Mixing
Product of Functions Holomorphic in Different Subsets of $mathbbC$
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Let $f:mathbbCtomathbbC$ be analytic in $CsubseteqmathbbC$ and let $g:mathbbCtomathbbC$ be analytic in $DsubseteqmathbbC$. Show that $fcdot g$ is analytic in $Ccap D$.
I know that the product of two functions analytic in the same subset of $mathbbC$ is analytic in that subset, so it seems reasonable to say that the product of functions analytic in different subsets would be analytic wherever $fleft(zright)$ and $gleft(zright)$ are both analytic. I’m trying to figure out how to show this rigorously. Or is there a counterexample to this?
complex-analysis
asked Jul 18 at 7:14
csch2
220211
add a comment |Â
up vote
0
down vote
favorite
Let $f:mathbbCtomathbbC$ be analytic in $CsubseteqmathbbC$ and let $g:mathbbCtomathbbC$ be analytic in $DsubseteqmathbbC$. Show that $fcdot g$ is analytic in $Ccap D$.
I know that the product of two functions analytic in the same subset of $mathbbC$ is analytic in that subset, so it seems reasonable to say that the product of functions analytic in different subsets would be analytic wherever $fleft(zright)$ and $gleft(zright)$ are both analytic. I’m trying to figure out how to show this rigorously. Or is there a counterexample to this?
complex-analysis
asked Jul 18 at 7:14
csch2
220211
2
If you know that $f$ is holomorphic on $C$, then it will also be holomorphic on the open subset $Ccap D$, and the same argument holds for $g$, which gives you two holomorphic functions on the same open subset
– asdq
Jul 18 at 7:17
@asdq thank you! Makes perfect sense now
– csch2
Jul 18 at 7:22
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f:mathbbCtomathbbC$ be analytic in $CsubseteqmathbbC$ and let $g:mathbbCtomathbbC$ be analytic in $DsubseteqmathbbC$. Show that $fcdot g$ is analytic in $Ccap D$.
I know that the product of two functions analytic in the same subset of $mathbbC$ is analytic in that subset, so it seems reasonable to say that the product of functions analytic in different subsets would be analytic wherever $fleft(zright)$ and $gleft(zright)$ are both analytic. I’m trying to figure out how to show this rigorously. Or is there a counterexample to this?
complex-analysis
asked Jul 18 at 7:14
csch2
220211
Let $f:mathbbCtomathbbC$ be analytic in $CsubseteqmathbbC$ and let $g:mathbbCtomathbbC$ be analytic in $DsubseteqmathbbC$. Show that $fcdot g$ is analytic in $Ccap D$.
I know that the product of two functions analytic in the same subset of $mathbbC$ is analytic in that subset, so it seems reasonable to say that the product of functions analytic in different subsets would be analytic wherever $fleft(zright)$ and $gleft(zright)$ are both analytic. I’m trying to figure out how to show this rigorously. Or is there a counterexample to this?
complex-analysis
asked Jul 18 at 7:14
csch2
220211
asked Jul 18 at 7:14
csch2
220211
asked Jul 18 at 7:14
csch2
220211
asked Jul 18 at 7:14
csch2
220211
220211
2
If you know that $f$ is holomorphic on $C$, then it will also be holomorphic on the open subset $Ccap D$, and the same argument holds for $g$, which gives you two holomorphic functions on the same open subset
– asdq
Jul 18 at 7:17
@asdq thank you! Makes perfect sense now
– csch2
Jul 18 at 7:22
add a comment |Â
2
If you know that $f$ is holomorphic on $C$, then it will also be holomorphic on the open subset $Ccap D$, and the same argument holds for $g$, which gives you two holomorphic functions on the same open subset
– asdq
Jul 18 at 7:17
@asdq thank you! Makes perfect sense now
– csch2
Jul 18 at 7:22
2
2
If you know that $f$ is holomorphic on $C$, then it will also be holomorphic on the open subset $Ccap D$, and the same argument holds for $g$, which gives you two holomorphic functions on the same open subset
– asdq
Jul 18 at 7:17
If you know that $f$ is holomorphic on $C$, then it will also be holomorphic on the open subset $Ccap D$, and the same argument holds for $g$, which gives you two holomorphic functions on the same open subset
– asdq
Jul 18 at 7:17
@asdq thank you! Makes perfect sense now
– csch2
Jul 18 at 7:22
@asdq thank you! Makes perfect sense now
– csch2
Jul 18 at 7:22
add a comment |Â
1 Answer
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Answering this question since I think I’ve figured it out now:
Since $left(Ccap Dright)subseteq C$, $f$ is analytic in $Ccap D$. Similarly, $g$ is also analytic in $Ccap D$. Since both functions are analytic in the same subset and the product of functions analytic in the same subset is also analytic in that subset, $fcdot g$ is analytic in $Ccap D$.
answered Jul 18 at 7:31
csch2
220211
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Answering this question since I think I’ve figured it out now:
Since $left(Ccap Dright)subseteq C$, $f$ is analytic in $Ccap D$. Similarly, $g$ is also analytic in $Ccap D$. Since both functions are analytic in the same subset and the product of functions analytic in the same subset is also analytic in that subset, $fcdot g$ is analytic in $Ccap D$.
answered Jul 18 at 7:31
csch2
220211
add a comment |Â
up vote
0
down vote
Answering this question since I think I’ve figured it out now:
Since $left(Ccap Dright)subseteq C$, $f$ is analytic in $Ccap D$. Similarly, $g$ is also analytic in $Ccap D$. Since both functions are analytic in the same subset and the product of functions analytic in the same subset is also analytic in that subset, $fcdot g$ is analytic in $Ccap D$.
answered Jul 18 at 7:31
csch2
220211
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Answering this question since I think I’ve figured it out now:
Since $left(Ccap Dright)subseteq C$, $f$ is analytic in $Ccap D$. Similarly, $g$ is also analytic in $Ccap D$. Since both functions are analytic in the same subset and the product of functions analytic in the same subset is also analytic in that subset, $fcdot g$ is analytic in $Ccap D$.
answered Jul 18 at 7:31
csch2
220211
Answering this question since I think I’ve figured it out now:
Since $left(Ccap Dright)subseteq C$, $f$ is analytic in $Ccap D$. Similarly, $g$ is also analytic in $Ccap D$. Since both functions are analytic in the same subset and the product of functions analytic in the same subset is also analytic in that subset, $fcdot g$ is analytic in $Ccap D$.
answered Jul 18 at 7:31
csch2
220211
answered Jul 18 at 7:31
csch2
220211
answered Jul 18 at 7:31
csch2
220211
answered Jul 18 at 7:31
csch2
220211
220211
add a comment |Â
add a comment |Â
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2
If you know that $f$ is holomorphic on $C$, then it will also be holomorphic on the open subset $Ccap D$, and the same argument holds for $g$, which gives you two holomorphic functions on the same open subset
– asdq
Jul 18 at 7:17
@asdq thank you! Makes perfect sense now
– csch2
Jul 18 at 7:22