Using The Argument Principle to Find How Many Zeros of a Function Are in a Region
$begingroup$
I'm a bit confused on the intuition on counting the number of zeros for a function in a given region using the Argument Principle.
For example:
Find the number of zeros of $f(z) = z^3 - 2z^2 + 4$ in the first quadrant.
Since the function is analytic, there doesn't exist any poles, so we can say
$frac{1}{2pi i}int_{gamma} frac{9z^8 + 10z}{z^9 + 5z^2 + 3} dz = N$
Where $gamma$ is the closed region encompassing the first quadrant, and $N$ is the number of zeros for $f(z)$
However, upon looking at the answers in my textbook I notice this integral is not explicitly used. In fact, the original function is parameterized for each part of the region and some information is derived from there.
For example:
On the Real line, we can substitute $z$ for $x$, and we see $f(x) geq 2$
But I'm not sure how this relates to the theorem at all, or what kind of analysis is to be done. Similarly, how could we figure out the number of zeros by just parameterizing the original function with each segment of the closed path?
Any help regarding intuition would be helpful.
Thanks.
complex-analysis
asked Dec 14 '18 at 3:53
jd94jd94
31017
$endgroup$
add a comment |
$begingroup$
I'm a bit confused on the intuition on counting the number of zeros for a function in a given region using the Argument Principle.
For example:
Find the number of zeros of $f(z) = z^3 - 2z^2 + 4$ in the first quadrant.
Since the function is analytic, there doesn't exist any poles, so we can say
$frac{1}{2pi i}int_{gamma} frac{9z^8 + 10z}{z^9 + 5z^2 + 3} dz = N$
Where $gamma$ is the closed region encompassing the first quadrant, and $N$ is the number of zeros for $f(z)$
However, upon looking at the answers in my textbook I notice this integral is not explicitly used. In fact, the original function is parameterized for each part of the region and some information is derived from there.
For example:
On the Real line, we can substitute $z$ for $x$, and we see $f(x) geq 2$
But I'm not sure how this relates to the theorem at all, or what kind of analysis is to be done. Similarly, how could we figure out the number of zeros by just parameterizing the original function with each segment of the closed path?
Any help regarding intuition would be helpful.
Thanks.
complex-analysis
asked Dec 14 '18 at 3:53
jd94jd94
31017
$endgroup$
1
$begingroup$
By the argument principle you should check how many times the image of the curve under f(z) winds around the origin. Splitting the curve into segments and analyzing its image under f(z) seperately makes the whole thing easier.
$endgroup$
– zokomoko
Dec 14 '18 at 15:40
add a comment |
$begingroup$
I'm a bit confused on the intuition on counting the number of zeros for a function in a given region using the Argument Principle.
For example:
Find the number of zeros of $f(z) = z^3 - 2z^2 + 4$ in the first quadrant.
Since the function is analytic, there doesn't exist any poles, so we can say
$frac{1}{2pi i}int_{gamma} frac{9z^8 + 10z}{z^9 + 5z^2 + 3} dz = N$
Where $gamma$ is the closed region encompassing the first quadrant, and $N$ is the number of zeros for $f(z)$
However, upon looking at the answers in my textbook I notice this integral is not explicitly used. In fact, the original function is parameterized for each part of the region and some information is derived from there.
For example:
On the Real line, we can substitute $z$ for $x$, and we see $f(x) geq 2$
But I'm not sure how this relates to the theorem at all, or what kind of analysis is to be done. Similarly, how could we figure out the number of zeros by just parameterizing the original function with each segment of the closed path?
Any help regarding intuition would be helpful.
Thanks.
complex-analysis
asked Dec 14 '18 at 3:53
jd94jd94
31017
$endgroup$
I'm a bit confused on the intuition on counting the number of zeros for a function in a given region using the Argument Principle.
For example:
Find the number of zeros of $f(z) = z^3 - 2z^2 + 4$ in the first quadrant.
Since the function is analytic, there doesn't exist any poles, so we can say
$frac{1}{2pi i}int_{gamma} frac{9z^8 + 10z}{z^9 + 5z^2 + 3} dz = N$
Where $gamma$ is the closed region encompassing the first quadrant, and $N$ is the number of zeros for $f(z)$
However, upon looking at the answers in my textbook I notice this integral is not explicitly used. In fact, the original function is parameterized for each part of the region and some information is derived from there.
For example:
On the Real line, we can substitute $z$ for $x$, and we see $f(x) geq 2$
But I'm not sure how this relates to the theorem at all, or what kind of analysis is to be done. Similarly, how could we figure out the number of zeros by just parameterizing the original function with each segment of the closed path?
Any help regarding intuition would be helpful.
Thanks.
complex-analysis
complex-analysis
asked Dec 14 '18 at 3:53
jd94jd94
31017
asked Dec 14 '18 at 3:53
jd94jd94
31017
asked Dec 14 '18 at 3:53
jd94jd94
31017
asked Dec 14 '18 at 3:53
jd94jd94
31017
asked Dec 14 '18 at 3:53
jd94jd94
31017
31017
1
$begingroup$
By the argument principle you should check how many times the image of the curve under f(z) winds around the origin. Splitting the curve into segments and analyzing its image under f(z) seperately makes the whole thing easier.
$endgroup$
– zokomoko
Dec 14 '18 at 15:40
add a comment |
1
$begingroup$
By the argument principle you should check how many times the image of the curve under f(z) winds around the origin. Splitting the curve into segments and analyzing its image under f(z) seperately makes the whole thing easier.
$endgroup$
– zokomoko
Dec 14 '18 at 15:40
1
1
$begingroup$
By the argument principle you should check how many times the image of the curve under f(z) winds around the origin. Splitting the curve into segments and analyzing its image under f(z) seperately makes the whole thing easier.
$endgroup$
– zokomoko
Dec 14 '18 at 15:40
$begingroup$
By the argument principle you should check how many times the image of the curve under f(z) winds around the origin. Splitting the curve into segments and analyzing its image under f(z) seperately makes the whole thing easier.
$endgroup$
– zokomoko
Dec 14 '18 at 15:40
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%2f3038921%2fusing-the-argument-principle-to-find-how-many-zeros-of-a-function-are-in-a-regio%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%2f3038921%2fusing-the-argument-principle-to-find-how-many-zeros-of-a-function-are-in-a-regio%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
1
$begingroup$
By the argument principle you should check how many times the image of the curve under f(z) winds around the origin. Splitting the curve into segments and analyzing its image under f(z) seperately makes the whole thing easier.
$endgroup$
– zokomoko
Dec 14 '18 at 15:40