Functions whose input is the same as the output?
$begingroup$
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
So the input variable is $tau$ and the output is also $tau$. Presumably these are identity functions $f(x)=x$?
Q: What are other not-so-trivial examples of identity functions?
P.S. There is a $f_1(tau)$ using $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
$endgroup$
add a comment |
$begingroup$
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
So the input variable is $tau$ and the output is also $tau$. Presumably these are identity functions $f(x)=x$?
Q: What are other not-so-trivial examples of identity functions?
P.S. There is a $f_1(tau)$ using $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
$endgroup$
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
add a comment |
$begingroup$
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
So the input variable is $tau$ and the output is also $tau$. Presumably these are identity functions $f(x)=x$?
Q: What are other not-so-trivial examples of identity functions?
P.S. There is a $f_1(tau)$ using $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
$endgroup$
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
So the input variable is $tau$ and the output is also $tau$. Presumably these are identity functions $f(x)=x$?
Q: What are other not-so-trivial examples of identity functions?
P.S. There is a $f_1(tau)$ using $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
complex-analysis functions terminology special-functions hypergeometric-function
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
edited Jan 4 at 12:05
Tito Piezas III
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
27.8k369178
27.8k369178
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
add a comment |
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
add a comment |
0
active
oldest
votes
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%2f3060410%2ffunctions-whose-input-is-the-same-as-the-output%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%2f3060410%2ffunctions-whose-input-is-the-same-as-the-output%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
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10