Calculate complex integral $int_-infty^infty e^-ix^2d x=?$ [duplicate]

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3












$begingroup$



This question already has an answer here:



  • Integration (Fourier transform)

    2 answers



How to calculate this elementary complex integral? This is what we would encounter if we are studying the Green's function for Schroedinger's equation.
$$int_-infty^infty e^-ix^2d x=?$$
However, I think there should be someone that posted similar question on Math SE, though I don't know how to search by equation.



Thank you very much if you can help me out! And I would be grateful if you can give more than one approach




P.S.: The equation $int _-infty^inftye^-kt^2d sqrtkt=sqrtpi$ surely comes to my mind, but I don't know why it holds for $kinmathbbC$, because for me, the above integral is over real line, however, the question here is like integral on $y=e^i pi/4x$ ( So I think it's the problem with my complex integral knowledge.)



I tried to rotate this integral path by $pi/4$, but the two arcs at $Rrightarrow infty$ seem not easy to handle either.










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marked as duplicate by Jyrki Lahtonen, GNUSupporter 8964民主女神 地下教會, Lee David Chung Lin, Song, Leucippus Feb 27 at 1:41


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    Hi Collin - Welcome to MSE - in order for the community to be able to assist you, you need to provide all working you have done so far. If you are looking for a starting point, please ask - but this site is not a 'homework for free' site.
    $endgroup$
    – DavidG
    Feb 25 at 3:46






  • 1




    $begingroup$
    Hint: This is all you need to solve: $$ int_-infty^infty e^-x^2:dx = sqrtpi$$
    $endgroup$
    – DavidG
    Feb 25 at 3:48











  • $begingroup$
    Assuming the jump into the complex domain is valid. I always do.
    $endgroup$
    – marty cohen
    Feb 25 at 5:39
















3












$begingroup$



This question already has an answer here:



  • Integration (Fourier transform)

    2 answers



How to calculate this elementary complex integral? This is what we would encounter if we are studying the Green's function for Schroedinger's equation.
$$int_-infty^infty e^-ix^2d x=?$$
However, I think there should be someone that posted similar question on Math SE, though I don't know how to search by equation.



Thank you very much if you can help me out! And I would be grateful if you can give more than one approach




P.S.: The equation $int _-infty^inftye^-kt^2d sqrtkt=sqrtpi$ surely comes to my mind, but I don't know why it holds for $kinmathbbC$, because for me, the above integral is over real line, however, the question here is like integral on $y=e^i pi/4x$ ( So I think it's the problem with my complex integral knowledge.)



I tried to rotate this integral path by $pi/4$, but the two arcs at $Rrightarrow infty$ seem not easy to handle either.










share|cite|improve this question











$endgroup$



marked as duplicate by Jyrki Lahtonen, GNUSupporter 8964民主女神 地下教會, Lee David Chung Lin, Song, Leucippus Feb 27 at 1:41


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    Hi Collin - Welcome to MSE - in order for the community to be able to assist you, you need to provide all working you have done so far. If you are looking for a starting point, please ask - but this site is not a 'homework for free' site.
    $endgroup$
    – DavidG
    Feb 25 at 3:46






  • 1




    $begingroup$
    Hint: This is all you need to solve: $$ int_-infty^infty e^-x^2:dx = sqrtpi$$
    $endgroup$
    – DavidG
    Feb 25 at 3:48











  • $begingroup$
    Assuming the jump into the complex domain is valid. I always do.
    $endgroup$
    – marty cohen
    Feb 25 at 5:39














3












3








3





$begingroup$



This question already has an answer here:



  • Integration (Fourier transform)

    2 answers



How to calculate this elementary complex integral? This is what we would encounter if we are studying the Green's function for Schroedinger's equation.
$$int_-infty^infty e^-ix^2d x=?$$
However, I think there should be someone that posted similar question on Math SE, though I don't know how to search by equation.



Thank you very much if you can help me out! And I would be grateful if you can give more than one approach




P.S.: The equation $int _-infty^inftye^-kt^2d sqrtkt=sqrtpi$ surely comes to my mind, but I don't know why it holds for $kinmathbbC$, because for me, the above integral is over real line, however, the question here is like integral on $y=e^i pi/4x$ ( So I think it's the problem with my complex integral knowledge.)



