Relationship Between CR equations and Complex Differentiability
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Consider a complex function $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real-valued functions.
In an attempt to better understand complex differentiation and the Cauchy-Riemann equations, I considered two questions.
If $f$ is complex differentiable at a point $z_o$, do the Cauchy-Riemann equations have to hold?
Yes (although not really sure on reasoning).
If the Cauchy-Riemann equations hold at $z_o$, does $f$ have to be complex differentiable there?
No, this not a sufficient condition. Further continuity conditions are required (e.g. partial derivatives are continuous at $z_0$).
Am I correct? Advice on these two questions would be really appreciated.
complex-analysis derivatives
asked Aug 17 at 5:00
Bell
716313
add a comment |Â
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0
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Consider a complex function $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real-valued functions.
In an attempt to better understand complex differentiation and the Cauchy-Riemann equations, I considered two questions.
If $f$ is complex differentiable at a point $z_o$, do the Cauchy-Riemann equations have to hold?
Yes (although not really sure on reasoning).
If the Cauchy-Riemann equations hold at $z_o$, does $f$ have to be complex differentiable there?
No, this not a sufficient condition. Further continuity conditions are required (e.g. partial derivatives are continuous at $z_0$).
Am I correct? Advice on these two questions would be really appreciated.
complex-analysis derivatives
asked Aug 17 at 5:00
Bell
716313
For the second see math.stackexchange.com/questions/496068/…
– Nosrati
Aug 17 at 5:11
Nice link, thanks for that. What about the first question?
– Bell
Aug 17 at 5:49
the first one is a theorem in complex analysis.If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold.
– Nosrati
Aug 17 at 7:47
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider a complex function $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real-valued functions.
In an attempt to better understand complex differentiation and the Cauchy-Riemann equations, I considered two questions.
If $f$ is complex differentiable at a point $z_o$, do the Cauchy-Riemann equations have to hold?
Yes (although not really sure on reasoning).
If the Cauchy-Riemann equations hold at $z_o$, does $f$ have to be complex differentiable there?
No, this not a sufficient condition. Further continuity conditions are required (e.g. partial derivatives are continuous at $z_0$).
Am I correct? Advice on these two questions would be really appreciated.
complex-analysis derivatives
asked Aug 17 at 5:00
Bell
716313
Consider a complex function $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real-valued functions.
In an attempt to better understand complex differentiation and the Cauchy-Riemann equations, I considered two questions.
If $f$ is complex differentiable at a point $z_o$, do the Cauchy-Riemann equations have to hold?
Yes (although not really sure on reasoning).
If the Cauchy-Riemann equations hold at $z_o$, does $f$ have to be complex differentiable there?
No, this not a sufficient condition. Further continuity conditions are required (e.g. partial derivatives are continuous at $z_0$).
Am I correct? Advice on these two questions would be really appreciated.
complex-analysis derivatives
asked Aug 17 at 5:00
Bell
716313
asked Aug 17 at 5:00
Bell
716313
asked Aug 17 at 5:00
Bell
716313
asked Aug 17 at 5:00
Bell
716313
716313
For the second see math.stackexchange.com/questions/496068/…
– Nosrati
Aug 17 at 5:11
Nice link, thanks for that. What about the first question?
– Bell
Aug 17 at 5:49
the first one is a theorem in complex analysis.If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold.
– Nosrati
Aug 17 at 7:47
add a comment |Â
For the second see math.stackexchange.com/questions/496068/…
– Nosrati
Aug 17 at 5:11
Nice link, thanks for that. What about the first question?
– Bell
Aug 17 at 5:49
the first one is a theorem in complex analysis.If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold.
– Nosrati
Aug 17 at 7:47
For the second see math.stackexchange.com/questions/496068/…
– Nosrati
Aug 17 at 5:11
For the second see math.stackexchange.com/questions/496068/…
– Nosrati
Aug 17 at 5:11
Nice link, thanks for that. What about the first question?
– Bell
Aug 17 at 5:49
Nice link, thanks for that. What about the first question?
– Bell
Aug 17 at 5:49
the first one is a theorem in complex analysis.
If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold.– Nosrati
Aug 17 at 7:47
the first one is a theorem in complex analysis.
If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold.– Nosrati
Aug 17 at 7:47
add a comment |Â
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For the second see math.stackexchange.com/questions/496068/…
– Nosrati
Aug 17 at 5:11
Nice link, thanks for that. What about the first question?
– Bell
Aug 17 at 5:49
the first one is a theorem in complex analysis.
If f is complex differentiable at a point z_0, then the Cauchy-Riemann equations hold.– Nosrati
Aug 17 at 7:47