracx^n+1n+1
endequation
where:
begineqnarray*
A_j,epsilon &=&frac2(-1)^(j-1)/2(j-2)(frac12(j-3))!sqrtpi %
text if jgeq 3text and jtext an odd integer; \
A_j,epsilon &=&varepsilon text otherwise (0<varepsilon <1)text.
endeqnarray*
Really that represnation can't give me the result because it is hard to conclude arctan of that complicated series , All my attempt can't give me the closed form , Only i know that integral could be close to $frac5pi$ close to the result shown by Wolfram alphasince i know that erf is the function which deals with $ pi$ incrementation , Now my question is :
Question: Is it possible to get its closed form value ? and if yes could we get also its series representation ?
integration probability-distributions closed-form trigonometric-series
integration probability-distributions closed-form trigonometric-series
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
edited Mar 30 at 22:05
zeraoulia rafik
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
asked Mar 30 at 21:12
zeraoulia rafikzeraoulia rafik
2,39211133
2,39211133
1
$begingroup$
Now what textbook would that be?
$endgroup$
– omegadot
Mar 31 at 1:48
add a comment |
1
$begingroup$
Now what textbook would that be?
$endgroup$
– omegadot
Mar 31 at 1:48
1
1
$begingroup$
Now what textbook would that be?
$endgroup$
– omegadot
Mar 31 at 1:48
$begingroup$
Now what textbook would that be?
$endgroup$
– omegadot
Mar 31 at 1:48
add a comment |
0
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$begingroup$
Now what textbook would that be?
$endgroup$
– omegadot
Mar 31 at 1:48