Integral of the reciprocal of a complex polynomial [duplicate]
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This question already has an answer here:
Proof that $frac {1} {2pi i} oint frac {{rm d}z} {P(z)} $ over a closed curve is zero.
2 answers
For a polynomial P(z) of degree $ngeq2$, show that there exists some $R_{1}>0$ such that for $R>R_{1}$ it holds that:
$$int_{C_{R}}frac{1}{P(z)}dz=0$$ where $C_{R}$ is a circle of radius R with the centre $0$.
I am already aware of the fact that $$lim_{Rto infty}int_{C_{R}}frac{1}{P(z)}dz=0$$ and how to go about proving it but I'm not sure how to evaluate the first integral written above without using any limit. Would I maybe use a similar approach as was used to show the latter integral?
complex-analysis complex-integration
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
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marked as duplicate by Martin R, Lord Shark the Unknown, user10354138, Cesareo, Rebellos Dec 9 '18 at 10:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Proof that $frac {1} {2pi i} oint frac {{rm d}z} {P(z)} $ over a closed curve is zero.
2 answers
For a polynomial P(z) of degree $ngeq2$, show that there exists some $R_{1}>0$ such that for $R>R_{1}$ it holds that:
$$int_{C_{R}}frac{1}{P(z)}dz=0$$ where $C_{R}$ is a circle of radius R with the centre $0$.
I am already aware of the fact that $$lim_{Rto infty}int_{C_{R}}frac{1}{P(z)}dz=0$$ and how to go about proving it but I'm not sure how to evaluate the first integral written above without using any limit. Would I maybe use a similar approach as was used to show the latter integral?
complex-analysis complex-integration
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
$endgroup$
marked as duplicate by Martin R, Lord Shark the Unknown, user10354138, Cesareo, Rebellos Dec 9 '18 at 10:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
Yes, I apologise if it is. I made a lot of research before I posted this question but I found nothing that was smiliar + simple enough for me to understand. Sorry!
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:19
add a comment |
$begingroup$
This question already has an answer here:
Proof that $frac {1} {2pi i} oint frac {{rm d}z} {P(z)} $ over a closed curve is zero.
2 answers
For a polynomial P(z) of degree $ngeq2$, show that there exists some $R_{1}>0$ such that for $R>R_{1}$ it holds that:
$$int_{C_{R}}frac{1}{P(z)}dz=0$$ where $C_{R}$ is a circle of radius R with the centre $0$.
I am already aware of the fact that $$lim_{Rto infty}int_{C_{R}}frac{1}{P(z)}dz=0$$ and how to go about proving it but I'm not sure how to evaluate the first integral written above without using any limit. Would I maybe use a similar approach as was used to show the latter integral?
complex-analysis complex-integration
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
$endgroup$
This question already has an answer here:
Proof that $frac {1} {2pi i} oint frac {{rm d}z} {P(z)} $ over a closed curve is zero.
2 answers
For a polynomial P(z) of degree $ngeq2$, show that there exists some $R_{1}>0$ such that for $R>R_{1}$ it holds that:
$$int_{C_{R}}frac{1}{P(z)}dz=0$$ where $C_{R}$ is a circle of radius R with the centre $0$.
I am already aware of the fact that $$lim_{Rto infty}int_{C_{R}}frac{1}{P(z)}dz=0$$ and how to go about proving it but I'm not sure how to evaluate the first integral written above without using any limit. Would I maybe use a similar approach as was used to show the latter integral?
This question already has an answer here:
Proof that $frac {1} {2pi i} oint frac {{rm d}z} {P(z)} $ over a closed curve is zero.
2 answers
complex-analysis complex-integration
complex-analysis complex-integration
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
asked Dec 8 '18 at 17:10
basic_ceremonybasic_ceremony
53
53
marked as duplicate by Martin R, Lord Shark the Unknown, user10354138, Cesareo, Rebellos Dec 9 '18 at 10:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Martin R, Lord Shark the Unknown, user10354138, Cesareo, Rebellos Dec 9 '18 at 10:24
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
Yes, I apologise if it is. I made a lot of research before I posted this question but I found nothing that was smiliar + simple enough for me to understand. Sorry!
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:19
add a comment |
$begingroup$
Yes, I apologise if it is. I made a lot of research before I posted this question but I found nothing that was smiliar + simple enough for me to understand. Sorry!