I tried to rotate this integral path by $pi/4$, but the two arcs at $Rrightarrow infty$ seem not easy to handle either.










share|cite|improve this question











$endgroup$





This question already has an answer here:



  • Integration (Fourier transform)

    2 answers



How to calculate this elementary complex integral? This is what we would encounter if we are studying the Green's function for Schroedinger's equation.
$$int_-infty^infty e^-ix^2d x=?$$
However, I think there should be someone that posted similar question on Math SE, though I don't know how to search by equation.



Thank you very much if you can help me out! And I would be grateful if you can give more than one approach




P.S.: The equation $int _-infty^inftye^-kt^2d sqrtkt=sqrtpi$ surely comes to my mind, but I don't know why it holds for $kinmathbbC$, because for me, the above integral is over real line, however, the question here is like integral on $y=e^i pi/4x$ ( So I think it's the problem with my complex integral knowledge.)



I tried to rotate this integral path by $pi/4$, but the two arcs at $Rrightarrow infty$ seem not easy to handle either.





This question already has an answer here:



  • Integration (Fourier transform)

    2 answers







integration complex-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 25 at 7:02









Andrews

1,2691421




1,2691421










asked Feb 25 at 3:31









CollinCollin

1447




1447




marked as duplicate by Jyrki Lahtonen, GNUSupporter 8964民主女神 地下教會, Lee David Chung Lin, Song, Leucippus Feb 27 at 1:41


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Jyrki Lahtonen, GNUSupporter 8964民主女神 地下教會, Lee David Chung Lin, Song, Leucippus Feb 27 at 1:41


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • $begingroup$
    Hi Collin - Welcome to MSE - in order for the community to be able to assist you, you need to provide all working you have done so far. If you are looking for a starting point, please ask - but this site is not a 'homework for free' site.
    $endgroup$
    – DavidG
    Feb 25 at 3:46






  • 1




    $begingroup$
    Hint: This is all you need to solve: $$ int_-infty^infty e^-x^2:dx = sqrtpi$$
    $endgroup$
    – DavidG
    Feb 25 at 3:48











  • $begingroup$
    Assuming the jump into the complex domain is valid. I always do.
    $endgroup$
    – marty cohen
    Feb 25 at 5:39

















  • $begingroup$
    Hi Collin - Welcome to MSE - in order for the community to be able to assist you, you need to provide all working you have done so far. If you are looking for a starting point, please ask - but this site is not a 'homework for free' site.
    $endgroup$
    – DavidG
    Feb 25 at 3:46






  • 1




    $begingroup$
    Hint: This is all you need to solve: $$ int_-infty^infty e^-x^2:dx = sqrtpi$$
    $endgroup$
    – DavidG
    Feb 25 at 3:48











  • $begingroup$
    Assuming the jump into the complex domain is valid. I always do.
    $endgroup$
    – marty cohen
    Feb 25 at 5:39
















$begingroup$
Hi Collin - Welcome to MSE - in order for the community to be able to assist you, you need to provide all working you have done so far. If you are looking for a starting point, please ask - but this site is not a 'homework for free' site.
$endgroup$
– DavidG
Feb 25 at 3:46




$begingroup$
Hi Collin - Welcome to MSE - in order for the community to be able to assist you, you need to provide all working you have done so far. If you are looking for a starting point, please ask - but this site is not a 'homework for free' site.
$endgroup$
– DavidG
Feb 25 at 3:46




1




1




$begingroup$
Hint: This is all you need to solve: $$ int_-infty^infty e^-x^2:dx = sqrtpi$$
$endgroup$
– DavidG
Feb 25 at 3:48





$begingroup$
Hint: This is all you need to solve: $$ int_-infty^infty e^-x^2:dx = sqrtpi$$
$endgroup$
– DavidG
Feb 25 at 3:48













$begingroup$
Assuming the jump into the complex domain is valid. I always do.
$endgroup$
– marty cohen
Feb 25 at 5:39





$begingroup$
Assuming the jump into the complex domain is valid. I always do.
$endgroup$
– marty cohen
Feb 25 at 5:39











3 Answers
3






active

oldest

votes


















5












$begingroup$

Trying to avoid complex funniness.



$beginarray\
int_-infty^infty e^-ix^2dx
&=int_-infty^infty (cos(x^2)-isin(x^2))dx\
&=2int_0^infty (cos(x^2)-isin(x^2))dx\
&=2int_0^infty cos(x^2)dx-2iint_0^inftysin(x^2))dx\
endarray
$



and these are the
Fresnel integrals
$C(x)$ and $S(x)$
both of which approach
$dfracsqrtpi8
$

as $x to infty$.



