Cauchy Residue Theorem and the Deformation Principle
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I am trying to use Cauchy Goursat to prove the Cauchy Residue theorem, and I am confused by what seems two viable routes in doing this.
Route 1
Here the region D sort of has holes pushed into itself, but is still connected
Route 2
Here the region D is split into two seperate pieces.
Both approaches seem reasonable to me. Is there a correct approach?
complex-analysis
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
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add a comment |
$begingroup$
I am trying to use Cauchy Goursat to prove the Cauchy Residue theorem, and I am confused by what seems two viable routes in doing this.
Route 1
Here the region D sort of has holes pushed into itself, but is still connected
Route 2
Here the region D is split into two seperate pieces.
Both approaches seem reasonable to me. Is there a correct approach?
complex-analysis
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
$endgroup$
$begingroup$
Contours enclosing simply connected domains are easier. That the Cauchy integral theorem for them implies the one for the boundary of any domains (under the winding number condition for the direction) is easy. So your two approaches are quite the same. The last one is when the function is meromorphic, you can start by substracting the poles to make it holomorphic then integrate the poles.
$endgroup$
– reuns
Dec 7 '18 at 13:49
add a comment |
$begingroup$
I am trying to use Cauchy Goursat to prove the Cauchy Residue theorem, and I am confused by what seems two viable routes in doing this.
Route 1
Here the region D sort of has holes pushed into itself, but is still connected
Route 2
Here the region D is split into two seperate pieces.
Both approaches seem reasonable to me. Is there a correct approach?
complex-analysis
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
$endgroup$
I am trying to use Cauchy Goursat to prove the Cauchy Residue theorem, and I am confused by what seems two viable routes in doing this.
Route 1
Here the region D sort of has holes pushed into itself, but is still connected
Route 2
Here the region D is split into two seperate pieces.
Both approaches seem reasonable to me. Is there a correct approach?
complex-analysis
complex-analysis
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
asked Dec 6 '18 at 22:12
JungleshrimpJungleshrimp
322111
322111
$begingroup$
Contours enclosing simply connected domains are easier. That the Cauchy integral theorem for them implies the one for the boundary of any domains (under the winding number condition for the direction) is easy. So your two approaches are quite the same. The last one is when the function is meromorphic, you can start by substracting the poles to make it holomorphic then integrate the poles.
$endgroup$
– reuns
Dec 7 '18 at 13:49
add a comment |
$begingroup$
Contours enclosing simply connected domains are easier. That the Cauchy integral theorem for them implies the one for the boundary of any domains (under the winding number condition for the direction) is easy. So your two approaches are quite the same. The last one is when the function is meromorphic, you can start by substracting the poles to make it holomorphic then integrate the poles.
$endgroup$
– reuns
Dec 7 '18 at 13:49
$begingroup$
Contours enclosing simply connected domains are easier. That the Cauchy integral theorem for them implies the one for the boundary of any domains (under the winding number condition for the direction) is easy. So your two approaches are quite the same. The last one is when the function is meromorphic, you can start by substracting the poles to make it holomorphic then integrate the poles.
$endgroup$
– reuns
Dec 7 '18 at 13:49
$begingroup$
Contours enclosing simply connected domains are easier. That the Cauchy integral theorem for them implies the one for the boundary of any domains (under the winding number condition for the direction) is easy. So your two approaches are quite the same. The last one is when the function is meromorphic, you can start by substracting the poles to make it holomorphic then integrate the poles.
$endgroup$
– reuns
Dec 7 '18 at 13:49
add a comment |
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$begingroup$
Contours enclosing simply connected domains are easier. That the Cauchy integral theorem for them implies the one for the boundary of any domains (under the winding number condition for the direction) is easy. So your two approaches are quite the same. The last one is when the function is meromorphic, you can start by substracting the poles to make it holomorphic then integrate the poles.
$endgroup$
– reuns
Dec 7 '18 at 13:49