Determining branch points and closed paths
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My question is: how do we determine, computationally, when a function changes it’s value after we complete a closed path around a potential branch point?
To be specific: take the function $f(z)=log(z^{2}-1)=log(z-1)+log(z+1)$. The branch points are $z=1, z=-1$, and $z=infty$. Now, when we show that $z=1$ is a branch point, we want to show that as we travel around the point $z=1$, on a closed path, $log(z-1)$ changes by a multiple of $2pi i$ but $log(z+1)$ returns to its original value. How do we, computationally, show that a closed path around $z=1$ changes $log(z-1)$ by a multiple of $2pi i$?
complex-analysis complex-numbers
asked yesterday
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410213
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My question is: how do we determine, computationally, when a function changes it’s value after we complete a closed path around a potential branch point?
To be specific: take the function $f(z)=log(z^{2}-1)=log(z-1)+log(z+1)$. The branch points are $z=1, z=-1$, and $z=infty$. Now, when we show that $z=1$ is a branch point, we want to show that as we travel around the point $z=1$, on a closed path, $log(z-1)$ changes by a multiple of $2pi i$ but $log(z+1)$ returns to its original value. How do we, computationally, show that a closed path around $z=1$ changes $log(z-1)$ by a multiple of $2pi i$?
complex-analysis complex-numbers
asked yesterday
Live Free or π Hard
410213
The function will have a discontinuity along some path. For instance, $log z$ computed as $logsqrt{x^2+y^2}+iarctan_2(y, x)$ will has a jump of $i2pi$ when crossing the negative axis.
– Yves Daoust
yesterday
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up vote
0
down vote
favorite
up vote
0
down vote
favorite
My question is: how do we determine, computationally, when a function changes it’s value after we complete a closed path around a potential branch point?
To be specific: take the function $f(z)=log(z^{2}-1)=log(z-1)+log(z+1)$. The branch points are $z=1, z=-1$, and $z=infty$. Now, when we show that $z=1$ is a branch point, we want to show that as we travel around the point $z=1$, on a closed path, $log(z-1)$ changes by a multiple of $2pi i$ but $log(z+1)$ returns to its original value. How do we, computationally, show that a closed path around $z=1$ changes $log(z-1)$ by a multiple of $2pi i$?
complex-analysis complex-numbers
asked yesterday
Live Free or π Hard
410213
My question is: how do we determine, computationally, when a function changes it’s value after we complete a closed path around a potential branch point?
To be specific: take the function $f(z)=log(z^{2}-1)=log(z-1)+log(z+1)$. The branch points are $z=1, z=-1$, and $z=infty$. Now, when we show that $z=1$ is a branch point, we want to show that as we travel around the point $z=1$, on a closed path, $log(z-1)$ changes by a multiple of $2pi i$ but $log(z+1)$ returns to its original value. How do we, computationally, show that a closed path around $z=1$ changes $log(z-1)$ by a multiple of $2pi i$?
complex-analysis complex-numbers
complex-analysis complex-numbers
asked yesterday
Live Free or π Hard
410213
asked yesterday
Live Free or π Hard
410213
asked yesterday
Live Free or π Hard
410213
asked yesterday
Live Free or π Hard
410213
asked yesterday
Live Free or π Hard
410213
410213
The function will have a discontinuity along some path. For instance, $log z$ computed as $logsqrt{x^2+y^2}+iarctan_2(y, x)$ will has a jump of $i2pi$ when crossing the negative axis.
– Yves Daoust
yesterday
add a comment |
The function will have a discontinuity along some path. For instance, $log z$ computed as $logsqrt{x^2+y^2}+iarctan_2(y, x)$ will has a jump of $i2pi$ when crossing the negative axis.
– Yves Daoust
yesterday
The function will have a discontinuity along some path. For instance, $log z$ computed as $logsqrt{x^2+y^2}+iarctan_2(y, x)$ will has a jump of $i2pi$ when crossing the negative axis.
– Yves Daoust
yesterday
The function will have a discontinuity along some path. For instance, $log z$ computed as $logsqrt{x^2+y^2}+iarctan_2(y, x)$ will has a jump of $i2pi$ when crossing the negative axis.
– Yves Daoust
yesterday
add a comment |
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The function will have a discontinuity along some path. For instance, $log z$ computed as $logsqrt{x^2+y^2}+iarctan_2(y, x)$ will has a jump of $i2pi$ when crossing the negative axis.
– Yves Daoust
yesterday