Real Part of the Dilogarithm
$begingroup$
It is well known that
$$frac{x-pi}{2}=-sum_{kgeq 1}frac{sin{kx}}{k}forall xin(0,tau),$$
which gives
$$frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}=sum_{kgeq 1}frac{cos(kx)}{k^2}.$$
Note that
$$textrm{Li}_2(e^{ix})=sum_{kgeq 1}frac{cos(kx)+isin(kx)}{k^2}$$
This means that
$$mathfrak{R}textrm{Li}_2(e^{ix})=frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}$$
unless I'm wrong on one of the above statements. Now, with the previous restrictions on $x$, we have that this formula is valid for all complex numbers lying on the complex unit circle as inputs. My question is: is there a way to define something along these lines (a finite degree polynomial, preferably) for complex inputs of the dilogarithm that don't necessarily lie on the complex unit circle? More specifically,
$$mathfrak{R}textrm{Li}_2(re^{ix})=?$$
complex-analysis fourier-series polylogarithm
asked Oct 9 '18 at 0:58
46andpi46andpi
788
$endgroup$
add a comment |
$begingroup$
It is well known that
$$frac{x-pi}{2}=-sum_{kgeq 1}frac{sin{kx}}{k}forall xin(0,tau),$$
which gives
$$frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}=sum_{kgeq 1}frac{cos(kx)}{k^2}.$$
Note that
$$textrm{Li}_2(e^{ix})=sum_{kgeq 1}frac{cos(kx)+isin(kx)}{k^2}$$
This means that
$$mathfrak{R}textrm{Li}_2(e^{ix})=frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}$$
unless I'm wrong on one of the above statements. Now, with the previous restrictions on $x$, we have that this formula is valid for all complex numbers lying on the complex unit circle as inputs. My question is: is there a way to define something along these lines (a finite degree polynomial, preferably) for complex inputs of the dilogarithm that don't necessarily lie on the complex unit circle? More specifically,
$$mathfrak{R}textrm{Li}_2(re^{ix})=?$$
complex-analysis fourier-series polylogarithm
asked Oct 9 '18 at 0:58
46andpi46andpi
788
$endgroup$
add a comment |
$begingroup$
It is well known that
$$frac{x-pi}{2}=-sum_{kgeq 1}frac{sin{kx}}{k}forall xin(0,tau),$$
which gives
$$frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}=sum_{kgeq 1}frac{cos(kx)}{k^2}.$$
Note that
$$textrm{Li}_2(e^{ix})=sum_{kgeq 1}frac{cos(kx)+isin(kx)}{k^2}$$
This means that
$$mathfrak{R}textrm{Li}_2(e^{ix})=frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}$$
unless I'm wrong on one of the above statements. Now, with the previous restrictions on $x$, we have that this formula is valid for all complex numbers lying on the complex unit circle as inputs. My question is: is there a way to define something along these lines (a finite degree polynomial, preferably) for complex inputs of the dilogarithm that don't necessarily lie on the complex unit circle? More specifically,
$$mathfrak{R}textrm{Li}_2(re^{ix})=?$$
complex-analysis fourier-series polylogarithm
asked Oct 9 '18 at 0:58
46andpi46andpi
788
$endgroup$
It is well known that
$$frac{x-pi}{2}=-sum_{kgeq 1}frac{sin{kx}}{k}forall xin(0,tau),$$
which gives
$$frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}=sum_{kgeq 1}frac{cos(kx)}{k^2}.$$
Note that
$$textrm{Li}_2(e^{ix})=sum_{kgeq 1}frac{cos(kx)+isin(kx)}{k^2}$$
This means that
$$mathfrak{R}textrm{Li}_2(e^{ix})=frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}$$
unless I'm wrong on one of the above statements. Now, with the previous restrictions on $x$, we have that this formula is valid for all complex numbers lying on the complex unit circle as inputs. My question is: is there a way to define something along these lines (a finite degree polynomial, preferably) for complex inputs of the dilogarithm that don't necessarily lie on the complex unit circle? More specifically,
$$mathfrak{R}textrm{Li}_2(re^{ix})=?$$
complex-analysis fourier-series polylogarithm
complex-analysis fourier-series polylogarithm
asked Oct 9 '18 at 0:58
46andpi46andpi
788
asked Oct 9 '18 at 0:58
46andpi46andpi
788
asked Oct 9 '18 at 0:58
46andpi46andpi
788
asked Oct 9 '18 at 0:58
46andpi46andpi
788
asked Oct 9 '18 at 0:58
46andpi46andpi
788
788
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%2f2947873%2freal-part-of-the-dilogarithm%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%2f2947873%2freal-part-of-the-dilogarithm%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
