Mixing
Solve for $m=0,1,2,…$ and $ninBbb{Z}$ the integral $int_{Gamma_1} z^n(1-z)^mdz$
$begingroup$
I need to solve the integral for $m=0,1,2, ...$, and $ninBbb{Z}$
$$int_{Gamma_1} z^n(1-z)^m,dz$$
where $Gamma_1$ is the circle centered in $0$ with radius $1$.
I'm struggling trying to see this integral in a way I can use Cauchy's Integral Formula or Cauchy's Integral Formula for derivatives, but so far I'm stuck trying to rewrite it. So could anyone give me any hint to point me in the right direction to solve this?
Thanks in advance.
complex-analysis complex-integration cauchy-integral-formula
edited Jan 8 at 13:42
amWhy
1
$endgroup$
add a comment |
$begingroup$
I need to solve the integral for $m=0,1,2, ...$, and $ninBbb{Z}$
$$int_{Gamma_1} z^n(1-z)^m,dz$$
where $Gamma_1$ is the circle centered in $0$ with radius $1$.
I'm struggling trying to see this integral in a way I can use Cauchy's Integral Formula or Cauchy's Integral Formula for derivatives, but so far I'm stuck trying to rewrite it. So could anyone give me any hint to point me in the right direction to solve this?
Thanks in advance.
complex-analysis complex-integration cauchy-integral-formula
edited Jan 8 at 13:42
amWhy
1
$endgroup$
$begingroup$
Do you know that $int_Gamma f(z) dz =0$ if $f$ has no pole inside $Gamma$?
$endgroup$
– user99914
Nov 14 '15 at 12:02
$begingroup$
@JohnMa Yes, but if I take $m=0$ and $n=-1$ then $int_{Gamma_1} z^{-1}dz$ is $2ipi$ which is not zero, that's where I'm confused.
$endgroup$
– user281593
Nov 14 '15 at 12:16
$begingroup$
Oh I missed that I thought $n.m$ are always in $mathbb N$. So can you use the residue?
$endgroup$
– user99914
Nov 14 '15 at 12:23
$begingroup$
@JohnMa I believe I can't, since in class we've not seen it yet...
$endgroup$
– user281593
Nov 14 '15 at 12:36
add a comment |
$begingroup$
I need to solve the integral for $m=0,1,2, ...$, and $ninBbb{Z}$
$$int_{Gamma_1} z^n(1-z)^m,dz$$
where $Gamma_1$ is the circle centered in $0$ with radius $1$.
I'm struggling trying to see this integral in a way I can use Cauchy's Integral Formula or Cauchy's Integral Formula for derivatives, but so far I'm stuck trying to rewrite it. So could anyone give me any hint to point me in the right direction to solve this?
Thanks in advance.
complex-analysis complex-integration cauchy-integral-formula
edited Jan 8 at 13:42
amWhy
1
$endgroup$
I need to solve the integral for $m=0,1,2, ...$, and $ninBbb{Z}$
$$int_{Gamma_1} z^n(1-z)^m,dz$$
where $Gamma_1$ is the circle centered in $0$ with radius $1$.
I'm struggling trying to see this integral in a way I can use Cauchy's Integral Formula or Cauchy's Integral Formula for derivatives, but so far I'm stuck trying to rewrite it. So could anyone give me any hint to point me in the right direction to solve this?
Thanks in advance.
complex-analysis complex-integration cauchy-integral-formula
complex-analysis complex-integration cauchy-integral-formula
edited Jan 8 at 13:42
amWhy
1
edited Jan 8 at 13:42
amWhy
1
edited Jan 8 at 13:42
amWhy
1
edited Jan 8 at 13:42
amWhy
1
edited Jan 8 at 13:42
amWhy
1
1
asked Nov 14 '15 at 11:59
user281593
$begingroup$
Do you know that $int_Gamma f(z) dz =0$ if $f$ has no pole inside $Gamma$?
$endgroup$
– user99914
Nov 14 '15 at 12:02
$begingroup$
@JohnMa Yes, but if I take $m=0$ and $n=-1$ then $int_{Gamma_1} z^{-1}dz$ is $2ipi$ which is not zero, that's where I'm confused.
$endgroup$
– user281593
Nov 14 '15 at 12:16
$begingroup$
Oh I missed that I thought $n.m$ are always in $mathbb N$. So can you use the residue?
$endgroup$
– user99914
Nov 14 '15 at 12:23
$begingroup$
@JohnMa I believe I can't, since in class we've not seen it yet...
$endgroup$
– user281593
Nov 14 '15 at 12:36
add a comment |
$begingroup$
Do you know that $int_Gamma f(z) dz =0$ if $f$ has no pole inside $Gamma$?
$endgroup$
– user99914
Nov 14 '15 at 12:02
$begingroup$
@JohnMa Yes, but if I take $m=0$ and $n=-1$ then $int_{Gamma_1} z^{-1}dz$ is $2ipi$ which is not zero, that's where I'm confused.
$endgroup$
– user281593
Nov 14 '15 at 12:16
$begingroup$
Oh I missed that I thought $n.m$ are always in $mathbb N$. So can you use the residue?
$endgroup$
– user99914
Nov 14 '15 at 12:23
$begingroup$
@JohnMa I believe I can't, since in class we've not seen it yet...
$endgroup$
– user281593
Nov 14 '15 at 12:36
$begingroup$
Do you know that $int_Gamma f(z) dz =0$ if $f$ has no pole inside $Gamma$?
$endgroup$
– user99914
Nov 14 '15 at 12:02
$begingroup$
Do you know that $int_Gamma f(z) dz =0$ if $f$ has no pole inside $Gamma$?
