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Is there a way to categorise the valleys of a holomorphic function?

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For an entire function $f$, the input space $mathbbC$ is split into hills and valleys about the saddle points of $Re(f)$ by the maximum modulus theorem. Visually it is quite obvious whether or not two points (say $z_1,z_2$) are in the same valley. Is there a way to formalise this idea? In addition to this, is there some way to categorise $mathbbC$ around this quality? Ideally I would like to find some test that extends to functions $f:mathbbC^2 rightarrow mathbbC$ i.e holomorphic functions of 2 complex variables

I had thought one idea would be to take the steepest descent curves of starting from $z_1,z_2$. Then if $arg(z_1),arg(z_2) rightarrow theta$ we can say the two points must be in the same valley. We need the added stipulation that $Re(f(z_1)),Re(f(z_2))leq Re f(z_0)$ for the relevant saddle point $z_0$. I think this would work pointwise; can we use it or some different idea to categorise $mathbbC$ based on which valley the point belongs to?

I hope this would then extend to functions of 2 complex variables by writing $z,w=(z_1,z_2),(w_1,w_2) in mathbbC$ and checking that the arguments of $z_1,w_1$ and $z_2,w_2$ tend to the same limit. Does this sound sensible? Is there a better way to do it?

real-analysis complex-analysis analysis numerical-methods complex-integration

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edited 2 days ago

D.Dog

asked 2 days ago

D.DogD.Dog

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0

1

$begingroup$

For an entire function $f$, the input space $mathbbC$ is split into hills and valleys about the saddle points of $Re(f)$ by the maximum modulus theorem. Visually it is quite obvious whether or not two points (say $z_1,z_2$) are in the same valley. Is there a way to formalise this idea? In addition to this, is there some way to categorise $mathbbC$ around this quality? Ideally I would like to find some test that extends to functions $f:mathbbC^2 rightarrow mathbbC$ i.e holomorphic functions of 2 complex variables

I had thought one idea would be to take the steepest descent curves of starting from $z_1,z_2$. Then if $arg(z_1),arg(z_2) rightarrow theta$ we can say the two points must be in the same valley. We need the added stipulation that $Re(f(z_1)),Re(f(z_2))leq Re f(z_0)$ for the relevant saddle point $z_0$. I think this would work pointwise; can we use it or some different idea to categorise $mathbbC$ based on which valley the point belongs to?

I hope this would then extend to functions of 2 complex variables by writing $z,w=(z_1,z_2),(w_1,w_2) in mathbbC$ and checking that the arguments of $z_1,w_1$ and $z_2,w_2$ tend to the same limit. Does this sound sensible? Is there a better way to do it?

real-analysis complex-analysis analysis numerical-methods complex-integration

share|cite|improve this question

edited 2 days ago

D.Dog

asked 2 days ago

D.DogD.Dog

207

$endgroup$

add a comment | 

$begingroup$

For an entire function $f$, the input space $mathbbC$ is split into hills and valleys about the saddle points of $Re(f)$ by the maximum modulus theorem. Visually it is quite obvious whether or not two points (say $z_1,z_2$) are in the same valley. Is there a way to formalise this idea? In addition to this, is there some way to categorise $mathbbC$ around this quality? Ideally I would like to find some test that extends to functions $f:mathbbC^2 rightarrow mathbbC$ i.e holomorphic functions of 2 complex variables

I had thought one idea would be to take the steepest descent curves of starting from $z_1,z_2$. Then if $arg(z_1),arg(z_2) rightarrow theta$ we can say the two points must be in the same valley. We need the added stipulation that $Re(f(z_1)),Re(f(z_2))leq Re f(z_0)$ for the relevant saddle point $z_0$. I think this would work pointwise; can we use it or some different idea to categorise $mathbbC$ based on which valley the point belongs to?

I hope this would then extend to functions of 2 complex variables by writing $z,w=(z_1,z_2),(w_1,w_2) in mathbbC$ and checking that the arguments of $z_1,w_1$ and $z_2,w_2$ tend to the same limit. Does this sound sensible? Is there a better way to do it?

real-analysis complex-analysis analysis numerical-methods complex-integration

share|cite|improve this question

edited 2 days ago

D.Dog

asked 2 days ago

D.DogD.Dog

207

$endgroup$

share|cite|improve this question

edited 2 days ago

D.Dog

asked 2 days ago

D.DogD.Dog

207

share|cite|improve this question

edited 2 days ago

D.Dog

asked 2 days ago

D.DogD.Dog

207

asked 2 days ago

D.DogD.Dog

207

asked 2 days ago

D.DogD.Dog

207