Therefore the result is
$(1-i)sqrtfracpi2
$

as Claude Leibovici
got.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    How did you avoid 'complexness' in this solution? :-)
    $endgroup$
    – DavidG
    Feb 25 at 9:41










  • $begingroup$
    By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
    $endgroup$
    – marty cohen
    Feb 25 at 14:20










  • $begingroup$
    Good way to avoid any further discussion about the complex ! Thanks.
    $endgroup$
    – Claude Leibovici
    Feb 26 at 3:21


















6












$begingroup$

Hint
$$int e^-k x^2,dx=fracsqrtpi 2 sqrtk,texterfleft(sqrtk xright)$$
$$f(k)=int_-infty^infty e^-k x^2,dx=fracsqrtpi sqrtk$$
$$f(i)=fracsqrtpi sqrti=(1-i) sqrtfracpi 2$$






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
    $endgroup$
    – Collin
    Feb 25 at 6:01











  • $begingroup$
    @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
    $endgroup$
    – Sangchul Lee
    Feb 25 at 6:24











  • $begingroup$
    @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
    $endgroup$
    – Collin
    Feb 25 at 19:57










  • $begingroup$
    See my answer that purposely avoids this problem.
    $endgroup$
    – marty cohen
    Feb 26 at 0:03


















2












$begingroup$

Hint:$$int_-infty^infty e^-kx^2dx=int_-infty^infty e^-left(xsqrt kright)^2dx$$
Use the $u$-substution $u=xsqrt k$ and this transforms the integral into the form given in DavidG's suggestion. Can you take it from here?






share|cite|improve this answer









$endgroup$



















    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    Trying to avoid complex funniness.



    $beginarray\
    int_-infty^infty e^-ix^2dx
    &=int_-infty^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty cos(x^2)dx-2iint_0^inftysin(x^2))dx\
    endarray
    $



    and these are the
    Fresnel integrals
    $C(x)$ and $S(x)$
    both of which approach
    $dfracsqrtpi8
    $

    as $x to infty$.



    Therefore the result is
    $(1-i)sqrtfracpi2
    $

    as Claude Leibovici
    got.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      How did you avoid 'complexness' in this solution? :-)
      $endgroup$
      – DavidG
      Feb 25 at 9:41










    • $begingroup$
      By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
      $endgroup$
      – marty cohen
      Feb 25 at 14:20










    • $begingroup$
      Good way to avoid any further discussion about the complex ! Thanks.
      $endgroup$
      – Claude Leibovici
      Feb 26 at 3:21















    5












    $begingroup$

    Trying to avoid complex funniness.



    $beginarray\
    int_-infty^infty e^-ix^2dx
    &=int_-infty^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty cos(x^2)dx-2iint_0^inftysin(x^2))dx\
    endarray
    $



    and these are the
    Fresnel integrals
    $C(x)$ and $S(x)$
    both of which approach
    $dfracsqrtpi8
    $

    as $x to infty$.



    Therefore the result is
    $(1-i)sqrtfracpi2
    $

    as Claude Leibovici
    got.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      How did you avoid 'complexness' in this solution? :-)
      $endgroup$
      – DavidG
      Feb 25 at 9:41










    • $begingroup$
      By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
      $endgroup$
      – marty cohen
      Feb 25 at 14:20










    • $begingroup$
      Good way to avoid any further discussion about the complex ! Thanks.
      $endgroup$
      – Claude Leibovici
      Feb 26 at 3:21













    5












    5








    5





    $begingroup$

    Trying to avoid complex funniness.



    $beginarray\
    int_-infty^infty e^-ix^2dx
    &=int_-infty^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty cos(x^2)dx-2iint_0^inftysin(x^2))dx\
    endarray
    $



    and these are the
    Fresnel integrals
    $C(x)$ and $S(x)$
    both of which approach
    $dfracsqrtpi8
    $

    as $x to infty$.



    Therefore the result is
    $(1-i)sqrtfracpi2
    $

    as Claude Leibovici
    got.






    share|cite|improve this answer









    $endgroup$



    Trying to avoid complex funniness.



    $beginarray\
    int_-infty^infty e^-ix^2dx
    &=int_-infty^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty (cos(x^2)-isin(x^2))dx\
    &=2int_0^infty cos(x^2)dx-2iint_0^inftysin(x^2))dx\
    endarray
    $



    and these are the
    Fresnel integrals
    $C(x)$ and $S(x)$
    both of which approach
    $dfracsqrtpi8
    $

    as $x to infty$.



