What is the harmonic conjugate of $v(x,y)=e^x$











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Letting $v(x,y)=e^x$, what is the complex conjugate of $v(x,y)=e^x$? i.e. find $u(x,y)$.



When I differentiate $e^x$ with respect to x and y, and use the Cauchy-Riemann equations, it seems impossible to then integrate either $dv/dx$ or $dv/dy$ to get the harmonic conjugate.










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    $e^x$ doesn't have a harmonic conjugate because it's not harmonic.
    – Qiaochu Yuan
    Nov 22 at 2:58















up vote
-2
down vote

favorite












Letting $v(x,y)=e^x$, what is the complex conjugate of $v(x,y)=e^x$? i.e. find $u(x,y)$.



When I differentiate $e^x$ with respect to x and y, and use the Cauchy-Riemann equations, it seems impossible to then integrate either $dv/dx$ or $dv/dy$ to get the harmonic conjugate.










share|cite|improve this question


















  • 6




    $e^x$ doesn't have a harmonic conjugate because it's not harmonic.
    – Qiaochu Yuan
    Nov 22 at 2:58













up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











Letting $v(x,y)=e^x$, what is the complex conjugate of $v(x,y)=e^x$? i.e. find $u(x,y)$.



When I differentiate $e^x$ with respect to x and y, and use the Cauchy-Riemann equations, it seems impossible to then integrate either $dv/dx$ or $dv/dy$ to get the harmonic conjugate.










share|cite|improve this question













Letting $v(x,y)=e^x$, what is the complex conjugate of $v(x,y)=e^x$? i.e. find $u(x,y)$.



When I differentiate $e^x$ with respect to x and y, and use the Cauchy-Riemann equations, it seems impossible to then integrate either $dv/dx$ or $dv/dy$ to get the harmonic conjugate.







complex-analysis harmonic-analysis






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asked Nov 22 at 2:39









M. Calculator

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  • 6




    $e^x$ doesn't have a harmonic conjugate because it's not harmonic.
    – Qiaochu Yuan
    Nov 22 at 2:58














  • 6




    $e^x$ doesn't have a harmonic conjugate because it's not harmonic.
    – Qiaochu Yuan
    Nov 22 at 2:58








6




6




$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 at 2:58




$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 at 2:58










1 Answer
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$u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.






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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    $u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      $u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        $u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.






        share|cite|improve this answer












        $u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 22 at 3:36









        Anupam

        2,3701823




        2,3701823






























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