About $ f(w,L) = int_1^w int_0^{2 L pi} frac{ ln(frac{sin(x) + sin(vx)}{2} + frac{5}{4})}{L(w - 1)} dx dv $
$begingroup$
Consider
$$ f(w,L) = int_1^w int_0^{2 L pi} frac{ ln(frac{sin(x) + sin(vx)}{2} + frac{5}{4})}{L(w - 1)} dx dv $$
For real $w > 1 $ and integer $ L > 1$
Conjecture :
$$ lim_{L to infty} f(w + 1, L) - f(w,L) = 0. $$
How to decide if this is true ?
Perhaps differentiation under the integral sign is the best method ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mickmick
5,09022064
$endgroup$
add a comment |
$begingroup$
Consider
$$ f(w,L) = int_1^w int_0^{2 L pi} frac{ ln(frac{sin(x) + sin(vx)}{2} + frac{5}{4})}{L(w - 1)} dx dv $$
For real $w > 1 $ and integer $ L > 1$
Conjecture :
$$ lim_{L to infty} f(w + 1, L) - f(w,L) = 0. $$
How to decide if this is true ?
Perhaps differentiation under the integral sign is the best method ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mickmick
5,09022064
$endgroup$
$begingroup$
Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ...
$endgroup$
– mick
Nov 25 '18 at 22:48
$begingroup$
I think the conjecture is false actually but hard to show false ?
$endgroup$
– mick
Nov 27 '18 at 20:21
add a comment |
$begingroup$
Consider
$$ f(w,L) = int_1^w int_0^{2 L pi} frac{ ln(frac{sin(x) + sin(vx)}{2} + frac{5}{4})}{L(w - 1)} dx dv $$
For real $w > 1 $ and integer $ L > 1$
Conjecture :
$$ lim_{L to infty} f(w + 1, L) - f(w,L) = 0. $$
How to decide if this is true ?
Perhaps differentiation under the integral sign is the best method ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mickmick
5,09022064
$endgroup$
Consider
$$ f(w,L) = int_1^w int_0^{2 L pi} frac{ ln(frac{sin(x) + sin(vx)}{2} + frac{5}{4})}{L(w - 1)} dx dv $$
For real $w > 1 $ and integer $ L > 1$
Conjecture :
$$ lim_{L to infty} f(w + 1, L) - f(w,L) = 0. $$
How to decide if this is true ?
Perhaps differentiation under the integral sign is the best method ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mickmick
5,09022064
asked Nov 25 '18 at 22:27
mickmick
5,09022064
edited Nov 25 '18 at 22:33
mick
asked Nov 25 '18 at 22:27
mickmick
5,09022064
asked Nov 25 '18 at 22:27
mickmick
5,09022064
asked Nov 25 '18 at 22:27
mickmick
5,09022064
5,09022064
$begingroup$
Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ...
$endgroup$
– mick
Nov 25 '18 at 22:48
$begingroup$
I think the conjecture is false actually but hard to show false ?
$endgroup$
– mick
Nov 27 '18 at 20:21
add a comment |
$begingroup$
Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ...
$endgroup$
– mick
Nov 25 '18 at 22:48
$begingroup$
I think the conjecture is false actually but hard to show false ?
$endgroup$
– mick
Nov 27 '18 at 20:21
$begingroup$
Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ...
$endgroup$
– mick
Nov 25 '18 at 22:48
$begingroup$
Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ...
$endgroup$
– mick
Nov 25 '18 at 22:48
$begingroup$
I think the conjecture is false actually but hard to show false ?
$endgroup$
– mick
Nov 27 '18 at 20:21
$begingroup$
I think the conjecture is false actually but hard to show false ?
$endgroup$
– mick
Nov 27 '18 at 20:21
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%2f3013502%2fabout-fw-l-int-1w-int-02-l-pi-frac-ln-frac-sinx-sinvx%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%2f3013502%2fabout-fw-l-int-1w-int-02-l-pi-frac-ln-frac-sinx-sinvx%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
$begingroup$
Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ...
$endgroup$
– mick
Nov 25 '18 at 22:48
$begingroup$
I think the conjecture is false actually but hard to show false ?
$endgroup$
– mick
Nov 27 '18 at 20:21