Are the components of a Cauchy-Riemann mapping harmonic?Cauchy-Riemann implies analyticityConsequence of Riemann mapping theoremCharacterization of Cauchy-Riemann operatorCauchy-Riemann equation and differentiability in $mathbb R^2$Are the Cauchy Riemann conditions sufficient for analyticityCauchy Riemann equation and Harmonic ConditionGiven u and v are harmonic in some region R prove the followingGeneralization of the Cauchy-Riemann equationsConverse of the Cauchy-Riemann conditionProving that the composition of a harmonic function and a Cauchy-Riemann mapping is harmonic
Smoothness of finite-dimensional functional calculus
To string or not to string
Do VLANs within a subnet need to have their own subnet for router on a stick?
Languages that we cannot (dis)prove to be Context-Free
Why, historically, did Gödel think CH was false?
Why doesn't H₄O²⁺ exist?
In Japanese, what’s the difference between “Tonari ni” (となりに) and “Tsugi” (つぎ)? When would you use one over the other?
What do three bars across the stem of a note mean?
What's the point of deactivating Num Lock on login screens?
What typically incentivizes a professor to change jobs to a lower ranking university?
Pattern match does not work in bash script
Why don't electron-positron collisions release infinite energy?
Schoenfled Residua test shows proportionality hazard assumptions holds but Kaplan-Meier plots intersect
Why can't I see bouncing of a switch on an oscilloscope?
Is a conference paper whose proceedings will be published in IEEE Xplore counted as a publication?
How can bays and straits be determined in a procedurally generated map?
Mage Armor with Defense fighting style (for Adventurers League bladeslinger)
A newer friend of my brother's gave him a load of baseball cards that are supposedly extremely valuable. Is this a scam?
Adding span tags within wp_list_pages list items
Why are electrically insulating heatsinks so rare? Is it just cost?
What do the dots in this tr command do: tr .............A-Z A-ZA-Z <<< "JVPQBOV" (with 13 dots)
What are these boxed doors outside store fronts in New York?
Have astronauts in space suits ever taken selfies? If so, how?
How can I prevent hyper evolved versions of regular creatures from wiping out their cousins?
Are the components of a Cauchy-Riemann mapping harmonic?
Cauchy-Riemann implies analyticityConsequence of Riemann mapping theoremCharacterization of Cauchy-Riemann operatorCauchy-Riemann equation and differentiability in $mathbb R^2$Are the Cauchy Riemann conditions sufficient for analyticityCauchy Riemann equation and Harmonic ConditionGiven u and v are harmonic in some region R prove the followingGeneralization of the Cauchy-Riemann equationsConverse of the Cauchy-Riemann conditionProving that the composition of a harmonic function and a Cauchy-Riemann mapping is harmonic
$begingroup$
Let $mathcalO$ be an open subset of the plane $mathbbR^2$ and
let the mapping $F : mathcalO rightarrow mathbbR^2$ be
represented by $F(x, y) = (u(x, y), v(x, y))$ for $(x, y)$ in
$mathcalO$. Then, we say the mapping $F : mathcalO rightarrow
mathbbR^2$ is called a Cauchy-Riemann mapping provided that
each of the functions $u : mathcalO rightarrow mathbbR$ and $v
: mathcalO rightarrow mathbbR$ has continuous second-order
partial derivatives and $$fracpartial upartial x(x, y) =
fracpartial vpartial y(x, y) hspace1em text and
hspace1em fracpartial upartial y = -fracpartial
vpartial x(x, y)$$
for all $(x, y)$ in $mathcalO$.
Is it necessarily true that $u(x, y)$ and $v(x, y)$ are harmonic?
real-analysis complex-analysis
$endgroup$
add a comment |
$begingroup$
Let $mathcalO$ be an open subset of the plane $mathbbR^2$ and
let the mapping $F : mathcalO rightarrow mathbbR^2$ be
represented by $F(x, y) = (u(x, y), v(x, y))$ for $(x, y)$ in
$mathcalO$. Then, we say the mapping $F : mathcalO rightarrow
mathbbR^2$ is called a Cauchy-Riemann mapping provided that
each of the functions $u : mathcalO rightarrow mathbbR$ and $v
: mathcalO rightarrow mathbbR$ has continuous second-order
partial derivatives and $$fracpartial upartial x(x, y) =
fracpartial vpartial y(x, y) hspace1em text and
hspace1em fracpartial upartial y = -fracpartial
vpartial x(x, y)$$
for all $(x, y)$ in $mathcalO$.
Is it necessarily true that $u(x, y)$ and $v(x, y)$ are harmonic?
real-analysis complex-analysis
$endgroup$
add a comment |
$begingroup$
Let $mathcalO$ be an open subset of the plane $mathbbR^2$ and
let the mapping $F : mathcalO rightarrow mathbbR^2$ be
represented by $F(x, y) = (u(x, y), v(x, y))$ for $(x, y)$ in
$mathcalO$. Then, we say the mapping $F : mathcalO rightarrow
mathbbR^2$ is called a Cauchy-Riemann mapping provided that
each of the functions $u : mathcalO rightarrow mathbbR$ and $v
: mathcalO rightarrow mathbbR$ has continuous second-order
partial derivatives and $$fracpartial upartial x(x, y) =
fracpartial vpartial y(x, y) hspace1em text and
hspace1em fracpartial upartial y = -fracpartial
vpartial x(x, y)$$
for all $(x, y)$ in $mathcalO$.
