Describing conformal maps in terms of a complex functional equation.
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Conformal maps have the interesting property that they map circles and points to points and circle.
I wonder if the only holomorphic functions with this property are conformal maps. But I realized I don’t know any way to write “mapping lines and circles to lines and circles†as a functional equation which can then be solved or approximated for a collection of functions (which would in principle be the conformal maps).
complex-analysis proof-writing functional-equations
asked Aug 16 at 1:44
frogeyedpeas
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Conformal maps have the interesting property that they map circles and points to points and circle.
I wonder if the only holomorphic functions with this property are conformal maps. But I realized I don’t know any way to write “mapping lines and circles to lines and circles†as a functional equation which can then be solved or approximated for a collection of functions (which would in principle be the conformal maps).
complex-analysis proof-writing functional-equations
asked Aug 16 at 1:44
frogeyedpeas
6,98471747
1
No, conformal maps do not have that property; not every conformal map is a linear-fractional transformation. (Also you stated the property wrong: linear-fractional transformations take lines and circles to lines and circles.)
– David C. Ullrich
Aug 16 at 17:20
Ah that’s a good clarification. Is it known that linear fractional transforms are the only such maps to do this?
– frogeyedpeas
Aug 16 at 17:22
I don't know, sorry. I doubt it. Hmm, the answer below says yes...
– David C. Ullrich
Aug 16 at 17:25
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Conformal maps have the interesting property that they map circles and points to points and circle.
I wonder if the only holomorphic functions with this property are conformal maps. But I realized I don’t know any way to write “mapping lines and circles to lines and circles†as a functional equation which can then be solved or approximated for a collection of functions (which would in principle be the conformal maps).
complex-analysis proof-writing functional-equations
asked Aug 16 at 1:44
frogeyedpeas
6,98471747
Conformal maps have the interesting property that they map circles and points to points and circle.
I wonder if the only holomorphic functions with this property are conformal maps. But I realized I don’t know any way to write “mapping lines and circles to lines and circles†as a functional equation which can then be solved or approximated for a collection of functions (which would in principle be the conformal maps).
complex-analysis proof-writing functional-equations
asked Aug 16 at 1:44
frogeyedpeas
6,98471747
asked Aug 16 at 1:44
frogeyedpeas
6,98471747
asked Aug 16 at 1:44
frogeyedpeas
6,98471747
asked Aug 16 at 1:44
frogeyedpeas
6,98471747
6,98471747
1
No, conformal maps do not have that property; not every conformal map is a linear-fractional transformation. (Also you stated the property wrong: linear-fractional transformations take lines and circles to lines and circles.)
– David C. Ullrich
Aug 16 at 17:20
Ah that’s a good clarification. Is it known that linear fractional transforms are the only such maps to do this?
– frogeyedpeas
Aug 16 at 17:22
I don't know, sorry. I doubt it. Hmm, the answer below says yes...
– David C. Ullrich
Aug 16 at 17:25
add a comment |Â
1
No, conformal maps do not have that property; not every conformal map is a linear-fractional transformation. (Also you stated the property wrong: linear-fractional transformations take lines and circles to lines and circles.)
– David C. Ullrich
Aug 16 at 17:20
Ah that’s a good clarification. Is it known that linear fractional transforms are the only such maps to do this?
– frogeyedpeas
Aug 16 at 17:22
I don't know, sorry. I doubt it. Hmm, the answer below says yes...
– David C. Ullrich
Aug 16 at 17:25
1
1
No, conformal maps do not have that property; not every conformal map is a linear-fractional transformation. (Also you stated the property wrong: linear-fractional transformations take lines and circles to lines and circles.)
– David C. Ullrich
Aug 16 at 17:20
No, conformal maps do not have that property; not every conformal map is a linear-fractional transformation. (Also you stated the property wrong: linear-fractional transformations take lines and circles to lines and circles.)
– David C. Ullrich
Aug 16 at 17:20
Ah that’s a good clarification. Is it known that linear fractional transforms are the only such maps to do this?
– frogeyedpeas
Aug 16 at 17:22
Ah that’s a good clarification. Is it known that linear fractional transforms are the only such maps to do this?
– frogeyedpeas
Aug 16 at 17:22
I don't know, sorry. I doubt it. Hmm, the answer below says yes...
– David C. Ullrich
Aug 16 at 17:25
I don't know, sorry. I doubt it. Hmm, the answer below says yes...
– David C. Ullrich
Aug 16 at 17:25
add a comment |Â
1 Answer
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The answer is that the functions that take lines and circles to lines and circles are precisely the Möbius transformations
$$f(z) = dfracaz+bcz+d,quad textwith ad-bcneq0.$$
I began writing an outline of the proof, but then I found this pdf, which is reasonably self-contained and goes into far more detail than I was willing to type here:
http://www.maths.qmul.ac.uk/~sb/CA_sectionIVnotes.pdf
answered Aug 16 at 4:36
Dosidis
39119
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The answer is that the functions that take lines and circles to lines and circles are precisely the Möbius transformations
$$f(z) = dfracaz+bcz+d,quad textwith ad-bcneq0.$$
I began writing an outline of the proof, but then I found this pdf, which is reasonably self-contained and goes into far more detail than I was willing to type here:
http://www.maths.qmul.ac.uk/~sb/CA_sectionIVnotes.pdf
answered Aug 16 at 4:36
Dosidis
39119
add a comment |Â
up vote
0
down vote
The answer is that the functions that take lines and circles to lines and circles are precisely the Möbius transformations
$$f(z) = dfracaz+bcz+d,quad textwith ad-bcneq0.$$
I began writing an outline of the proof, but then I found this pdf, which is reasonably self-contained and goes into far more detail than I was willing to type here:
http://www.maths.qmul.ac.uk/~sb/CA_sectionIVnotes.pdf
answered Aug 16 at 4:36
Dosidis
39119
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The answer is that the functions that take lines and circles to lines and circles are precisely the Möbius transformations
$$f(z) = dfracaz+bcz+d,quad textwith ad-bcneq0.$$
I began writing an outline of the proof, but then I found this pdf, which is reasonably self-contained and goes into far more detail than I was willing to type here:
http://www.maths.qmul.ac.uk/~sb/CA_sectionIVnotes.pdf
answered Aug 16 at 4:36
Dosidis
39119
The answer is that the functions that take lines and circles to lines and circles are precisely the Möbius transformations
$$f(z) = dfracaz+bcz+d,quad textwith ad-bcneq0.$$
I began writing an outline of the proof, but then I found this pdf, which is reasonably self-contained and goes into far more detail than I was willing to type here:
http://www.maths.qmul.ac.uk/~sb/CA_sectionIVnotes.pdf
answered Aug 16 at 4:36
Dosidis
39119
answered Aug 16 at 4:36
Dosidis
39119
answered Aug 16 at 4:36
Dosidis
39119
answered Aug 16 at 4:36
Dosidis
39119
39119
add a comment |Â
add a comment |Â
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1
No, conformal maps do not have that property; not every conformal map is a linear-fractional transformation. (Also you stated the property wrong: linear-fractional transformations take lines and circles to lines and circles.)
– David C. Ullrich
Aug 16 at 17:20
Ah that’s a good clarification. Is it known that linear fractional transforms are the only such maps to do this?
– frogeyedpeas
Aug 16 at 17:22
I don't know, sorry. I doubt it. Hmm, the answer below says yes...
– David C. Ullrich
Aug 16 at 17:25