Polylogarithmic integrals
$begingroup$
I'm a physicist looking at the Fredholm inverse of some integral equation. In attempting to solve the equation I stumbled upon a type of integral of the form
begin{equation}
int frac{prod_{i=1}^N operatorname{Li}_{s_i}(x)}{x+alpha} mathrm dx.
end{equation}
At first glance the above may be integrated by elementary methods, using the relation
begin{equation}
xfrac{mathrm doperatorname{Li}_s(x)}{mathrm dx} = operatorname{Li}_{s-1}(x).
end{equation}
However, in the above it spawns expressions with products of $N+1$ (!!) polylogarithms under the integral sign, after many partial integrations.
I ask this question hoping someone could point me to some reading material that would allow me to solve the above integral.
integration polylogarithm
asked Jan 8 at 17:39
10100110101010011010
23929
$endgroup$
add a comment |
$begingroup$
I'm a physicist looking at the Fredholm inverse of some integral equation. In attempting to solve the equation I stumbled upon a type of integral of the form
begin{equation}
int frac{prod_{i=1}^N operatorname{Li}_{s_i}(x)}{x+alpha} mathrm dx.
end{equation}
At first glance the above may be integrated by elementary methods, using the relation
begin{equation}
xfrac{mathrm doperatorname{Li}_s(x)}{mathrm dx} = operatorname{Li}_{s-1}(x).
end{equation}
However, in the above it spawns expressions with products of $N+1$ (!!) polylogarithms under the integral sign, after many partial integrations.
I ask this question hoping someone could point me to some reading material that would allow me to solve the above integral.
integration polylogarithm
asked Jan 8 at 17:39
10100110101010011010
23929
$endgroup$
$begingroup$
Note that $$mathrm{Li}_{s}(z)=int_0^zmathrm{Li}_{s-1}(t)frac{mathrm dt}t$$ which may help you to turn the product into the form $$mathrm{Li}^N_{K}(z)$$
$endgroup$
– clathratus
Jan 8 at 22:20
add a comment |
$begingroup$
I'm a physicist looking at the Fredholm inverse of some integral equation. In attempting to solve the equation I stumbled upon a type of integral of the form
begin{equation}
int frac{prod_{i=1}^N operatorname{Li}_{s_i}(x)}{x+alpha} mathrm dx.
end{equation}
At first glance the above may be integrated by elementary methods, using the relation
begin{equation}
xfrac{mathrm doperatorname{Li}_s(x)}{mathrm dx} = operatorname{Li}_{s-1}(x).
end{equation}
However, in the above it spawns expressions with products of $N+1$ (!!) polylogarithms under the integral sign, after many partial integrations.
I ask this question hoping someone could point me to some reading material that would allow me to solve the above integral.
integration polylogarithm
asked Jan 8 at 17:39
10100110101010011010
23929
$endgroup$
I'm a physicist looking at the Fredholm inverse of some integral equation. In attempting to solve the equation I stumbled upon a type of integral of the form
begin{equation}
int frac{prod_{i=1}^N operatorname{Li}_{s_i}(x)}{x+alpha} mathrm dx.
end{equation}
At first glance the above may be integrated by elementary methods, using the relation
begin{equation}
xfrac{mathrm doperatorname{Li}_s(x)}{mathrm dx} = operatorname{Li}_{s-1}(x).
end{equation}
However, in the above it spawns expressions with products of $N+1$ (!!) polylogarithms under the integral sign, after many partial integrations.
I ask this question hoping someone could point me to some reading material that would allow me to solve the above integral.
integration polylogarithm
integration polylogarithm
asked Jan 8 at 17:39
10100110101010011010
23929
asked Jan 8 at 17:39
10100110101010011010
23929
asked Jan 8 at 17:39
10100110101010011010
23929
asked Jan 8 at 17:39
10100110101010011010
23929
asked Jan 8 at 17:39
10100110101010011010
23929
23929
$begingroup$
Note that $$mathrm{Li}_{s}(z)=int_0^zmathrm{Li}_{s-1}(t)frac{mathrm dt}t$$ which may help you to turn the product into the form $$mathrm{Li}^N_{K}(z)$$
$endgroup$
– clathratus
Jan 8 at 22:20
add a comment |
$begingroup$
Note that $$mathrm{Li}_{s}(z)=int_0^zmathrm{Li}_{s-1}(t)frac{mathrm dt}t$$ which may help you to turn the product into the form $$mathrm{Li}^N_{K}(z)$$
$endgroup$
– clathratus
Jan 8 at 22:20
$begingroup$
Note that $$mathrm{Li}_{s}(z)=int_0^zmathrm{Li}_{s-1}(t)frac{mathrm dt}t$$ which may help you to turn the product into the form $$mathrm{Li}^N_{K}(z)$$
$endgroup$
– clathratus
Jan 8 at 22:20
$begingroup$
Note that $$mathrm{Li}_{s}(z)=int_0^zmathrm{Li}_{s-1}(t)frac{mathrm dt}t$$ which may help you to turn the product into the form $$mathrm{Li}^N_{K}(z)$$
$endgroup$
– clathratus
Jan 8 at 22:20
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%2f3066489%2fpolylogarithmic-integrals%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%2f3066489%2fpolylogarithmic-integrals%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
$begingroup$
Note that $$mathrm{Li}_{s}(z)=int_0^zmathrm{Li}_{s-1}(t)frac{mathrm dt}t$$ which may help you to turn the product into the form $$mathrm{Li}^N_{K}(z)$$
$endgroup$
– clathratus
Jan 8 at 22:20