Integral of the difference between a function and its floorElementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni ConstantContinued fraction of $gamma+1$ using recursionWhy is $-gamma = int_0^1 frace^-z-1zdz+int_1^infty frace^-zzdz$Infinite Series $sumlimits_n=1^inftyfrac(-1)^nnleftlfloorfraclog(n)log(2)rightrfloor$An integral representation for $psi$A fractional part integral giving $fracF_n-1F_n-frac(-1)^nF_n^2lnleft(!fracF_n+2-F_ngammaF_n+1-F_ngammaright)$Series for Stieltjes constants from $gamma= sum_n=1^infty left(frac2n-sum_j=n(n-1)+1^n(n+1) frac1jright)$Proving $lim_ntoinftysum_k=n^inftyBig(frac1nleftlfloorfrac knrightrfloor-frac1kBig)=gamma$On the series $sum_n=1^infty fracleft lfloor log_2 n right rfloor2n(2n+1)(2n+2)$Fractional part of the floor function Integral
How strong is the axiom of well-ordered choice?
What is the purpose of a disclaimer like "this is not legal advice"?
Vector-transposing function
Why is there an extra space when I type "ls" on the Desktop?
What is the best index strategy or query SELECT when performing a search/lookup BETWEEN IP address (IPv4 and IPv6) ranges?
Why is my explanation wrong?
Why restrict private health insurance?
How would an energy-based "projectile" blow up a spaceship?
How does learning spells work when leveling a multiclass character?
Averaging over columns while ignoring zero entries
Was this cameo in Captain Marvel computer generated?
Is there a math expression equivalent to the conditional ternary operator?
How do you make a gun that shoots melee weapons and/or swords?
Should I file my taxes? No income, unemployed, but paid 2k in student loan interest
Why does a car's steering wheel get lighter with increasing speed
Will the concrete slab in a partially heated shed conduct a lot of heat to the unconditioned area?
Who has more? Ireland or Iceland?
Create chunks from an array
How can I paste into a layer with a layer mask?
What does *dead* mean in *What do you mean, dead?*?
Can I negotiate a patent idea for a raise, under French law?
What does it take to become a wilderness skills guide as a business?
How to write a chaotic neutral protagonist and prevent my readers from thinking they are evil?
Why does this boat have a landing pad? (SpaceX's GO Searcher) Any plans for propulsive capsule landings?
Integral of the difference between a function and its floor
Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni ConstantContinued fraction of $gamma+1$ using recursionWhy is $-gamma = int_0^1 frace^-z-1zdz+int_1^infty frace^-zzdz$Infinite Series $sumlimits_n=1^inftyfrac(-1)^nnleftlfloorfraclog(n)log(2)rightrfloor$An integral representation for $psi$A fractional part integral giving $fracF_n-1F_n-frac(-1)^nF_n^2lnleft(!fracF_n+2-F_ngammaF_n+1-F_ngammaright)$Series for Stieltjes constants from $gamma= sum_n=1^infty left(frac2n-sum_j=n(n-1)+1^n(n+1) frac1jright)$Proving $lim_ntoinftysum_k=n^inftyBig(frac1nleftlfloorfrac knrightrfloor-frac1kBig)=gamma$On the series $sum_n=1^infty fracleft lfloor log_2 n right rfloor2n(2n+1)(2n+2)$Fractional part of the floor function Integral
$begingroup$
I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$
Thanks
eulers-constant
$endgroup$
add a comment |
$begingroup$
I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$
Thanks
eulers-constant
$endgroup$
add a comment |
$begingroup$
I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$
Thanks
eulers-constant
$endgroup$
I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$
Thanks
eulers-constant
eulers-constant
asked 15 hours ago
RTnRTn
186
186
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%2f3139976%2fintegral-of-the-difference-between-a-function-and-its-floor%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%2f3139976%2fintegral-of-the-difference-between-a-function-and-its-floor%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