Mixing
How do I prove that any second-order equation with three regular singular points can be transformed into a...
$begingroup$
Consider the hypergeometric equation $$z(1-z)u''+(c-(a+b+1)z)u'-abu=0.$$ I read that it should be possible to show that any second-order equation with three regular singular points can be transformed into a hypergeometric equation by a combination of Möbius maps $tilde{z}=(az+b)/(cz+d)$ (changing the variable), and multiplication operations $tilde{u}=L(z)u(z)$ for fixed functions $L$. But I could not find any explanation about how/why anywhere on the internet. How does one prove that?
ordinary-differential-equations hypergeometric-function
asked Jan 27 at 2:51
user638526user638526
213
$endgroup$
add a comment |
$begingroup$
Consider the hypergeometric equation $$z(1-z)u''+(c-(a+b+1)z)u'-abu=0.$$ I read that it should be possible to show that any second-order equation with three regular singular points can be transformed into a hypergeometric equation by a combination of Möbius maps $tilde{z}=(az+b)/(cz+d)$ (changing the variable), and multiplication operations $tilde{u}=L(z)u(z)$ for fixed functions $L$. But I could not find any explanation about how/why anywhere on the internet. How does one prove that?
ordinary-differential-equations hypergeometric-function
asked Jan 27 at 2:51
user638526user638526
213
$endgroup$
add a comment |
$begingroup$
Consider the hypergeometric equation $$z(1-z)u''+(c-(a+b+1)z)u'-abu=0.$$ I read that it should be possible to show that any second-order equation with three regular singular points can be transformed into a hypergeometric equation by a combination of Möbius maps $tilde{z}=(az+b)/(cz+d)$ (changing the variable), and multiplication operations $tilde{u}=L(z)u(z)$ for fixed functions $L$. But I could not find any explanation about how/why anywhere on the internet. How does one prove that?
ordinary-differential-equations hypergeometric-function
asked Jan 27 at 2:51
user638526user638526
213
$endgroup$
Consider the hypergeometric equation $$z(1-z)u''+(c-(a+b+1)z)u'-abu=0.$$ I read that it should be possible to show that any second-order equation with three regular singular points can be transformed into a hypergeometric equation by a combination of Möbius maps $tilde{z}=(az+b)/(cz+d)$ (changing the variable), and multiplication operations $tilde{u}=L(z)u(z)$ for fixed functions $L$. But I could not find any explanation about how/why anywhere on the internet. How does one prove that?
ordinary-differential-equations hypergeometric-function
ordinary-differential-equations hypergeometric-function
asked Jan 27 at 2:51
user638526user638526
213
asked Jan 27 at 2:51
user638526user638526
213
asked Jan 27 at 2:51
user638526user638526
213
asked Jan 27 at 2:51
user638526user638526
213
asked Jan 27 at 2:51
user638526user638526
213
213
add a comment |
add a comment |
0
active
oldest
votes
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%2f3089081%2fhow-do-i-prove-that-any-second-order-equation-with-three-regular-singular-points%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3089081%2fhow-do-i-prove-that-any-second-order-equation-with-three-regular-singular-points%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
