Graph of complex exponential function
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I need to graph this set on a complex plane: $$A={zin C :|e^{iz}|ge|e^z|}$$
I know that: $$|e^{ix-y}|ge|e^{x+iy}|$$ $$|e^{-y}e^{ix}|ge|e^xe^{iy}|$$ $$|e^{-y}(cos x+isin x)|ge|e^x(cos y+isin y)|$$ $$e^{-y}ge e^x$$ $$-yge x$$
But I'm not sure how to graph it since $e^{x+iy}=e^x(cos y+isin y)$ and both $x$ and $y$ are changing. Can you give me any hints? Thank you in advance.
complex-analysis
edited Dec 20 '18 at 13:57
amWhy
1
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add a comment |
$begingroup$
I need to graph this set on a complex plane: $$A={zin C :|e^{iz}|ge|e^z|}$$
I know that: $$|e^{ix-y}|ge|e^{x+iy}|$$ $$|e^{-y}e^{ix}|ge|e^xe^{iy}|$$ $$|e^{-y}(cos x+isin x)|ge|e^x(cos y+isin y)|$$ $$e^{-y}ge e^x$$ $$-yge x$$
But I'm not sure how to graph it since $e^{x+iy}=e^x(cos y+isin y)$ and both $x$ and $y$ are changing. Can you give me any hints? Thank you in advance.
complex-analysis
edited Dec 20 '18 at 13:57
amWhy
1
$endgroup$
add a comment |
$begingroup$
I need to graph this set on a complex plane: $$A={zin C :|e^{iz}|ge|e^z|}$$
I know that: $$|e^{ix-y}|ge|e^{x+iy}|$$ $$|e^{-y}e^{ix}|ge|e^xe^{iy}|$$ $$|e^{-y}(cos x+isin x)|ge|e^x(cos y+isin y)|$$ $$e^{-y}ge e^x$$ $$-yge x$$
But I'm not sure how to graph it since $e^{x+iy}=e^x(cos y+isin y)$ and both $x$ and $y$ are changing. Can you give me any hints? Thank you in advance.
complex-analysis
edited Dec 20 '18 at 13:57
amWhy
1
$endgroup$
I need to graph this set on a complex plane: $$A={zin C :|e^{iz}|ge|e^z|}$$
I know that: $$|e^{ix-y}|ge|e^{x+iy}|$$ $$|e^{-y}e^{ix}|ge|e^xe^{iy}|$$ $$|e^{-y}(cos x+isin x)|ge|e^x(cos y+isin y)|$$ $$e^{-y}ge e^x$$ $$-yge x$$
But I'm not sure how to graph it since $e^{x+iy}=e^x(cos y+isin y)$ and both $x$ and $y$ are changing. Can you give me any hints? Thank you in advance.
complex-analysis
complex-analysis
edited Dec 20 '18 at 13:57
amWhy
1
edited Dec 20 '18 at 13:57
amWhy
1
edited Dec 20 '18 at 13:57
amWhy
1
edited Dec 20 '18 at 13:57
amWhy
1
edited Dec 20 '18 at 13:57
amWhy
1
1
asked Dec 20 '18 at 12:48
user609637
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1 Answer
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Hint: You are graphing the set $A$, not a function. $-y = x$ is a line in the $xy$ plane, and $-y ge x$ is the region on one side of that line.
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
add a comment |
Your Answer
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1 Answer
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oldest
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1 Answer
1
active
oldest
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active
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active
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votes
$begingroup$
Hint: You are graphing the set $A$, not a function. $-y = x$ is a line in the $xy$ plane, and $-y ge x$ is the region on one side of that line.
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
add a comment |
$begingroup$
Hint: You are graphing the set $A$, not a function. $-y = x$ is a line in the $xy$ plane, and $-y ge x$ is the region on one side of that line.
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
add a comment |
$begingroup$
Hint: You are graphing the set $A$, not a function. $-y = x$ is a line in the $xy$ plane, and $-y ge x$ is the region on one side of that line.
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
$endgroup$
Hint: You are graphing the set $A$, not a function. $-y = x$ is a line in the $xy$ plane, and $-y ge x$ is the region on one side of that line.
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
answered Dec 20 '18 at 13:01
Robert IsraelRobert Israel
321k23210463
321k23210463
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
add a comment |
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
Hello, thank you for your help. So if an exp function transforms a vertical line in a complex plane to a ray, then in this case it would be just an inverse? I'm referring to this: mathonline.wikidot.com/…
$endgroup$
– user609637
Dec 20 '18 at 13:10
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
$begingroup$
What would be an inverse of what? I repeat, you are graphing the set $A$. You already found what that set is.
$endgroup$
– Robert Israel
Dec 20 '18 at 14:56
add a comment |
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