Laurent expansion of the given function
$begingroup$
I have the following question :
Find a Laurent expansion of $f(z) = dfrac{z}{(z^2 + 1)}$, valid for $|z-3| > 2$.
I have learnt finding the Laurent expansion of functions where the denominator is of the form $(z+a)^n$, but this form in the given question seems to be unfamiliar to me. I guess I need to transform it to the $(z+a)^n$ form, but how do I utilise the condition given (that it is valid for $|z-3| > 2$) ?
Thanks in advance.
laurent-series
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
asked Dec 5 '18 at 6:31
JasmineJasmine
283
$endgroup$
add a comment |
$begingroup$
I have the following question :
Find a Laurent expansion of $f(z) = dfrac{z}{(z^2 + 1)}$, valid for $|z-3| > 2$.
I have learnt finding the Laurent expansion of functions where the denominator is of the form $(z+a)^n$, but this form in the given question seems to be unfamiliar to me. I guess I need to transform it to the $(z+a)^n$ form, but how do I utilise the condition given (that it is valid for $|z-3| > 2$) ?
Thanks in advance.
laurent-series
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
asked Dec 5 '18 at 6:31
JasmineJasmine
283
$endgroup$
$begingroup$
That question makes no sense. The poles of $f$ are $pm i$ and the distance from $3$ to $pm i$ is $sqrt{10}>2$. So, there is no single Laurent series that works in the region $lvert z-3rvert>2$.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 6:42
$begingroup$
May I then assume that the condition given in the question has been misprinted?
$endgroup$
– Jasmine
Dec 5 '18 at 7:11
$begingroup$
It sure looks like.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 7:25
add a comment |
$begingroup$
I have the following question :
Find a Laurent expansion of $f(z) = dfrac{z}{(z^2 + 1)}$, valid for $|z-3| > 2$.
I have learnt finding the Laurent expansion of functions where the denominator is of the form $(z+a)^n$, but this form in the given question seems to be unfamiliar to me. I guess I need to transform it to the $(z+a)^n$ form, but how do I utilise the condition given (that it is valid for $|z-3| > 2$) ?
Thanks in advance.
laurent-series
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
asked Dec 5 '18 at 6:31
JasmineJasmine
283
$endgroup$
I have the following question :
Find a Laurent expansion of $f(z) = dfrac{z}{(z^2 + 1)}$, valid for $|z-3| > 2$.
I have learnt finding the Laurent expansion of functions where the denominator is of the form $(z+a)^n$, but this form in the given question seems to be unfamiliar to me. I guess I need to transform it to the $(z+a)^n$ form, but how do I utilise the condition given (that it is valid for $|z-3| > 2$) ?
Thanks in advance.
laurent-series
laurent-series
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
asked Dec 5 '18 at 6:31
JasmineJasmine
283
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
asked Dec 5 '18 at 6:31
JasmineJasmine
283
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
edited Dec 5 '18 at 6:40
Yadati Kiran
1,751619
1,751619
asked Dec 5 '18 at 6:31
JasmineJasmine
283
asked Dec 5 '18 at 6:31
JasmineJasmine
283
asked Dec 5 '18 at 6:31
JasmineJasmine
283
283
$begingroup$
That question makes no sense. The poles of $f$ are $pm i$ and the distance from $3$ to $pm i$ is $sqrt{10}>2$. So, there is no single Laurent series that works in the region $lvert z-3rvert>2$.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 6:42
$begingroup$
May I then assume that the condition given in the question has been misprinted?
$endgroup$
– Jasmine
Dec 5 '18 at 7:11
$begingroup$
It sure looks like.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 7:25
add a comment |
$begingroup$
That question makes no sense. The poles of $f$ are $pm i$ and the distance from $3$ to $pm i$ is $sqrt{10}>2$. So, there is no single Laurent series that works in the region $lvert z-3rvert>2$.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 6:42
$begingroup$
May I then assume that the condition given in the question has been misprinted?
$endgroup$
– Jasmine
Dec 5 '18 at 7:11
$begingroup$
It sure looks like.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 7:25
$begingroup$
That question makes no sense. The poles of $f$ are $pm i$ and the distance from $3$ to $pm i$ is $sqrt{10}>2$. So, there is no single Laurent series that works in the region $lvert z-3rvert>2$.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 6:42
$begingroup$
That question makes no sense. The poles of $f$ are $pm i$ and the distance from $3$ to $pm i$ is $sqrt{10}>2$. So, there is no single Laurent series that works in the region $lvert z-3rvert>2$.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 6:42
$begingroup$
May I then assume that the condition given in the question has been misprinted?
$endgroup$
– Jasmine
Dec 5 '18 at 7:11
$begingroup$
May I then assume that the condition given in the question has been misprinted?
$endgroup$
– Jasmine
Dec 5 '18 at 7:11
$begingroup$
It sure looks like.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 7:25
$begingroup$
It sure looks like.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 7:25
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%2f3026723%2flaurent-expansion-of-the-given-function%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%2f3026723%2flaurent-expansion-of-the-given-function%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
$begingroup$
That question makes no sense. The poles of $f$ are $pm i$ and the distance from $3$ to $pm i$ is $sqrt{10}>2$. So, there is no single Laurent series that works in the region $lvert z-3rvert>2$.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 6:42
$begingroup$
May I then assume that the condition given in the question has been misprinted?
$endgroup$
– Jasmine
Dec 5 '18 at 7:11
$begingroup$
It sure looks like.
$endgroup$
– José Carlos Santos
Dec 5 '18 at 7:25