    Therefore the result is
    $(1-i)sqrtfracpi2
    $

    as Claude Leibovici
    got.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Feb 25 at 5:38









    marty cohenmarty cohen

    74.5k549129




    74.5k549129











    • $begingroup$
      How did you avoid 'complexness' in this solution? :-)
      $endgroup$
      – DavidG
      Feb 25 at 9:41










    • $begingroup$
      By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
      $endgroup$
      – marty cohen
      Feb 25 at 14:20










    • $begingroup$
      Good way to avoid any further discussion about the complex ! Thanks.
      $endgroup$
      – Claude Leibovici
      Feb 26 at 3:21
















    • $begingroup$
      How did you avoid 'complexness' in this solution? :-)
      $endgroup$
      – DavidG
      Feb 25 at 9:41










    • $begingroup$
      By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
      $endgroup$
      – marty cohen
      Feb 25 at 14:20










    • $begingroup$
      Good way to avoid any further discussion about the complex ! Thanks.
      $endgroup$
      – Claude Leibovici
      Feb 26 at 3:21















    $begingroup$
    How did you avoid 'complexness' in this solution? :-)
    $endgroup$
    – DavidG
    Feb 25 at 9:41




    $begingroup$
    How did you avoid 'complexness' in this solution? :-)
    $endgroup$
    – DavidG
    Feb 25 at 9:41












    $begingroup$
    By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
    $endgroup$
    – marty cohen
    Feb 25 at 14:20




    $begingroup$
    By not letting $k$ be complex in. $int e^-kx^2 dx$ when the result had only been proven for real $k$.
    $endgroup$
    – marty cohen
    Feb 25 at 14:20












    $begingroup$
    Good way to avoid any further discussion about the complex ! Thanks.
    $endgroup$
    – Claude Leibovici
    Feb 26 at 3:21




    $begingroup$
    Good way to avoid any further discussion about the complex ! Thanks.
    $endgroup$
    – Claude Leibovici
    Feb 26 at 3:21











    6












    $begingroup$

    Hint
    $$int e^-k x^2,dx=fracsqrtpi 2 sqrtk,texterfleft(sqrtk xright)$$
    $$f(k)=int_-infty^infty e^-k x^2,dx=fracsqrtpi sqrtk$$
    $$f(i)=fracsqrtpi sqrti=(1-i) sqrtfracpi 2$$






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
      $endgroup$
      – Collin
      Feb 25 at 6:01











    • $begingroup$
      @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
      $endgroup$
      – Sangchul Lee
      Feb 25 at 6:24











    • $begingroup$
      @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
      $endgroup$
      – Collin
      Feb 25 at 19:57










    • $begingroup$
      See my answer that purposely avoids this problem.
      $endgroup$
      – marty cohen
      Feb 26 at 0:03















    6












    $begingroup$

    Hint
    $$int e^-k x^2,dx=fracsqrtpi 2 sqrtk,texterfleft(sqrtk xright)$$
    $$f(k)=int_-infty^infty e^-k x^2,dx=fracsqrtpi sqrtk$$
    $$f(i)=fracsqrtpi sqrti=(1-i) sqrtfracpi 2$$






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
      $endgroup$
      – Collin
      Feb 25 at 6:01











    • $begingroup$
      @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
      $endgroup$
      – Sangchul Lee
      Feb 25 at 6:24











    • $begingroup$
      @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
      $endgroup$
      – Collin
      Feb 25 at 19:57










    • $begingroup$
      See my answer that purposely avoids this problem.
      $endgroup$
      – marty cohen
      Feb 26 at 0:03













    6












    6








    6





    $begingroup$

    Hint
    $$int e^-k x^2,dx=fracsqrtpi 2 sqrtk,texterfleft(sqrtk xright)$$
    $$f(k)=int_-infty^infty e^-k x^2,dx=fracsqrtpi sqrtk$$
    $$f(i)=fracsqrtpi sqrti=(1-i) sqrtfracpi 2$$






    share|cite|improve this answer









    $endgroup$



    Hint
    $$int e^-k x^2,dx=fracsqrtpi 2 sqrtk,texterfleft(sqrtk xright)$$
    $$f(k)=int_-infty^infty e^-k x^2,dx=fracsqrtpi sqrtk$$
    $$f(i)=fracsqrtpi sqrti=(1-i) sqrtfracpi 2$$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Feb 25 at 4:52