Is it necessarily true that $u(x, y)$ and $v(x, y)$ are harmonic?
real-analysis complex-analysis
$endgroup$
Let $mathcalO$ be an open subset of the plane $mathbbR^2$ and
let the mapping $F : mathcalO rightarrow mathbbR^2$ be
represented by $F(x, y) = (u(x, y), v(x, y))$ for $(x, y)$ in
$mathcalO$. Then, we say the mapping $F : mathcalO rightarrow
mathbbR^2$ is called a Cauchy-Riemann mapping provided that
each of the functions $u : mathcalO rightarrow mathbbR$ and $v
: mathcalO rightarrow mathbbR$ has continuous second-order
partial derivatives and $$fracpartial upartial x(x, y) =
fracpartial vpartial y(x, y) hspace1em text and
hspace1em fracpartial upartial y = -fracpartial
vpartial x(x, y)$$
for all $(x, y)$ in $mathcalO$.
Is it necessarily true that $u(x, y)$ and $v(x, y)$ are harmonic?
real-analysis complex-analysis
real-analysis complex-analysis
asked Mar 29 at 15:33
user658798
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Yes. For instance,
$$
fracpartial^2 upartial^2 x + fracpartial^2 upartial^2 y =
fracpartial partial xfracpartial upartial x + fracpartial partial yfracpartial upartial y
= fracpartial partial xfracpartial vpartial y - fracpartial partial yfracpartial vpartial x = fracpartial^2 vpartial xpartial y - fracpartial^2 vpartial xpartial y = 0.
$$
The calculation for $v$ is similar.
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
$endgroup$
add a comment |
$begingroup$
The real and imaginary parts of a holomorphic function are harmonic. Source: Cartan's Elementary Theory of Analytic Functions.
answered Mar 29 at 15:36
avsavs
3,894515
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3167261%2fare-the-components-of-a-cauchy-riemann-mapping-harmonic%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes. For instance,
$$
fracpartial^2 upartial^2 x + fracpartial^2 upartial^2 y =
fracpartial partial xfracpartial upartial x + fracpartial partial yfracpartial upartial y
= fracpartial partial xfracpartial vpartial y - fracpartial partial yfracpartial vpartial x = fracpartial^2 vpartial xpartial y - fracpartial^2 vpartial xpartial y = 0.
$$
The calculation for $v$ is similar.
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
$endgroup$
add a comment |
$begingroup$
Yes. For instance,
$$
fracpartial^2 upartial^2 x + fracpartial^2 upartial^2 y =
fracpartial partial xfracpartial upartial x + fracpartial partial yfracpartial upartial y
= fracpartial partial xfracpartial vpartial y - fracpartial partial yfracpartial vpartial x = fracpartial^2 vpartial xpartial y - fracpartial^2 vpartial xpartial y = 0.
$$
The calculation for $v$ is similar.
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
$endgroup$
add a comment |
$begingroup$
Yes. For instance,
$$
fracpartial^2 upartial^2 x + fracpartial^2 upartial^2 y =
fracpartial partial xfracpartial upartial x + fracpartial partial yfracpartial upartial y
= fracpartial partial xfracpartial vpartial y - fracpartial partial yfracpartial vpartial x = fracpartial^2 vpartial xpartial y - fracpartial^2 vpartial xpartial y = 0.
$$
The calculation for $v$ is similar.
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
$endgroup$
Yes. For instance,
$$
fracpartial^2 upartial^2 x + fracpartial^2 upartial^2 y =
fracpartial partial xfracpartial upartial x + fracpartial partial yfracpartial upartial y
= fracpartial partial xfracpartial vpartial y - fracpartial partial yfracpartial vpartial x = fracpartial^2 vpartial xpartial y - fracpartial^2 vpartial xpartial y = 0.
$$
The calculation for $v$ is similar.
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
answered Mar 29 at 15:38
Ernie060Ernie060
2,940719
2,940719
add a comment |
add a comment |
$begingroup$
The real and imaginary parts of a holomorphic function are harmonic. Source: Cartan's Elementary Theory of Analytic Functions.
answered Mar 29 at 15:36
avsavs
3,894515
$endgroup$
add a comment |
$begingroup$
The real and imaginary parts of a holomorphic function are harmonic. Source: Cartan's Elementary Theory of Analytic Functions.
answered Mar 29 at 15:36
avsavs
3,894515
$endgroup$
add a comment |
$begingroup$
The real and imaginary parts of a holomorphic function are harmonic. Source: Cartan's Elementary Theory of Analytic Functions.
answered Mar 29 at 15:36
avsavs
3,894515
$endgroup$
The real and imaginary parts of a holomorphic function are harmonic. Source: Cartan's Elementary Theory of Analytic Functions.
answered Mar 29 at 15:36
avsavs
3,894515
answered Mar 29 at 15:36
avsavs
3,894515
answered Mar 29 at 15:36
avsavs
3,894515
answered Mar 29 at 15:36
avsavs
3,894515
3,894515
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3167261%2fare-the-components-of-a-cauchy-riemann-mapping-harmonic%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
