Find the residue of the function $f(z) = frac{z^3}{(z-1)(z^4+2)}$ at $z=0$
1
Problem: Calculate the residue of the function $f(z) = frac{z^3}{(z-1)(z^4+2)}$ at $z=0$
I am confused on where to even start on this question since there is no pole at $z=0$. Is $z=0$ even a singularity of this function?
complex-analysis residue-calculus
asked Jul 24 at 22:01
Fiticous
17218
add a comment |
1
Problem: Calculate the residue of the function $f(z) = frac{z^3}{(z-1)(z^4+2)}$ at $z=0$
I am confused on where to even start on this question since there is no pole at $z=0$. Is $z=0$ even a singularity of this function?
complex-analysis residue-calculus
asked Jul 24 at 22:01
Fiticous
17218
Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole?
– Count Iblis
Jul 24 at 22:07
1
@CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = sum_{n=-infty}^{infty} a_n(z - z_0)^n$ is always going to be zero?
– Fiticous
Jul 24 at 22:17
That's right! :)
– Count Iblis
Jul 24 at 23:12
add a comment |
1
1
1
Problem: Calculate the residue of the function $f(z) = frac{z^3}{(z-1)(z^4+2)}$ at $z=0$
I am confused on where to even start on this question since there is no pole at $z=0$. Is $z=0$ even a singularity of this function?
complex-analysis residue-calculus
asked Jul 24 at 22:01
Fiticous
17218
Problem: Calculate the residue of the function $f(z) = frac{z^3}{(z-1)(z^4+2)}$ at $z=0$
I am confused on where to even start on this question since there is no pole at $z=0$. Is $z=0$ even a singularity of this function?
complex-analysis residue-calculus
complex-analysis residue-calculus
asked Jul 24 at 22:01
Fiticous
17218
asked Jul 24 at 22:01
Fiticous
17218
edited Nov 30 at 20:19
asked Jul 24 at 22:01
Fiticous
17218
asked Jul 24 at 22:01
Fiticous
17218
asked Jul 24 at 22:01
Fiticous
17218
17218
Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole?
– Count Iblis
Jul 24 at 22:07
1
@CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = sum_{n=-infty}^{infty} a_n(z - z_0)^n$ is always going to be zero?
– Fiticous
Jul 24 at 22:17
That's right! :)
– Count Iblis
Jul 24 at 23:12
add a comment |
Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole?
– Count Iblis
Jul 24 at 22:07
1
@CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = sum_{n=-infty}^{infty} a_n(z - z_0)^n$ is always going to be zero?
– Fiticous
Jul 24 at 22:17
That's right! :)
– Count Iblis
Jul 24 at 23:12
Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole?
– Count Iblis
Jul 24 at 22:07
Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole?
– Count Iblis
Jul 24 at 22:07
1
1
@CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = sum_{n=-infty}^{infty} a_n(z - z_0)^n$ is always going to be zero?
– Fiticous
Jul 24 at 22:17
@CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = sum_{n=-infty}^{infty} a_n(z - z_0)^n$ is always going to be zero?
– Fiticous
Jul 24 at 22:17
That's right! :)
– Count Iblis
Jul 24 at 23:12
That's right! :)
– Count Iblis
Jul 24 at 23:12
add a comment |
1 Answer
1
active
oldest
votes
1
No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
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
});
}
});
draft saved
draft discarded
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%2f2861806%2ffind-the-residue-of-the-function-fz-fracz3z-1z42-at-z-0%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
1
No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
add a comment |
1
No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
add a comment |
1
1
1
No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
answered Jul 24 at 22:04
José Carlos Santos
149k22119221
149k22119221
add a comment |
add a comment |
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f2861806%2ffind-the-residue-of-the-function-fz-fracz3z-1z42-at-z-0%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
Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole?
– Count Iblis
Jul 24 at 22:07
1
@CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = sum_{n=-infty}^{infty} a_n(z - z_0)^n$ is always going to be zero?
– Fiticous
Jul 24 at 22:17
That's right! :)
– Count Iblis
Jul 24 at 23:12