Cauchy-Riemann equations for $z=x+iy$ and $f(z)=R(x,y)e^{itheta(x,y)}$
I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake):
$frac{partial u}{partial x} = frac{partial v}{partial y}$, $frac{partial u}{partial y} = -frac{partial v}{partial x}$, where $z=x+iy$, $f(z)=u(x,y)+iv(x,y)$,
$rfrac{partial u}{partial r} = frac{partial v}{partial theta}$, $frac{partial u}{partial theta} = -rfrac{partial v}{partial r}$, where $z=re^{itheta}$, $f(z)=u(r,theta)+iv(r,theta)$,
$frac{partial ln R}{partial ln r} = frac{partial varphi}{partial theta}$, $frac{partial ln R}{partial theta} = -frac{partial varphi}{partial ln r}$, where $z=re^{itheta}$, $f(z)=R(r,theta)e^{ivarphi(r,theta)}$.
Can someone show me the fourth form, i.e. with $z=x+iy$ and $f(z)=R(x,y)e^{ivarphi(x,y)}$, preferably with a proof if it is not too bothersome?
complex-analysis partial-derivative analytic-functions cauchy-riemann-equation
asked Nov 28 '18 at 20:07
Haris Gusic
132
add a comment |
I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake):
$frac{partial u}{partial x} = frac{partial v}{partial y}$, $frac{partial u}{partial y} = -frac{partial v}{partial x}$, where $z=x+iy$, $f(z)=u(x,y)+iv(x,y)$,
$rfrac{partial u}{partial r} = frac{partial v}{partial theta}$, $frac{partial u}{partial theta} = -rfrac{partial v}{partial r}$, where $z=re^{itheta}$, $f(z)=u(r,theta)+iv(r,theta)$,
$frac{partial ln R}{partial ln r} = frac{partial varphi}{partial theta}$, $frac{partial ln R}{partial theta} = -frac{partial varphi}{partial ln r}$, where $z=re^{itheta}$, $f(z)=R(r,theta)e^{ivarphi(r,theta)}$.
Can someone show me the fourth form, i.e. with $z=x+iy$ and $f(z)=R(x,y)e^{ivarphi(x,y)}$, preferably with a proof if it is not too bothersome?
complex-analysis partial-derivative analytic-functions cauchy-riemann-equation
asked Nov 28 '18 at 20:07
Haris Gusic
132
add a comment |
I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake):
$frac{partial u}{partial x} = frac{partial v}{partial y}$, $frac{partial u}{partial y} = -frac{partial v}{partial x}$, where $z=x+iy$, $f(z)=u(x,y)+iv(x,y)$,
$rfrac{partial u}{partial r} = frac{partial v}{partial theta}$, $frac{partial u}{partial theta} = -rfrac{partial v}{partial r}$, where $z=re^{itheta}$, $f(z)=u(r,theta)+iv(r,theta)$,
$frac{partial ln R}{partial ln r} = frac{partial varphi}{partial theta}$, $frac{partial ln R}{partial theta} = -frac{partial varphi}{partial ln r}$, where $z=re^{itheta}$, $f(z)=R(r,theta)e^{ivarphi(r,theta)}$.
Can someone show me the fourth form, i.e. with $z=x+iy$ and $f(z)=R(x,y)e^{ivarphi(x,y)}$, preferably with a proof if it is not too bothersome?
complex-analysis partial-derivative analytic-functions cauchy-riemann-equation
asked Nov 28 '18 at 20:07
Haris Gusic
132
I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake):
$frac{partial u}{partial x} = frac{partial v}{partial y}$, $frac{partial u}{partial y} = -frac{partial v}{partial x}$, where $z=x+iy$, $f(z)=u(x,y)+iv(x,y)$,
$rfrac{partial u}{partial r} = frac{partial v}{partial theta}$, $frac{partial u}{partial theta} = -rfrac{partial v}{partial r}$, where $z=re^{itheta}$, $f(z)=u(r,theta)+iv(r,theta)$,
$frac{partial ln R}{partial ln r} = frac{partial varphi}{partial theta}$, $frac{partial ln R}{partial theta} = -frac{partial varphi}{partial ln r}$, where $z=re^{itheta}$, $f(z)=R(r,theta)e^{ivarphi(r,theta)}$.
Can someone show me the fourth form, i.e. with $z=x+iy$ and $f(z)=R(x,y)e^{ivarphi(x,y)}$, preferably with a proof if it is not too bothersome?
complex-analysis partial-derivative analytic-functions cauchy-riemann-equation
complex-analysis partial-derivative analytic-functions cauchy-riemann-equation
asked Nov 28 '18 at 20:07
Haris Gusic
132
asked Nov 28 '18 at 20:07
Haris Gusic
132
asked Nov 28 '18 at 20:07
Haris Gusic
132
asked Nov 28 '18 at 20:07
Haris Gusic
132
asked Nov 28 '18 at 20:07
Haris Gusic
132
132
add a comment |
add a comment |
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