$endgroup$
– user99914
Nov 14 '15 at 12:02
$begingroup$
@JohnMa Yes, but if I take $m=0$ and $n=-1$ then $int_{Gamma_1} z^{-1}dz$ is $2ipi$ which is not zero, that's where I'm confused.
$endgroup$
– user281593
Nov 14 '15 at 12:16
$begingroup$
@JohnMa Yes, but if I take $m=0$ and $n=-1$ then $int_{Gamma_1} z^{-1}dz$ is $2ipi$ which is not zero, that's where I'm confused.
$endgroup$
– user281593
Nov 14 '15 at 12:16
$begingroup$
Oh I missed that I thought $n.m$ are always in $mathbb N$. So can you use the residue?
$endgroup$
– user99914
Nov 14 '15 at 12:23
$begingroup$
Oh I missed that I thought $n.m$ are always in $mathbb N$. So can you use the residue?
$endgroup$
– user99914
Nov 14 '15 at 12:23
$begingroup$
@JohnMa I believe I can't, since in class we've not seen it yet...
$endgroup$
– user281593
Nov 14 '15 at 12:36
$begingroup$
@JohnMa I believe I can't, since in class we've not seen it yet...
$endgroup$
– user281593
Nov 14 '15 at 12:36
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Just use Cauchy's Integral formula: When $ngeq 0$, $z^n (1-z)^m$ is analytic and so the integral is zero. When $n<0$, set $a=0$ and $f(z) = (1-z)^m$, the Cauchy integral formula is
$$f^{(k-1)}(0) = frac{k!}{2pi i} int_Gamma frac{f(z)}{z^k} dz.$$
and so putting $k = -n$ gives
$$int_Gamma z^n(1-z)^m dz = frac{2pi i}{|n|!}frac{d^{|n|-1}}{dz^{|n|-1}} (1-z)^mbigg|_{z=0}.$$
$endgroup$
1
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
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%2f1528467%2fsolve-for-m-0-1-2-and-n-in-bbbz-the-integral-int-gamma-1-zn1-z%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
$begingroup$
Just use Cauchy's Integral formula: When $ngeq 0$, $z^n (1-z)^m$ is analytic and so the integral is zero. When $n<0$, set $a=0$ and $f(z) = (1-z)^m$, the Cauchy integral formula is
$$f^{(k-1)}(0) = frac{k!}{2pi i} int_Gamma frac{f(z)}{z^k} dz.$$
and so putting $k = -n$ gives
$$int_Gamma z^n(1-z)^m dz = frac{2pi i}{|n|!}frac{d^{|n|-1}}{dz^{|n|-1}} (1-z)^mbigg|_{z=0}.$$
$endgroup$
1
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
add a comment |
$begingroup$
Just use Cauchy's Integral formula: When $ngeq 0$, $z^n (1-z)^m$ is analytic and so the integral is zero. When $n<0$, set $a=0$ and $f(z) = (1-z)^m$, the Cauchy integral formula is
$$f^{(k-1)}(0) = frac{k!}{2pi i} int_Gamma frac{f(z)}{z^k} dz.$$
and so putting $k = -n$ gives
$$int_Gamma z^n(1-z)^m dz = frac{2pi i}{|n|!}frac{d^{|n|-1}}{dz^{|n|-1}} (1-z)^mbigg|_{z=0}.$$
$endgroup$
1
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
add a comment |
$begingroup$
Just use Cauchy's Integral formula: When $ngeq 0$, $z^n (1-z)^m$ is analytic and so the integral is zero. When $n<0$, set $a=0$ and $f(z) = (1-z)^m$, the Cauchy integral formula is
$$f^{(k-1)}(0) = frac{k!}{2pi i} int_Gamma frac{f(z)}{z^k} dz.$$
and so putting $k = -n$ gives
$$int_Gamma z^n(1-z)^m dz = frac{2pi i}{|n|!}frac{d^{|n|-1}}{dz^{|n|-1}} (1-z)^mbigg|_{z=0}.$$
$endgroup$
Just use Cauchy's Integral formula: When $ngeq 0$, $z^n (1-z)^m$ is analytic and so the integral is zero. When $n<0$, set $a=0$ and $f(z) = (1-z)^m$, the Cauchy integral formula is
$$f^{(k-1)}(0) = frac{k!}{2pi i} int_Gamma frac{f(z)}{z^k} dz.$$
and so putting $k = -n$ gives
$$int_Gamma z^n(1-z)^m dz = frac{2pi i}{|n|!}frac{d^{|n|-1}}{dz^{|n|-1}} (1-z)^mbigg|_{z=0}.$$
edited Nov 14 '15 at 17:34
answered Nov 14 '15 at 12:46
user99914
1
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
add a comment |
1
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
1
1
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
$begingroup$
Thanks! I didn't see it that way. But just one more thing, shouldn't it be $f^{(n-1)}(a)$ on the first equation?
$endgroup$
– user281593
Nov 14 '15 at 15:06
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.
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%2f1528467%2fsolve-for-m-0-1-2-and-n-in-bbbz-the-integral-int-gamma-1-zn1-z%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
$begingroup$
Do you know that $int_Gamma f(z) dz =0$ if $f$ has no pole inside $Gamma$?
$endgroup$
– user99914
Nov 14 '15 at 12:02
$begingroup$
@JohnMa Yes, but if I take $m=0$ and $n=-1$ then $int_{Gamma_1} z^{-1}dz$ is $2ipi$ which is not zero, that's where I'm confused.
$endgroup$
– user281593
Nov 14 '15 at 12:16
$begingroup$
Oh I missed that I thought $n.m$ are always in $mathbb N$. So can you use the residue?
$endgroup$
– user99914
Nov 14 '15 at 12:23
$begingroup$
@JohnMa I believe I can't, since in class we've not seen it yet...
$endgroup$
– user281593
Nov 14 '15 at 12:36