Conditions on the Cauchy residue theorem [on hold]
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Are there any conditions to apply Cauchy-Residue theorem? or it's always valid for analytic functions with a singularity within a given closed contour?
complex-analysis
asked Nov 16 at 21:16
Khaled Yasein
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put on hold as off-topic by Did, Leucippus, Cesareo, user10354138, Brahadeesh 2 days ago
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Are there any conditions to apply Cauchy-Residue theorem? or it's always valid for analytic functions with a singularity within a given closed contour?
complex-analysis
asked Nov 16 at 21:16
Khaled Yasein
41
put on hold as off-topic by Did, Leucippus, Cesareo, user10354138, Brahadeesh 2 days ago
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Leucippus, Cesareo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
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up vote
0
down vote
favorite
up vote
0
down vote
favorite
Are there any conditions to apply Cauchy-Residue theorem? or it's always valid for analytic functions with a singularity within a given closed contour?
complex-analysis
asked Nov 16 at 21:16
Khaled Yasein
41
Are there any conditions to apply Cauchy-Residue theorem? or it's always valid for analytic functions with a singularity within a given closed contour?
complex-analysis
complex-analysis
asked Nov 16 at 21:16
Khaled Yasein
41
asked Nov 16 at 21:16
Khaled Yasein
41
asked Nov 16 at 21:16
Khaled Yasein
41
asked Nov 16 at 21:16
Khaled Yasein
41
asked Nov 16 at 21:16
Khaled Yasein
41
41
put on hold as off-topic by Did, Leucippus, Cesareo, user10354138, Brahadeesh 2 days ago
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Leucippus, Cesareo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by Did, Leucippus, Cesareo, user10354138, Brahadeesh 2 days ago
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Leucippus, Cesareo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
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2 Answers
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The only conditions are that the curve be closed and over a region on which the function is analytic except possibly at a finite set of points, say $A$, where the residue are nonzero.
The function being integrated need not be analytic over all of $mathbb{C}$, just an open region $D$ containing the curve and its interior (except of course for points in $A$).
answered Nov 16 at 21:19
Leland Reardon
688
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According to wikipedia, the curve needs to be
- positively oriented
- simple and
- closed
The function needs be meromorphic on the inside of this curve.
answered Nov 16 at 21:57
mathreadler
14.6k72160
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The only conditions are that the curve be closed and over a region on which the function is analytic except possibly at a finite set of points, say $A$, where the residue are nonzero.
The function being integrated need not be analytic over all of $mathbb{C}$, just an open region $D$ containing the curve and its interior (except of course for points in $A$).
answered Nov 16 at 21:19
Leland Reardon
688
add a comment |
up vote
0
down vote
The only conditions are that the curve be closed and over a region on which the function is analytic except possibly at a finite set of points, say $A$, where the residue are nonzero.
The function being integrated need not be analytic over all of $mathbb{C}$, just an open region $D$ containing the curve and its interior (except of course for points in $A$).
answered Nov 16 at 21:19
Leland Reardon
688
add a comment |
up vote
0
down vote
up vote
0
down vote
The only conditions are that the curve be closed and over a region on which the function is analytic except possibly at a finite set of points, say $A$, where the residue are nonzero.
The function being integrated need not be analytic over all of $mathbb{C}$, just an open region $D$ containing the curve and its interior (except of course for points in $A$).
answered Nov 16 at 21:19
Leland Reardon
688
The only conditions are that the curve be closed and over a region on which the function is analytic except possibly at a finite set of points, say $A$, where the residue are nonzero.
The function being integrated need not be analytic over all of $mathbb{C}$, just an open region $D$ containing the curve and its interior (except of course for points in $A$).
answered Nov 16 at 21:19
Leland Reardon
688
edited Nov 16 at 21:37
answered Nov 16 at 21:19
Leland Reardon
688
answered Nov 16 at 21:19
Leland Reardon
688
answered Nov 16 at 21:19
Leland Reardon
688
688
add a comment |
add a comment |
up vote
0
down vote
According to wikipedia, the curve needs to be
- positively oriented
- simple and
- closed
The function needs be meromorphic on the inside of this curve.
answered Nov 16 at 21:57
mathreadler
14.6k72160
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
add a comment |
up vote
0
down vote
According to wikipedia, the curve needs to be
- positively oriented
- simple and
- closed
The function needs be meromorphic on the inside of this curve.
answered Nov 16 at 21:57
mathreadler
14.6k72160
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
add a comment |
up vote
0
down vote
up vote
0
down vote
According to wikipedia, the curve needs to be
- positively oriented
- simple and
- closed
The function needs be meromorphic on the inside of this curve.
answered Nov 16 at 21:57
mathreadler
14.6k72160
According to wikipedia, the curve needs to be
- positively oriented
- simple and
- closed
The function needs be meromorphic on the inside of this curve.
answered Nov 16 at 21:57
mathreadler
14.6k72160
answered Nov 16 at 21:57
mathreadler
14.6k72160
answered Nov 16 at 21:57
mathreadler
14.6k72160
answered Nov 16 at 21:57
mathreadler
14.6k72160
14.6k72160
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
add a comment |
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
If the curve is negatively oriented, the integral will just be the negative of if the curve were positively oriented.
– Leland Reardon
Nov 16 at 21:59
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
@LelandReardon yes. But if it isn't simple, then it gets more complicated.
– mathreadler
Nov 16 at 22:19
add a comment |