    Claude LeiboviciClaude Leibovici

    124k1158135




    124k1158135











    • $begingroup$
      Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
      $endgroup$
      – Collin
      Feb 25 at 6:01











    • $begingroup$
      @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
      $endgroup$
      – Sangchul Lee
      Feb 25 at 6:24











    • $begingroup$
      @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
      $endgroup$
      – Collin
      Feb 25 at 19:57










    • $begingroup$
      See my answer that purposely avoids this problem.
      $endgroup$
      – marty cohen
      Feb 26 at 0:03
















    • $begingroup$
      Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
      $endgroup$
      – Collin
      Feb 25 at 6:01











    • $begingroup$
      @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
      $endgroup$
      – Sangchul Lee
      Feb 25 at 6:24











    • $begingroup$
      @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
      $endgroup$
      – Collin
      Feb 25 at 19:57










    • $begingroup$
      See my answer that purposely avoids this problem.
      $endgroup$
      – marty cohen
      Feb 26 at 0:03















    $begingroup$
    Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
    $endgroup$
    – Collin
    Feb 25 at 6:01





    $begingroup$
    Hi! I think it's because I'm not familiar with complex integral; so how can you guys play so freely on the $mathbbC$ plane, without worrying much? The very reason I ask this is because I don't know why your second equation holds for k being a complex number. I tried to rotate this integral path by $pi/4$ on the complex plane, but the two arcs at $|x|rightarrow infty$ seem not trying to vanish.
    $endgroup$
    – Collin
    Feb 25 at 6:01













    $begingroup$
    @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
    $endgroup$
    – Sangchul Lee
    Feb 25 at 6:24





    $begingroup$
    @Collin, The reasoning behind this 'extrapolation' is that $f$ defines a holomorphic function on the right half-plane $H=k:operatornameRe(k) > 0$ and a continuous function on $overlineHsetminus0 = k : operatornameRe(k) geq 0, k neq 0 $. Now, in view of identity theorem, if you can identify another holomorphic function $g$ on $H$ such that $f = g$ on $(0,infty)$, then $f = g$ on all of $H$ as well. We can check that this argument works with $g(k)=sqrtpi/k$, and so, $f = g$ on $H$ by identity theorem and on all of $overlineHsetminus0$ by continuity.
    $endgroup$
    – Sangchul Lee
    Feb 25 at 6:24













    $begingroup$
    @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
    $endgroup$
    – Collin
    Feb 25 at 19:57




    $begingroup$
    @SangchulLee Hi! So this is the material covered by complex analysis? I have encountered many times similar problems when doing complex integral. As a physicist, complex analysis is only usually covered by Math-physics course within a chapter.
    $endgroup$
    – Collin
    Feb 25 at 19:57












    $begingroup$
    See my answer that purposely avoids this problem.
    $endgroup$
    – marty cohen
    Feb 26 at 0:03




    $begingroup$
    See my answer that purposely avoids this problem.
    $endgroup$
    – marty cohen
    Feb 26 at 0:03











    2












    $begingroup$

    Hint:$$int_-infty^infty e^-kx^2dx=int_-infty^infty e^-left(xsqrt kright)^2dx$$
    Use the $u$-substution $u=xsqrt k$ and this transforms the integral into the form given in DavidG's suggestion. Can you take it from here?






    share|cite|improve this answer









    $endgroup$

















      2












      $begingroup$

      Hint:$$int_-infty^infty e^-kx^2dx=int_-infty^infty e^-left(xsqrt kright)^2dx$$
      Use the $u$-substution $u=xsqrt k$ and this transforms the integral into the form given in DavidG's suggestion. Can you take it from here?






      share|cite|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        Hint:$$int_-infty^infty e^-kx^2dx=int_-infty^infty e^-left(xsqrt kright)^2dx$$
        Use the $u$-substution $u=xsqrt k$ and this transforms the integral into the form given in DavidG's suggestion. Can you take it from here?






        share|cite|improve this answer









        $endgroup$



        Hint:$$int_-infty^infty e^-kx^2dx=int_-infty^infty e^-left(xsqrt kright)^2dx$$
        Use the $u$-substution $u=xsqrt k$ and this transforms the integral into the form given in DavidG's suggestion. Can you take it from here?







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 25 at 5:35









        csch2csch2

        6031314




        6031314












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