Composite of Holomorphic Functions is Holomorphic
$begingroup$
Let $f(z)$ and $g(w)$ be holomorphic functions. I want to prove that the composite $g(f(z))$ is also holomorphic by using Cauchy-Riemann Equation directly.
Let $g(f(z))=s(u(x,y), v(x,y)),+,i t(u(x,y), v(x,y))$.
Then we have $left{ begin{array} frac{partial s}{partial x}=frac{partial s}{partial u}frac{partial u}{partial x}+frac{partial s}{partial v}frac{partial v}{partial x}\ frac{partial t}{partial y}=frac{partial t}{partial u}frac{partial u}{partial y}+frac{partial t}{partial v}frac{partial v}{partial y}end{array}right.$.
If we can show that $frac{partial s}{partial x}=frac{partial t}{partial y}$ and $frac{partial s}{partial y}=-frac{partial t}{partial x}$, we can conclude that the composite $g(f(z))$ is holomorphic. Thus I tried to compute $frac{partial s}{partial x}-frac{partial t}{partial y}$, but I found it difficult to yield the desired result $0$ using Cauchy-Riemann equations of $f$ and $g$. Can I get some advices or hints to proceed more? Thank you.
complex-analysis cauchy-riemann-equations
asked Jan 17 at 19:33
SH LeeSH Lee
235
$endgroup$
add a comment |
$begingroup$
Let $f(z)$ and $g(w)$ be holomorphic functions. I want to prove that the composite $g(f(z))$ is also holomorphic by using Cauchy-Riemann Equation directly.
Let $g(f(z))=s(u(x,y), v(x,y)),+,i t(u(x,y), v(x,y))$.
Then we have $left{ begin{array} frac{partial s}{partial x}=frac{partial s}{partial u}frac{partial u}{partial x}+frac{partial s}{partial v}frac{partial v}{partial x}\ frac{partial t}{partial y}=frac{partial t}{partial u}frac{partial u}{partial y}+frac{partial t}{partial v}frac{partial v}{partial y}end{array}right.$.
If we can show that $frac{partial s}{partial x}=frac{partial t}{partial y}$ and $frac{partial s}{partial y}=-frac{partial t}{partial x}$, we can conclude that the composite $g(f(z))$ is holomorphic. Thus I tried to compute $frac{partial s}{partial x}-frac{partial t}{partial y}$, but I found it difficult to yield the desired result $0$ using Cauchy-Riemann equations of $f$ and $g$. Can I get some advices or hints to proceed more? Thank you.
complex-analysis cauchy-riemann-equations
asked Jan 17 at 19:33
SH LeeSH Lee
235
$endgroup$
add a comment |
$begingroup$
Let $f(z)$ and $g(w)$ be holomorphic functions. I want to prove that the composite $g(f(z))$ is also holomorphic by using Cauchy-Riemann Equation directly.
Let $g(f(z))=s(u(x,y), v(x,y)),+,i t(u(x,y), v(x,y))$.
Then we have $left{ begin{array} frac{partial s}{partial x}=frac{partial s}{partial u}frac{partial u}{partial x}+frac{partial s}{partial v}frac{partial v}{partial x}\ frac{partial t}{partial y}=frac{partial t}{partial u}frac{partial u}{partial y}+frac{partial t}{partial v}frac{partial v}{partial y}end{array}right.$.
If we can show that $frac{partial s}{partial x}=frac{partial t}{partial y}$ and $frac{partial s}{partial y}=-frac{partial t}{partial x}$, we can conclude that the composite $g(f(z))$ is holomorphic. Thus I tried to compute $frac{partial s}{partial x}-frac{partial t}{partial y}$, but I found it difficult to yield the desired result $0$ using Cauchy-Riemann equations of $f$ and $g$. Can I get some advices or hints to proceed more? Thank you.
complex-analysis cauchy-riemann-equations
asked Jan 17 at 19:33
SH LeeSH Lee
235
$endgroup$
Let $f(z)$ and $g(w)$ be holomorphic functions. I want to prove that the composite $g(f(z))$ is also holomorphic by using Cauchy-Riemann Equation directly.
Let $g(f(z))=s(u(x,y), v(x,y)),+,i t(u(x,y), v(x,y))$.
