Is $ln|z|$ harmonic in the punctured disk [closed]
$begingroup$
1.
how can i show that $ln|z|$ is harmonic in punctured disk ?
- also $ln|z|$ has no harmonic conjugate in $Bbb Csetminus{0 }$ but has in $Bbb Csetminus[0, infty)$.
complex-analysis harmonic-functions multivalued-functions
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
asked Jan 9 at 14:55
HenryHenry
327
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closed as off-topic by José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh Jan 10 at 7:23
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
1.
how can i show that $ln|z|$ is harmonic in punctured disk ?
- also $ln|z|$ has no harmonic conjugate in $Bbb Csetminus{0 }$ but has in $Bbb Csetminus[0, infty)$.
complex-analysis harmonic-functions multivalued-functions
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
asked Jan 9 at 14:55
HenryHenry
327
$endgroup$
closed as off-topic by José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh Jan 10 at 7:23
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
1.
how can i show that $ln|z|$ is harmonic in punctured disk ?
- also $ln|z|$ has no harmonic conjugate in $Bbb Csetminus{0 }$ but has in $Bbb Csetminus[0, infty)$.
complex-analysis harmonic-functions multivalued-functions
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
asked Jan 9 at 14:55
HenryHenry
327
$endgroup$
1.
how can i show that $ln|z|$ is harmonic in punctured disk ?
- also $ln|z|$ has no harmonic conjugate in $Bbb Csetminus{0 }$ but has in $Bbb Csetminus[0, infty)$.
complex-analysis harmonic-functions multivalued-functions
complex-analysis harmonic-functions multivalued-functions
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
asked Jan 9 at 14:55
HenryHenry
327
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
asked Jan 9 at 14:55
HenryHenry
327
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
edited Jan 9 at 15:45
David C. Ullrich
61.6k43995
61.6k43995
asked Jan 9 at 14:55
HenryHenry
327
asked Jan 9 at 14:55
HenryHenry
327
asked Jan 9 at 14:55
HenryHenry
327
327
closed as off-topic by José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh Jan 10 at 7:23
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh Jan 10 at 7:23
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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The second part of (2) should be clear, since $Bbb Csetminus[0,infty)$ is simply connected. Hint for the first part:
Lemma 1 If $f$ is holomorphic in an open set $V$ then $int_gamma f'(z),dz=0$ for any smooth closed curve $gamma$ in $V$.
Lemma 2. If $f=u+iv$ is holomorphic in some open set, where $u(z)=ln|z|$, then $f'(z)=1/z$.
Hint for Lemma 2: The Cauchy-Riemann equations show that $$f=frac12(u_x-iu_y).$$(You can simplfy the calculation of $u_x$ and $u_y$ by writing $u(z)=frac12ln(x^2+y^2)$.)
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
$endgroup$
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
add a comment |
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$z mapsto ln |z|$ is locally the real part of a determination of the complex logarithm, so it is locally harmonic, therefore harmonic on the whole punctured disc.
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The second part of (2) should be clear, since $Bbb Csetminus[0,infty)$ is simply connected. Hint for the first part:
Lemma 1 If $f$ is holomorphic in an open set $V$ then $int_gamma f'(z),dz=0$ for any smooth closed curve $gamma$ in $V$.
Lemma 2. If $f=u+iv$ is holomorphic in some open set, where $u(z)=ln|z|$, then $f'(z)=1/z$.
Hint for Lemma 2: The Cauchy-Riemann equations show that $$f=frac12(u_x-iu_y).$$(You can simplfy the calculation of $u_x$ and $u_y$ by writing $u(z)=frac12ln(x^2+y^2)$.)
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
$endgroup$
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
add a comment |
$begingroup$
The second part of (2) should be clear, since $Bbb Csetminus[0,infty)$ is simply connected. Hint for the first part:
Lemma 1 If $f$ is holomorphic in an open set $V$ then $int_gamma f'(z),dz=0$ for any smooth closed curve $gamma$ in $V$.
Lemma 2. If $f=u+iv$ is holomorphic in some open set, where $u(z)=ln|z|$, then $f'(z)=1/z$.
Hint for Lemma 2: The Cauchy-Riemann equations show that $$f=frac12(u_x-iu_y).$$(You can simplfy the calculation of $u_x$ and $u_y$ by writing $u(z)=frac12ln(x^2+y^2)$.)
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
$endgroup$
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
add a comment |
$begingroup$
The second part of (2) should be clear, since $Bbb Csetminus[0,infty)$ is simply connected. Hint for the first part:
Lemma 1 If $f$ is holomorphic in an open set $V$ then $int_gamma f'(z),dz=0$ for any smooth closed curve $gamma$ in $V$.
Lemma 2. If $f=u+iv$ is holomorphic in some open set, where $u(z)=ln|z|$, then $f'(z)=1/z$.
Hint for Lemma 2: The Cauchy-Riemann equations show that $$f=frac12(u_x-iu_y).$$(You can simplfy the calculation of $u_x$ and $u_y$ by writing $u(z)=frac12ln(x^2+y^2)$.)
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
$endgroup$
The second part of (2) should be clear, since $Bbb Csetminus[0,infty)$ is simply connected. Hint for the first part:
Lemma 1 If $f$ is holomorphic in an open set $V$ then $int_gamma f'(z),dz=0$ for any smooth closed curve $gamma$ in $V$.
Lemma 2. If $f=u+iv$ is holomorphic in some open set, where $u(z)=ln|z|$, then $f'(z)=1/z$.
Hint for Lemma 2: The Cauchy-Riemann equations show that $$f=frac12(u_x-iu_y).$$(You can simplfy the calculation of $u_x$ and $u_y$ by writing $u(z)=frac12ln(x^2+y^2)$.)
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
answered Jan 9 at 16:06
David C. UllrichDavid C. Ullrich
61.6k43995
61.6k43995
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
add a comment |
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
$begingroup$
thankyou i got it
$endgroup$
– Henry
Jan 9 at 16:09
add a comment |
$begingroup$
$z mapsto ln |z|$ is locally the real part of a determination of the complex logarithm, so it is locally harmonic, therefore harmonic on the whole punctured disc.
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
$endgroup$
add a comment |
$begingroup$
$z mapsto ln |z|$ is locally the real part of a determination of the complex logarithm, so it is locally harmonic, therefore harmonic on the whole punctured disc.
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
$endgroup$
add a comment |
$begingroup$
$z mapsto ln |z|$ is locally the real part of a determination of the complex logarithm, so it is locally harmonic, therefore harmonic on the whole punctured disc.
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
$endgroup$
$z mapsto ln |z|$ is locally the real part of a determination of the complex logarithm, so it is locally harmonic, therefore harmonic on the whole punctured disc.
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
answered Jan 9 at 15:15
A. BailleulA. Bailleul
1587
1587
add a comment |
add a comment |
