complex harmonic functions$u(z)$ harmonic if and only if $u(overlinez)$ harmonicRelation Between Complex GradientsHarmonic Maximum modulusShow that h is harmonic iff $fracpartial hpartial overline z$ is conjugate harmonicShow composition of harmonic function and analytic function is harmonic without calculating 2nd derivativesProof about Harmonic functionFor a point in a disk, how does it show that the following equation defines a harmonic conjugate?Dirichlet problem and Brownian motionWhat is the partial derivative of the composition of two complex functions?Harmonic Functions and Partial Derivatives with Chain Rule (Complex Variables)
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complex harmonic functions
$u(z)$ harmonic if and only if $u(overlinez)$ harmonicRelation Between Complex GradientsHarmonic Maximum modulusShow that h is harmonic iff $fracpartial hpartial overline z$ is conjugate harmonicShow composition of harmonic function and analytic function is harmonic without calculating 2nd derivativesProof about Harmonic functionFor a point in a disk, how does it show that the following equation defines a harmonic conjugate?Dirichlet problem and Brownian motionWhat is the partial derivative of the composition of two complex functions?Harmonic Functions and Partial Derivatives with Chain Rule (Complex Variables)
$begingroup$
I have not been able to do this exercise since I do not understand well how to use the chain rule in this exercise, they could help me with the way to solve it...
Let $phi :Dto D’$ be analytic and even twice continuously differentiable and $eta: D’ to mathbbR $ twice continuously partial differentiable show
if $phi$ is conformal then $eta$ is harmonic if and only if $eta circ phi$ is harmonic.
complex-analysis
asked Mar 15 at 4:30
J9360J9360
1
$endgroup$
add a comment |
$begingroup$
I have not been able to do this exercise since I do not understand well how to use the chain rule in this exercise, they could help me with the way to solve it...
Let $phi :Dto D’$ be analytic and even twice continuously differentiable and $eta: D’ to mathbbR $ twice continuously partial differentiable show
if $phi$ is conformal then $eta$ is harmonic if and only if $eta circ phi$ is harmonic.
complex-analysis
asked Mar 15 at 4:30
J9360J9360
1
$endgroup$
$begingroup$
What are $D$ and $D'$?
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:31
$begingroup$
$D,D′ subseteq mathbbC$
$endgroup$
– J9360
Mar 15 at 5:43
$begingroup$
Do you know that analytic functions on open sets in $mathbb C$ are infinitely differentiable and have continuous partial derivatives of all orders? You are assumption are superfluous.
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:50
add a comment |
$begingroup$
I have not been able to do this exercise since I do not understand well how to use the chain rule in this exercise, they could help me with the way to solve it...
Let $phi :Dto D’$ be analytic and even twice continuously differentiable and $eta: D’ to mathbbR $ twice continuously partial differentiable show
if $phi$ is conformal then $eta$ is harmonic if and only if $eta circ phi$ is harmonic.
complex-analysis
asked Mar 15 at 4:30
J9360J9360
1
$endgroup$
I have not been able to do this exercise since I do not understand well how to use the chain rule in this exercise, they could help me with the way to solve it...
Let $phi :Dto D’$ be analytic and even twice continuously differentiable and $eta: D’ to mathbbR $ twice continuously partial differentiable show
if $phi$ is conformal then $eta$ is harmonic if and only if $eta circ phi$ is harmonic.
complex-analysis
complex-analysis
asked Mar 15 at 4:30
J9360J9360
1
asked Mar 15 at 4:30
J9360J9360
1
asked Mar 15 at 4:30
J9360J9360
1
asked Mar 15 at 4:30
J9360J9360
1
asked Mar 15 at 4:30
J9360J9360
1
1
$begingroup$
What are $D$ and $D'$?
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:31
$begingroup$
$D,D′ subseteq mathbbC$
$endgroup$
– J9360
Mar 15 at 5:43
$begingroup$
Do you know that analytic functions on open sets in $mathbb C$ are infinitely differentiable and have continuous partial derivatives of all orders? You are assumption are superfluous.
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:50
add a comment |
$begingroup$
What are $D$ and $D'$?
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:31
$begingroup$
$D,D′ subseteq mathbbC$
$endgroup$
– J9360
Mar 15 at 5:43
$begingroup$
Do you know that analytic functions on open sets in $mathbb C$ are infinitely differentiable and have continuous partial derivatives of all orders? You are assumption are superfluous.
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:50
$begingroup$
What are $D$ and $D'$?
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:31
$begingroup$
What are $D$ and $D'$?
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:31
$begingroup$
$D,D′ subseteq mathbbC$
$endgroup$
– J9360
Mar 15 at 5:43
$begingroup$
$D,D′ subseteq mathbbC$
$endgroup$
– J9360
Mar 15 at 5:43
$begingroup$
Do you know that analytic functions on open sets in $mathbb C$ are infinitely differentiable and have continuous partial derivatives of all orders? You are assumption are superfluous.
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:50
$begingroup$
Do you know that analytic functions on open sets in $mathbb C$ are infinitely differentiable and have continuous partial derivatives of all orders? You are assumption are superfluous.
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:50
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint :harmonicity is a local property: a function is harmonic in an open set iff it is harmonic in some disk around each point. In an disk a function is harmonic iff it is the real part of an anlytic function. The result follows from this and the fact that composition if two analytic functions is analytic. [If $phi$ is a conformal mapping then $phi ^-1$ is also analytic].
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Hint :harmonicity is a local property: a function is harmonic in an open set iff it is harmonic in some disk around each point. In an disk a function is harmonic iff it is the real part of an anlytic function. The result follows from this and the fact that composition if two analytic functions is analytic. [If $phi$ is a conformal mapping then $phi ^-1$ is also analytic].
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
Hint :harmonicity is a local property: a function is harmonic in an open set iff it is harmonic in some disk around each point. In an disk a function is harmonic iff it is the real part of an anlytic function. The result follows from this and the fact that composition if two analytic functions is analytic. [If $phi$ is a conformal mapping then $phi ^-1$ is also analytic].
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
Hint :harmonicity is a local property: a function is harmonic in an open set iff it is harmonic in some disk around each point. In an disk a function is harmonic iff it is the real part of an anlytic function. The result follows from this and the fact that composition if two analytic functions is analytic. [If $phi$ is a conformal mapping then $phi ^-1$ is also analytic].
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
Hint :harmonicity is a local property: a function is harmonic in an open set iff it is harmonic in some disk around each point. In an disk a function is harmonic iff it is the real part of an anlytic function. The result follows from this and the fact that composition if two analytic functions is analytic. [If $phi$ is a conformal mapping then $phi ^-1$ is also analytic].
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Mar 15 at 5:49
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
69.1k53169
add a comment |
add a comment |
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$begingroup$
What are $D$ and $D'$?
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:31
$begingroup$
$D,D′ subseteq mathbbC$
$endgroup$
– J9360
Mar 15 at 5:43
$begingroup$
Do you know that analytic functions on open sets in $mathbb C$ are infinitely differentiable and have continuous partial derivatives of all orders? You are assumption are superfluous.
$endgroup$
– Kavi Rama Murthy
Mar 15 at 5:50