Using The Argument Principle to Find How Many Zeros of a Function Are in a Region
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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
user2965071user2965071
1356
$endgroup$
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
user2965071user2965071
1356
$endgroup$
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
user2965071user2965071
1356
$endgroup$
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
user2965071user2965071
1356
asked Dec 14 '18 at 3:53
user2965071user2965071
1356
asked Dec 14 '18 at 3:53
user2965071user2965071
1356
asked Dec 14 '18 at 3:53
user2965071user2965071
1356
asked Dec 14 '18 at 3:53
user2965071user2965071
1356
1356
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 |
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$begingroup$
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