Then we have $left{ begin{array} frac{partial s}{partial x}=frac{partial s}{partial u}frac{partial u}{partial x}+frac{partial s}{partial v}frac{partial v}{partial x}\ frac{partial t}{partial y}=frac{partial t}{partial u}frac{partial u}{partial y}+frac{partial t}{partial v}frac{partial v}{partial y}end{array}right.$.
If we can show that $frac{partial s}{partial x}=frac{partial t}{partial y}$ and $frac{partial s}{partial y}=-frac{partial t}{partial x}$, we can conclude that the composite $g(f(z))$ is holomorphic. Thus I tried to compute $frac{partial s}{partial x}-frac{partial t}{partial y}$, but I found it difficult to yield the desired result $0$ using Cauchy-Riemann equations of $f$ and $g$. Can I get some advices or hints to proceed more? Thank you.
complex-analysis cauchy-riemann-equations
complex-analysis cauchy-riemann-equations
asked Jan 17 at 19:33
SH LeeSH Lee
235
asked Jan 17 at 19:33
SH LeeSH Lee
235
asked Jan 17 at 19:33
SH LeeSH Lee
235
asked Jan 17 at 19:33
SH LeeSH Lee
235
asked Jan 17 at 19:33
SH LeeSH Lee
235
235
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since $f$ and $g$ are holomorphic, you know that $partial s/partial u=partial t/partial v$ and $partial u/partial x=partial v/partial y$. Therefore the first term in the expression of $partial s/partial x$ equals the second one in the expression for $partial t/partial y$. Using the other CR equations for $f$ and $g$ you can show that the other two terms are equal as well (the minus signs will cancel)
answered Jan 17 at 19:49
GReyesGReyes
2,52815
$endgroup$
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
add a comment |
Your Answer
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%2f3077415%2fcomposite-of-holomorphic-functions-is-holomorphic%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
$begingroup$
Since $f$ and $g$ are holomorphic, you know that $partial s/partial u=partial t/partial v$ and $partial u/partial x=partial v/partial y$. Therefore the first term in the expression of $partial s/partial x$ equals the second one in the expression for $partial t/partial y$. Using the other CR equations for $f$ and $g$ you can show that the other two terms are equal as well (the minus signs will cancel)
answered Jan 17 at 19:49
GReyesGReyes
2,52815
$endgroup$
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
add a comment |
$begingroup$
Since $f$ and $g$ are holomorphic, you know that $partial s/partial u=partial t/partial v$ and $partial u/partial x=partial v/partial y$. Therefore the first term in the expression of $partial s/partial x$ equals the second one in the expression for $partial t/partial y$. Using the other CR equations for $f$ and $g$ you can show that the other two terms are equal as well (the minus signs will cancel)
answered Jan 17 at 19:49
GReyesGReyes
2,52815
$endgroup$
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
add a comment |
$begingroup$
Since $f$ and $g$ are holomorphic, you know that $partial s/partial u=partial t/partial v$ and $partial u/partial x=partial v/partial y$. Therefore the first term in the expression of $partial s/partial x$ equals the second one in the expression for $partial t/partial y$. Using the other CR equations for $f$ and $g$ you can show that the other two terms are equal as well (the minus signs will cancel)
answered Jan 17 at 19:49
GReyesGReyes
2,52815
$endgroup$
Since $f$ and $g$ are holomorphic, you know that $partial s/partial u=partial t/partial v$ and $partial u/partial x=partial v/partial y$. Therefore the first term in the expression of $partial s/partial x$ equals the second one in the expression for $partial t/partial y$. Using the other CR equations for $f$ and $g$ you can show that the other two terms are equal as well (the minus signs will cancel)
answered Jan 17 at 19:49
GReyesGReyes
2,52815
answered Jan 17 at 19:49
GReyesGReyes
2,52815
answered Jan 17 at 19:49
GReyesGReyes
2,52815
answered Jan 17 at 19:49
GReyesGReyes
2,52815
2,52815
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
add a comment |
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
$begingroup$
Oh, I got it. Thanks a lot!
$endgroup$
– SH Lee
Jan 17 at 19:54
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%2f3077415%2fcomposite-of-holomorphic-functions-is-holomorphic%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
