Denoting the domain of a complex function
$begingroup$
If I have a function defined as so:
$$f(z)=z^2$$
Can I limit its domain based on its argument (angle in its $re^{itheta}$ form) like this?
$$f(re^{itheta}) = r^2e^{2itheta}, 0 leq theta < pi$$
Is this the correct notation and does this function become bijective if defined like this?
complex-analysis functions complex-numbers notation
asked Dec 17 '18 at 10:12
MatthewMatthew
316
$endgroup$
add a comment |
$begingroup$
If I have a function defined as so:
$$f(z)=z^2$$
Can I limit its domain based on its argument (angle in its $re^{itheta}$ form) like this?
$$f(re^{itheta}) = r^2e^{2itheta}, 0 leq theta < pi$$
Is this the correct notation and does this function become bijective if defined like this?
complex-analysis functions complex-numbers notation
asked Dec 17 '18 at 10:12
MatthewMatthew
316
$endgroup$
2
$begingroup$
Add $r>0$ and it seems good to me
$endgroup$
– N74
Dec 17 '18 at 10:16
add a comment |
$begingroup$
If I have a function defined as so:
$$f(z)=z^2$$
Can I limit its domain based on its argument (angle in its $re^{itheta}$ form) like this?
$$f(re^{itheta}) = r^2e^{2itheta}, 0 leq theta < pi$$
Is this the correct notation and does this function become bijective if defined like this?
complex-analysis functions complex-numbers notation
asked Dec 17 '18 at 10:12
MatthewMatthew
316
$endgroup$
If I have a function defined as so:
$$f(z)=z^2$$
Can I limit its domain based on its argument (angle in its $re^{itheta}$ form) like this?
$$f(re^{itheta}) = r^2e^{2itheta}, 0 leq theta < pi$$
Is this the correct notation and does this function become bijective if defined like this?
complex-analysis functions complex-numbers notation
complex-analysis functions complex-numbers notation
asked Dec 17 '18 at 10:12
MatthewMatthew
316
asked Dec 17 '18 at 10:12
MatthewMatthew
316
asked Dec 17 '18 at 10:12
MatthewMatthew
316
asked Dec 17 '18 at 10:12
MatthewMatthew
316
asked Dec 17 '18 at 10:12
MatthewMatthew
316
316
2
$begingroup$
Add $r>0$ and it seems good to me
$endgroup$
– N74
Dec 17 '18 at 10:16
add a comment |
2
$begingroup$
Add $r>0$ and it seems good to me
$endgroup$
– N74
Dec 17 '18 at 10:16
2
2
$begingroup$
Add $r>0$ and it seems good to me
$endgroup$
– N74
Dec 17 '18 at 10:16
$begingroup$
Add $r>0$ and it seems good to me
$endgroup$
– N74
Dec 17 '18 at 10:16
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
It is a bijection from upper half plane to (including non-negative real axis) to $mathbb C $.
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
$endgroup$
1
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It is a bijection from upper half plane to (including non-negative real axis) to $mathbb C $.
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
$endgroup$
1
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
add a comment |
$begingroup$
It is a bijection from upper half plane to (including non-negative real axis) to $mathbb C $.
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
$endgroup$
1
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
add a comment |
$begingroup$
It is a bijection from upper half plane to (including non-negative real axis) to $mathbb C $.
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
$endgroup$
It is a bijection from upper half plane to (including non-negative real axis) to $mathbb C $.
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
edited Dec 17 '18 at 10:27
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
answered Dec 17 '18 at 10:15
Kavi Rama MurthyKavi Rama Murthy
68k53068
68k53068
1
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
add a comment |
1
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
1
1
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
$begingroup$
@YadatiKiran You are right. I have corrected my answer. Though $theta$ is arbitrary when $r=0$ the function is still well defined. Thanks for the comment.
$endgroup$
– Kavi Rama Murthy
Dec 17 '18 at 10:29
add a comment |
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$begingroup$
Add $r>0$ and it seems good to me
$endgroup$
– N74
Dec 17 '18 at 10:16