Residue theorem with winding numbers
$begingroup$
In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue Theorem.
I've been reading discussions about complex integrals, where some people suggest solutions that take winding numbers into account. Is there a general advantage in doing so?
complex-analysis
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
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migrated from mathoverflow.net Jan 31 at 18:02
This question came from our site for professional mathematicians.
add a comment |
$begingroup$
In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue Theorem.
I've been reading discussions about complex integrals, where some people suggest solutions that take winding numbers into account. Is there a general advantage in doing so?
complex-analysis
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
$endgroup$
migrated from mathoverflow.net Jan 31 at 18:02
This question came from our site for professional mathematicians.
add a comment |
$begingroup$
In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue Theorem.
I've been reading discussions about complex integrals, where some people suggest solutions that take winding numbers into account. Is there a general advantage in doing so?
complex-analysis
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
$endgroup$
In my studies, I've been uniquely using the Residue Theorem with no more than a single winding around each singularity. Actually, my professor has never mentioned winding-numbers in the Residue Theorem.
I've been reading discussions about complex integrals, where some people suggest solutions that take winding numbers into account. Is there a general advantage in doing so?
complex-analysis
complex-analysis
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
asked Jan 31 at 15:36
Minamoto YoshitsuneMinamoto Yoshitsune
82
82
migrated from mathoverflow.net Jan 31 at 18:02
This question came from our site for professional mathematicians.
migrated from mathoverflow.net Jan 31 at 18:02
This question came from our site for professional mathematicians.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I taught a basic complex analysis class 16 times so far. Initially I did not include the winding number, and restricted the residue theorem to simple closed curves. This is visually satisfactory and usually sufficient for applications, but it gives a hard time for the teacher and the careful student alike. The reason is that one needs to talk about "the interior" and "counterclockwise orientation" of a simple closed curve. However, I found that discussing these concepts rigorously is beyond an introductory course. So, after a while, I decided to include the winding number: it simplified the proofs and made the theorem more general. Here is the version I currently teach in my course:
Residue Theorem. Let $Msubsetmathbb{C}$ be open. Let $gammasubset M$ be a closed curve freely homotopic to a constant curve within $M$. Assume that $f$ is holomorphic on $Msetminus S$, where $Ssubset M$ is a discrete subset of isolated singularities of $f$ disjoint from $gamma$. Then
$$int_gamma f=2pi isum_{pin S} w_p(gamma)mathrm{res}_p(f),$$
where the winding number $w_p(gamma)$ vanishes for all but finitely many points $pin S$.
$endgroup$
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I taught a basic complex analysis class 16 times so far. Initially I did not include the winding number, and restricted the residue theorem to simple closed curves. This is visually satisfactory and usually sufficient for applications, but it gives a hard time for the teacher and the careful student alike. The reason is that one needs to talk about "the interior" and "counterclockwise orientation" of a simple closed curve. However, I found that discussing these concepts rigorously is beyond an introductory course. So, after a while, I decided to include the winding number: it simplified the proofs and made the theorem more general. Here is the version I currently teach in my course:
Residue Theorem. Let $Msubsetmathbb{C}$ be open. Let $gammasubset M$ be a closed curve freely homotopic to a constant curve within $M$. Assume that $f$ is holomorphic on $Msetminus S$, where $Ssubset M$ is a discrete subset of isolated singularities of $f$ disjoint from $gamma$. Then
$$int_gamma f=2pi isum_{pin S} w_p(gamma)mathrm{res}_p(f),$$
where the winding number $w_p(gamma)$ vanishes for all but finitely many points $pin S$.
