If $f[mathbb{T}]subset mathbb{R}$ then $f$ is constant
$begingroup$
If $f:overline{mathbb{D}}longrightarrowmathbb{C}$ is a holomorphic function over $mathbb{D}$ and $f(mathbb{T})subset mathbb{R}$ then is $f$ constant?
Consider:
$f:overline{mathbb{D}}longrightarrowmathbb{C}$ continuous
$mathbb{D}={zinmathbb{C}:|z|<1}$
$overline{mathbb{D}}={zinmathbb{C}:|z|le1}$
$mathbb{T}={zinmathbb{C}:|z|=1}$
Any hint would be appreciated.
complex-analysis
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
$endgroup$
add a comment |
$begingroup$
If $f:overline{mathbb{D}}longrightarrowmathbb{C}$ is a holomorphic function over $mathbb{D}$ and $f(mathbb{T})subset mathbb{R}$ then is $f$ constant?
Consider:
$f:overline{mathbb{D}}longrightarrowmathbb{C}$ continuous
$mathbb{D}={zinmathbb{C}:|z|<1}$
$overline{mathbb{D}}={zinmathbb{C}:|z|le1}$
$mathbb{T}={zinmathbb{C}:|z|=1}$
Any hint would be appreciated.
complex-analysis
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
$endgroup$
add a comment |
$begingroup$
If $f:overline{mathbb{D}}longrightarrowmathbb{C}$ is a holomorphic function over $mathbb{D}$ and $f(mathbb{T})subset mathbb{R}$ then is $f$ constant?
Consider:
$f:overline{mathbb{D}}longrightarrowmathbb{C}$ continuous
$mathbb{D}={zinmathbb{C}:|z|<1}$
$overline{mathbb{D}}={zinmathbb{C}:|z|le1}$
$mathbb{T}={zinmathbb{C}:|z|=1}$
Any hint would be appreciated.
complex-analysis
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
$endgroup$
If $f:overline{mathbb{D}}longrightarrowmathbb{C}$ is a holomorphic function over $mathbb{D}$ and $f(mathbb{T})subset mathbb{R}$ then is $f$ constant?
Consider:
$f:overline{mathbb{D}}longrightarrowmathbb{C}$ continuous
$mathbb{D}={zinmathbb{C}:|z|<1}$
$overline{mathbb{D}}={zinmathbb{C}:|z|le1}$
$mathbb{T}={zinmathbb{C}:|z|=1}$
Any hint would be appreciated.
complex-analysis
complex-analysis
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
asked Aug 30 '14 at 23:35
felipeunifelipeuni
1,99111431
1,99111431
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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The imaginary part of a holomorphic function is harmonic, and it attains its maximum on the border of any compact set contained in $mathbb{D}$, taking the limit on all such compact sets we get that that maximum is $0$, that is $Im{f}=0$. You now have an holomorphic function with zero imaginary part, so by the Cauchy Riemann equation you conclude.
answered Aug 30 '14 at 23:57
LolmanLolman
970613
$endgroup$
add a comment |
$begingroup$
I don't know if this will work, but this is what I think when I see the problem given.
Complex analysis gives us a number of tools for reconstructing a function given its values on various sets; in this case, we have a direct formula (Cauchy's Integral formula, I think the name is) for obtaining the value of a function inside of a loop from its values on the loop.
So I would start by looking at writing down said formula and seeing if it implies anything.
My other thought from the problem is that $log(z)$ is aligned with these sorts of domains. I would spend some time experimenting with functions related to $log(z)$ to see if I could construct one that is holomorphic in the disk and real on the boundary.
