Understanding the change in $arg(z-i)$ around a closed curve containing the point $i$.
I understand why, for any complex number $z=re^{itheta}$, as $z$ goes once around a closed curved $C$, counterclockwise, enclosing the point 0, the principal argument of $z$ increases by $2pi$.
I need help in understanding why the argument of $z-i=Re^{iphi}$ changes by $2pi$ when $z$ goes once around a closed curved enclosing the point $i$.
complex-analysis complex-numbers
edited Nov 29 '18 at 21:55
Bernard
118k639112
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
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add a comment |
I understand why, for any complex number $z=re^{itheta}$, as $z$ goes once around a closed curved $C$, counterclockwise, enclosing the point 0, the principal argument of $z$ increases by $2pi$.
I need help in understanding why the argument of $z-i=Re^{iphi}$ changes by $2pi$ when $z$ goes once around a closed curved enclosing the point $i$.
complex-analysis complex-numbers
edited Nov 29 '18 at 21:55
Bernard
118k639112
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
1
Imagine a translation which sends $0$ to $i$. Set $w=z-i.$ In fact is $w$ the number from the first part (going around $0$), the situation you understand.
– user376343
Nov 29 '18 at 21:44
1
Use your understanding of the first to prove the second.
– N74
Nov 29 '18 at 21:45
add a comment |
I understand why, for any complex number $z=re^{itheta}$, as $z$ goes once around a closed curved $C$, counterclockwise, enclosing the point 0, the principal argument of $z$ increases by $2pi$.
I need help in understanding why the argument of $z-i=Re^{iphi}$ changes by $2pi$ when $z$ goes once around a closed curved enclosing the point $i$.
complex-analysis complex-numbers
edited Nov 29 '18 at 21:55
Bernard
118k639112
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
I understand why, for any complex number $z=re^{itheta}$, as $z$ goes once around a closed curved $C$, counterclockwise, enclosing the point 0, the principal argument of $z$ increases by $2pi$.
I need help in understanding why the argument of $z-i=Re^{iphi}$ changes by $2pi$ when $z$ goes once around a closed curved enclosing the point $i$.
complex-analysis complex-numbers
complex-analysis complex-numbers
edited Nov 29 '18 at 21:55
Bernard
118k639112
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
edited Nov 29 '18 at 21:55
Bernard
118k639112
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
edited Nov 29 '18 at 21:55
Bernard
118k639112
edited Nov 29 '18 at 21:55
Bernard
118k639112
edited Nov 29 '18 at 21:55
Bernard
118k639112
118k639112
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
asked Nov 29 '18 at 21:33
Live Free or π HardLive Free or π Hard
479213
479213
1
Imagine a translation which sends $0$ to $i$. Set $w=z-i.$ In fact is $w$ the number from the first part (going around $0$), the situation you understand.
– user376343
Nov 29 '18 at 21:44
1
Use your understanding of the first to prove the second.
– N74
Nov 29 '18 at 21:45
add a comment |
1
Imagine a translation which sends $0$ to $i$. Set $w=z-i.$ In fact is $w$ the number from the first part (going around $0$), the situation you understand.
– user376343
Nov 29 '18 at 21:44
1
Use your understanding of the first to prove the second.
– N74
Nov 29 '18 at 21:45
1
1
Imagine a translation which sends $0$ to $i$. Set $w=z-i.$ In fact is $w$ the number from the first part (going around $0$), the situation you understand.
– user376343
Nov 29 '18 at 21:44
Imagine a translation which sends $0$ to $i$. Set $w=z-i.$ In fact is $w$ the number from the first part (going around $0$), the situation you understand.
– user376343
Nov 29 '18 at 21:44
1
1
Use your understanding of the first to prove the second.
– N74
Nov 29 '18 at 21:45
Use your understanding of the first to prove the second.
– N74
Nov 29 '18 at 21:45
add a comment |
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Imagine a translation which sends $0$ to $i$. Set $w=z-i.$ In fact is $w$ the number from the first part (going around $0$), the situation you understand.
– user376343
Nov 29 '18 at 21:44
1
Use your understanding of the first to prove the second.
– N74
Nov 29 '18 at 21:45