Harmonic holomorphic function in Ω
$begingroup$
$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$
$$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$
Prove that there exists a holomorphic function $u+iv$ in $Ω$
I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.
Photo/solution
holomorphic-functions
asked Dec 4 '18 at 10:29
malilinimalilini
44
$endgroup$
add a comment |
$begingroup$
$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$
$$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$
Prove that there exists a holomorphic function $u+iv$ in $Ω$
I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.
Photo/solution
holomorphic-functions
asked Dec 4 '18 at 10:29
malilinimalilini
44
$endgroup$
add a comment |
$begingroup$
$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$
$$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$
Prove that there exists a holomorphic function $u+iv$ in $Ω$
I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.
Photo/solution
holomorphic-functions
asked Dec 4 '18 at 10:29
malilinimalilini
44
$endgroup$
$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$
$$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$
Prove that there exists a holomorphic function $u+iv$ in $Ω$
I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.
Photo/solution
holomorphic-functions
holomorphic-functions
asked Dec 4 '18 at 10:29
malilinimalilini
44
asked Dec 4 '18 at 10:29
malilinimalilini
44
asked Dec 4 '18 at 10:29
malilinimalilini
44
asked Dec 4 '18 at 10:29
malilinimalilini
44
asked Dec 4 '18 at 10:29
malilinimalilini
44
44
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $g(t):= tu_y(tx,ty)$, then
$ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
$endgroup$
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
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%2f3025401%2fharmonic-holomorphic-function-in-%25ce%25a9%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $g(t):= tu_y(tx,ty)$, then
$ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
$endgroup$
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
add a comment |
$begingroup$
Let $g(t):= tu_y(tx,ty)$, then
$ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
$endgroup$
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
add a comment |
$begingroup$
Let $g(t):= tu_y(tx,ty)$, then
$ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
$endgroup$
Let $g(t):= tu_y(tx,ty)$, then
$ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
answered Dec 4 '18 at 10:40
FredFred
44.7k1846
44.7k1846
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
add a comment |
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
$endgroup$
– malilini
Dec 4 '18 at 10:53
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
$begingroup$
I didnt think about tagging you @Fred
$endgroup$
– malilini
Dec 4 '18 at 19:45
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%2f3025401%2fharmonic-holomorphic-function-in-%25ce%25a9%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