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The imaginary part of a holomorphic function is harmonic, and it attains its maximum on the border of any compact set contained in $mathbb{D}$, taking the limit on all such compact sets we get that that maximum is $0$, that is $Im{f}=0$. You now have an holomorphic function with zero imaginary part, so by the Cauchy Riemann equation you conclude.
answered Aug 30 '14 at 23:57
LolmanLolman
970613
$endgroup$
add a comment |
$begingroup$
The imaginary part of a holomorphic function is harmonic, and it attains its maximum on the border of any compact set contained in $mathbb{D}$, taking the limit on all such compact sets we get that that maximum is $0$, that is $Im{f}=0$. You now have an holomorphic function with zero imaginary part, so by the Cauchy Riemann equation you conclude.
answered Aug 30 '14 at 23:57
LolmanLolman
970613
$endgroup$
add a comment |
$begingroup$
The imaginary part of a holomorphic function is harmonic, and it attains its maximum on the border of any compact set contained in $mathbb{D}$, taking the limit on all such compact sets we get that that maximum is $0$, that is $Im{f}=0$. You now have an holomorphic function with zero imaginary part, so by the Cauchy Riemann equation you conclude.
answered Aug 30 '14 at 23:57
LolmanLolman
970613
$endgroup$
The imaginary part of a holomorphic function is harmonic, and it attains its maximum on the border of any compact set contained in $mathbb{D}$, taking the limit on all such compact sets we get that that maximum is $0$, that is $Im{f}=0$. You now have an holomorphic function with zero imaginary part, so by the Cauchy Riemann equation you conclude.
answered Aug 30 '14 at 23:57
LolmanLolman
970613
answered Aug 30 '14 at 23:57
LolmanLolman
970613
answered Aug 30 '14 at 23:57
LolmanLolman
970613
answered Aug 30 '14 at 23:57
LolmanLolman
970613
970613
add a comment |
add a comment |
$begingroup$
I don't know if this will work, but this is what I think when I see the problem given.
Complex analysis gives us a number of tools for reconstructing a function given its values on various sets; in this case, we have a direct formula (Cauchy's Integral formula, I think the name is) for obtaining the value of a function inside of a loop from its values on the loop.
So I would start by looking at writing down said formula and seeing if it implies anything.
My other thought from the problem is that $log(z)$ is aligned with these sorts of domains. I would spend some time experimenting with functions related to $log(z)$ to see if I could construct one that is holomorphic in the disk and real on the boundary.
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
$endgroup$
add a comment |
$begingroup$
I don't know if this will work, but this is what I think when I see the problem given.
Complex analysis gives us a number of tools for reconstructing a function given its values on various sets; in this case, we have a direct formula (Cauchy's Integral formula, I think the name is) for obtaining the value of a function inside of a loop from its values on the loop.
So I would start by looking at writing down said formula and seeing if it implies anything.
My other thought from the problem is that $log(z)$ is aligned with these sorts of domains. I would spend some time experimenting with functions related to $log(z)$ to see if I could construct one that is holomorphic in the disk and real on the boundary.
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
$endgroup$
add a comment |
$begingroup$
I don't know if this will work, but this is what I think when I see the problem given.
Complex analysis gives us a number of tools for reconstructing a function given its values on various sets; in this case, we have a direct formula (Cauchy's Integral formula, I think the name is) for obtaining the value of a function inside of a loop from its values on the loop.
So I would start by looking at writing down said formula and seeing if it implies anything.
My other thought from the problem is that $log(z)$ is aligned with these sorts of domains. I would spend some time experimenting with functions related to $log(z)$ to see if I could construct one that is holomorphic in the disk and real on the boundary.
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
$endgroup$
I don't know if this will work, but this is what I think when I see the problem given.
Complex analysis gives us a number of tools for reconstructing a function given its values on various sets; in this case, we have a direct formula (Cauchy's Integral formula, I think the name is) for obtaining the value of a function inside of a loop from its values on the loop.
So I would start by looking at writing down said formula and seeing if it implies anything.
My other thought from the problem is that $log(z)$ is aligned with these sorts of domains. I would spend some time experimenting with functions related to $log(z)$ to see if I could construct one that is holomorphic in the disk and real on the boundary.
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
answered Aug 30 '14 at 23:41
HurkylHurkyl
111k9119262
111k9119262
add a comment |
add a comment |
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