$endgroup$
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
add a comment |
$begingroup$
I taught a basic complex analysis class 16 times so far. Initially I did not include the winding number, and restricted the residue theorem to simple closed curves. This is visually satisfactory and usually sufficient for applications, but it gives a hard time for the teacher and the careful student alike. The reason is that one needs to talk about "the interior" and "counterclockwise orientation" of a simple closed curve. However, I found that discussing these concepts rigorously is beyond an introductory course. So, after a while, I decided to include the winding number: it simplified the proofs and made the theorem more general. Here is the version I currently teach in my course:
Residue Theorem. Let $Msubsetmathbb{C}$ be open. Let $gammasubset M$ be a closed curve freely homotopic to a constant curve within $M$. Assume that $f$ is holomorphic on $Msetminus S$, where $Ssubset M$ is a discrete subset of isolated singularities of $f$ disjoint from $gamma$. Then
$$int_gamma f=2pi isum_{pin S} w_p(gamma)mathrm{res}_p(f),$$
where the winding number $w_p(gamma)$ vanishes for all but finitely many points $pin S$.
$endgroup$
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
add a comment |
$begingroup$
I taught a basic complex analysis class 16 times so far. Initially I did not include the winding number, and restricted the residue theorem to simple closed curves. This is visually satisfactory and usually sufficient for applications, but it gives a hard time for the teacher and the careful student alike. The reason is that one needs to talk about "the interior" and "counterclockwise orientation" of a simple closed curve. However, I found that discussing these concepts rigorously is beyond an introductory course. So, after a while, I decided to include the winding number: it simplified the proofs and made the theorem more general. Here is the version I currently teach in my course:
Residue Theorem. Let $Msubsetmathbb{C}$ be open. Let $gammasubset M$ be a closed curve freely homotopic to a constant curve within $M$. Assume that $f$ is holomorphic on $Msetminus S$, where $Ssubset M$ is a discrete subset of isolated singularities of $f$ disjoint from $gamma$. Then
$$int_gamma f=2pi isum_{pin S} w_p(gamma)mathrm{res}_p(f),$$
where the winding number $w_p(gamma)$ vanishes for all but finitely many points $pin S$.
$endgroup$
I taught a basic complex analysis class 16 times so far. Initially I did not include the winding number, and restricted the residue theorem to simple closed curves. This is visually satisfactory and usually sufficient for applications, but it gives a hard time for the teacher and the careful student alike. The reason is that one needs to talk about "the interior" and "counterclockwise orientation" of a simple closed curve. However, I found that discussing these concepts rigorously is beyond an introductory course. So, after a while, I decided to include the winding number: it simplified the proofs and made the theorem more general. Here is the version I currently teach in my course:
Residue Theorem. Let $Msubsetmathbb{C}$ be open. Let $gammasubset M$ be a closed curve freely homotopic to a constant curve within $M$. Assume that $f$ is holomorphic on $Msetminus S$, where $Ssubset M$ is a discrete subset of isolated singularities of $f$ disjoint from $gamma$. Then
$$int_gamma f=2pi isum_{pin S} w_p(gamma)mathrm{res}_p(f),$$
where the winding number $w_p(gamma)$ vanishes for all but finitely many points $pin S$.
answered Jan 31 at 16:12
GH from MO
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
add a comment |
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
What does freely homotopic mean?
$endgroup$
– Vít Tuček
Jan 31 at 17:02
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: see en.wikipedia.org/wiki/Free_loop
$endgroup$
– Ben McKay
Jan 31 at 17:17
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
@VítTuček: Free homotopy is homotopy without a base point (see e.g. proofwiki.org/wiki/Definition:Homotopy/Free) Fun fact: it took me several years of teaching complex analysis before I learned the proper terminology. For a while I just told my students "homotopy without a base point".
$endgroup$
– GH from MO
Jan 31 at 17:41
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
$begingroup$
That's interesting. I've studied complex analysis as part of a course in "advanced mathematics for engineers", where homotopy is never once mentioned. It's nice to extend my knowledge. What I was really wondering is whether winding numbers may help in the evaluation of integrals via the residue theorem.
$endgroup$
– Minamoto Yoshitsune
Feb 1 at 16:20
add a comment |
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