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:19
$begingroup$
Yes, I apologise if it is. I made a lot of research before I posted this question but I found nothing that was smiliar + simple enough for me to understand. Sorry!
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:19
$begingroup$
Yes, I apologise if it is. I made a lot of research before I posted this question but I found nothing that was smiliar + simple enough for me to understand. Sorry!
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:19
add a comment |
1 Answer
1
active
oldest
votes
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If $R$ and $S$ are greater than the absolute value of every root of $P$, then, by Cauchy's theorem$$oint_{C_R}frac1{P(z)},mathrm dz=oint_{C_S}frac1{P(z)},mathrm dz.$$This, together with the fact that that limit that you mentioned is $0$; is enogh to prove that $oint_{C_R}frac1{P(z)},mathrm dz$ is $0$ if $R$ is large enough.
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
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$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
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– basic_ceremony
Dec 8 '18 at 17:17
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Yes, you are right.
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– José Carlos Santos
Dec 8 '18 at 17:30
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Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If $R$ and $S$ are greater than the absolute value of every root of $P$, then, by Cauchy's theorem$$oint_{C_R}frac1{P(z)},mathrm dz=oint_{C_S}frac1{P(z)},mathrm dz.$$This, together with the fact that that limit that you mentioned is $0$; is enogh to prove that $oint_{C_R}frac1{P(z)},mathrm dz$ is $0$ if $R$ is large enough.
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
$endgroup$
$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:17
$begingroup$
Yes, you are right.
$endgroup$
– José Carlos Santos
Dec 8 '18 at 17:30
$begingroup$
Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
add a comment |
$begingroup$
If $R$ and $S$ are greater than the absolute value of every root of $P$, then, by Cauchy's theorem$$oint_{C_R}frac1{P(z)},mathrm dz=oint_{C_S}frac1{P(z)},mathrm dz.$$This, together with the fact that that limit that you mentioned is $0$; is enogh to prove that $oint_{C_R}frac1{P(z)},mathrm dz$ is $0$ if $R$ is large enough.
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
$endgroup$
$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:17
$begingroup$
Yes, you are right.
$endgroup$
– José Carlos Santos
Dec 8 '18 at 17:30
$begingroup$
Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
add a comment |
$begingroup$
If $R$ and $S$ are greater than the absolute value of every root of $P$, then, by Cauchy's theorem$$oint_{C_R}frac1{P(z)},mathrm dz=oint_{C_S}frac1{P(z)},mathrm dz.$$This, together with the fact that that limit that you mentioned is $0$; is enogh to prove that $oint_{C_R}frac1{P(z)},mathrm dz$ is $0$ if $R$ is large enough.
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
$endgroup$
If $R$ and $S$ are greater than the absolute value of every root of $P$, then, by Cauchy's theorem$$oint_{C_R}frac1{P(z)},mathrm dz=oint_{C_S}frac1{P(z)},mathrm dz.$$This, together with the fact that that limit that you mentioned is $0$; is enogh to prove that $oint_{C_R}frac1{P(z)},mathrm dz$ is $0$ if $R$ is large enough.
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
answered Dec 8 '18 at 17:14
José Carlos SantosJosé Carlos Santos
161k22127232
161k22127232
$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:17
$begingroup$
Yes, you are right.
$endgroup$
– José Carlos Santos
Dec 8 '18 at 17:30
$begingroup$
Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
add a comment |
$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:17
$begingroup$
Yes, you are right.
$endgroup$
– José Carlos Santos
Dec 8 '18 at 17:30
$begingroup$
Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:17
$begingroup$
Would 'If R and S are greater than the absolute value of every root of P' be another way of saying that all roots of P are contained in R and S? I'm confident the answer is yes but I'd like to be certain
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:17
$begingroup$
Yes, you are right.
$endgroup$
– José Carlos Santos
Dec 8 '18 at 17:30
$begingroup$
Yes, you are right.
$endgroup$
– José Carlos Santos
Dec 8 '18 at 17:30
$begingroup$
Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
$begingroup$
Thanks for the great responses + respone times, as usual :)
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:33
add a comment |
$begingroup$
Yes, I apologise if it is. I made a lot of research before I posted this question but I found nothing that was smiliar + simple enough for me to understand. Sorry!
$endgroup$
– basic_ceremony
Dec 8 '18 at 17:19