What is the harmonic conjugate of $v(x,y)=e^x$
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
asked Nov 22 '18 at 2:39
M. Calculator
376
add a comment |
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
asked Nov 22 '18 at 2:39
M. Calculator
376
6
$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 '18 at 2:58
add a comment |
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
asked Nov 22 '18 at 2:39
M. Calculator
376
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
complex-analysis harmonic-analysis
asked Nov 22 '18 at 2:39
M. Calculator
376
asked Nov 22 '18 at 2:39
M. Calculator
376
asked Nov 22 '18 at 2:39
M. Calculator
376
asked Nov 22 '18 at 2:39
M. Calculator
376
asked Nov 22 '18 at 2:39
M. Calculator
376
376
6
$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 '18 at 2:58
add a comment |
6
$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 '18 at 2:58
6
6
$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 '18 at 2:58
$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 '18 at 2:58
add a comment |
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.
answered Nov 22 '18 at 3:36
Anupam
2,3741823
add a comment |
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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.
answered Nov 22 '18 at 3:36
Anupam
2,3741823
add a comment |
$u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.
answered Nov 22 '18 at 3:36
Anupam
2,3741823
add a comment |
$u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.
answered Nov 22 '18 at 3:36
Anupam
2,3741823
$u_{xx}+u_{yy}=e^xneq 0$. Thus $u$ is not a harmonic function and so it can not have a harmonic conjugate.
answered Nov 22 '18 at 3:36
Anupam
2,3741823
answered Nov 22 '18 at 3:36
Anupam
2,3741823
answered Nov 22 '18 at 3:36
Anupam
2,3741823
answered Nov 22 '18 at 3:36
Anupam
2,3741823
2,3741823
add a comment |
add a comment |
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6
$e^x$ doesn't have a harmonic conjugate because it's not harmonic.
– Qiaochu Yuan
Nov 22 '18 at 2:58