Proof that β-function ∈ C^∞
I need to find correct proof that β-function is smooth on its domain.
Is there some feature of such functions, I guess that we need to prove the continuity of all n-derivatives, or their partial derivatives, but how it will be eventually I hadn't got yet. I will very glad your help.
calculus integration special-functions smooth-functions
asked Nov 29 at 6:19
user2952487
224
add a comment |
I need to find correct proof that β-function is smooth on its domain.
Is there some feature of such functions, I guess that we need to prove the continuity of all n-derivatives, or their partial derivatives, but how it will be eventually I hadn't got yet. I will very glad your help.
calculus integration special-functions smooth-functions
asked Nov 29 at 6:19
user2952487
224
You need to write in latex, click on edit in the other post to see how. That the $beta$ function is $k$-times differentiable in each of its variable is not hard. To see it is analytic you can expand $t^{x-1}$ as a series in powers of $x+c$.
– reuns
Nov 29 at 6:58
Which function do you mean: Dirichlet's $beta$ or Euler's $B$?
– gammatester
Nov 29 at 8:22
Euler's Β function
– user2952487
Nov 29 at 11:39
add a comment |
I need to find correct proof that β-function is smooth on its domain.
Is there some feature of such functions, I guess that we need to prove the continuity of all n-derivatives, or their partial derivatives, but how it will be eventually I hadn't got yet. I will very glad your help.
calculus integration special-functions smooth-functions
asked Nov 29 at 6:19
user2952487
224
I need to find correct proof that β-function is smooth on its domain.
Is there some feature of such functions, I guess that we need to prove the continuity of all n-derivatives, or their partial derivatives, but how it will be eventually I hadn't got yet. I will very glad your help.
calculus integration special-functions smooth-functions
calculus integration special-functions smooth-functions
asked Nov 29 at 6:19
user2952487
224
asked Nov 29 at 6:19
user2952487
224
edited Nov 29 at 6:56
asked Nov 29 at 6:19
user2952487
224
asked Nov 29 at 6:19
user2952487
224
asked Nov 29 at 6:19
user2952487
224
224
You need to write in latex, click on edit in the other post to see how. That the $beta$ function is $k$-times differentiable in each of its variable is not hard. To see it is analytic you can expand $t^{x-1}$ as a series in powers of $x+c$.
– reuns
Nov 29 at 6:58
Which function do you mean: Dirichlet's $beta$ or Euler's $B$?
– gammatester
Nov 29 at 8:22
Euler's Β function
– user2952487
Nov 29 at 11:39
add a comment |
You need to write in latex, click on edit in the other post to see how. That the $beta$ function is $k$-times differentiable in each of its variable is not hard. To see it is analytic you can expand $t^{x-1}$ as a series in powers of $x+c$.
– reuns
Nov 29 at 6:58
Which function do you mean: Dirichlet's $beta$ or Euler's $B$?
– gammatester
Nov 29 at 8:22
Euler's Β function
– user2952487
Nov 29 at 11:39
You need to write in latex, click on edit in the other post to see how. That the $beta$ function is $k$-times differentiable in each of its variable is not hard. To see it is analytic you can expand $t^{x-1}$ as a series in powers of $x+c$.
– reuns
Nov 29 at 6:58
You need to write in latex, click on edit in the other post to see how. That the $beta$ function is $k$-times differentiable in each of its variable is not hard. To see it is analytic you can expand $t^{x-1}$ as a series in powers of $x+c$.
– reuns
Nov 29 at 6:58
Which function do you mean: Dirichlet's $beta$ or Euler's $B$?
– gammatester
Nov 29 at 8:22
Which function do you mean: Dirichlet's $beta$ or Euler's $B$?
– gammatester
Nov 29 at 8:22
Euler's Β function
– user2952487
Nov 29 at 11:39
Euler's Β function
– user2952487
Nov 29 at 11:39
add a comment |
1 Answer
1
active
oldest
votes
You have more than enough to differentiate through the integral sign (the Leibniz rule.) For example, thinking of $x,y>0,$ we have
$$frac{d}{dx}int_0^1t^{x-1}(1-t)^{y-1},dt = int_0^1(ln t)t^{x-1}(1-t)^{y-1},dt.$$
You can keep going, piling up factors like $(ln t)^m ln (1-t)^n$ in the integral. None of these factors will destroy integrability.
answered Nov 29 at 18:02
zhw.
71.5k43075
add a comment |
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%2f3018270%2fproof-that-%25ce%25b2-function-%25e2%2588%2588-c%25e2%2588%259e%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
You have more than enough to differentiate through the integral sign (the Leibniz rule.) For example, thinking of $x,y>0,$ we have
$$frac{d}{dx}int_0^1t^{x-1}(1-t)^{y-1},dt = int_0^1(ln t)t^{x-1}(1-t)^{y-1},dt.$$
You can keep going, piling up factors like $(ln t)^m ln (1-t)^n$ in the integral. None of these factors will destroy integrability.
answered Nov 29 at 18:02
zhw.
71.5k43075
add a comment |
You have more than enough to differentiate through the integral sign (the Leibniz rule.) For example, thinking of $x,y>0,$ we have
$$frac{d}{dx}int_0^1t^{x-1}(1-t)^{y-1},dt = int_0^1(ln t)t^{x-1}(1-t)^{y-1},dt.$$
You can keep going, piling up factors like $(ln t)^m ln (1-t)^n$ in the integral. None of these factors will destroy integrability.
answered Nov 29 at 18:02
zhw.
71.5k43075
add a comment |
You have more than enough to differentiate through the integral sign (the Leibniz rule.) For example, thinking of $x,y>0,$ we have
$$frac{d}{dx}int_0^1t^{x-1}(1-t)^{y-1},dt = int_0^1(ln t)t^{x-1}(1-t)^{y-1},dt.$$
You can keep going, piling up factors like $(ln t)^m ln (1-t)^n$ in the integral. None of these factors will destroy integrability.
answered Nov 29 at 18:02
zhw.
71.5k43075
You have more than enough to differentiate through the integral sign (the Leibniz rule.) For example, thinking of $x,y>0,$ we have
$$frac{d}{dx}int_0^1t^{x-1}(1-t)^{y-1},dt = int_0^1(ln t)t^{x-1}(1-t)^{y-1},dt.$$
You can keep going, piling up factors like $(ln t)^m ln (1-t)^n$ in the integral. None of these factors will destroy integrability.
answered Nov 29 at 18:02
zhw.
71.5k43075
answered Nov 29 at 18:02
zhw.
71.5k43075
answered Nov 29 at 18:02
zhw.
71.5k43075
answered Nov 29 at 18:02
zhw.
71.5k43075
71.5k43075
add a comment |
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3018270%2fproof-that-%25ce%25b2-function-%25e2%2588%2588-c%25e2%2588%259e%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
You need to write in latex, click on edit in the other post to see how. That the $beta$ function is $k$-times differentiable in each of its variable is not hard. To see it is analytic you can expand $t^{x-1}$ as a series in powers of $x+c$.
– reuns
Nov 29 at 6:58
Which function do you mean: Dirichlet's $beta$ or Euler's $B$?
– gammatester
Nov 29 at 8:22
Euler's Β function
– user2952487
Nov 29 at 11:39