To show that the function is not holomorphic at origin but satisfies the Cauchy-Reimann equation
$begingroup$
Let $$ f(z) = left{ begin{align}
&e^{-frac{1}{z^4}} &hspace{1mm} mbox{if} hspace{1mm} z neq 0 \
&0 &hspace{1mm} mbox{if} hspace{1mm} z = 0 \
end{align} right. $$
I have to show that this function satisfies the Cauchy-Riemann equation at $z=0$, but is not holomorphic.
I have been able to prove that it satisfies Cauchy-Riemann. But I am struggling with proving that it is not holomorphic at $z=0$.
My attempt:
Let $z=x+iy$ where $x,yinmathbb{R}$. Then
$$frac{1}{z}=frac{x-iy}{x^2+y^2}.$$
So
$$frac{1}{z^4}=frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}+ifrac{4xy^3-4x^3y}{(x^2+y^2)^4}.$$
Then
$$e^{-frac{1}{z^4}}=e^{-frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}}left(cosleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)-i sinleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)right)$$
My idea is the following:
If we can show that $f$ is not continuous at $z=0$, then it will follow automatically that $f$ is not holomorphic.
To show that $f$ is not continuous, I am trying to show that $f$ has different limits as $(x,y)to (0,0)$ along different paths. I have tried the paths $y=x$, $y=-x$, $y=x^2$. But all these paths give the limit to be $0$, which is the value of $f$ at $z=0$.
(1) Is my idea wrong and the function is continuous at $z=0$?
(2) If $f$ is not continuous, which path can be used to prove it?
complex-analysis holomorphic-functions
$endgroup$
|
show 5 more comments
$begingroup$
Let $$ f(z) = left{ begin{align}
&e^{-frac{1}{z^4}} &hspace{1mm} mbox{if} hspace{1mm} z neq 0 \
&0 &hspace{1mm} mbox{if} hspace{1mm} z = 0 \
end{align} right. $$
I have to show that this function satisfies the Cauchy-Riemann equation at $z=0$, but is not holomorphic.
I have been able to prove that it satisfies Cauchy-Riemann. But I am struggling with proving that it is not holomorphic at $z=0$.
My attempt:
Let $z=x+iy$ where $x,yinmathbb{R}$. Then
$$frac{1}{z}=frac{x-iy}{x^2+y^2}.$$
So
$$frac{1}{z^4}=frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}+ifrac{4xy^3-4x^3y}{(x^2+y^2)^4}.$$
Then
$$e^{-frac{1}{z^4}}=e^{-frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}}left(cosleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)-i sinleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)right)$$
My idea is the following:
If we can show that $f$ is not continuous at $z=0$, then it will follow automatically that $f$ is not holomorphic.
To show that $f$ is not continuous, I am trying to show that $f$ has different limits as $(x,y)to (0,0)$ along different paths. I have tried the paths $y=x$, $y=-x$, $y=x^2$. But all these paths give the limit to be $0$, which is the value of $f$ at $z=0$.
(1) Is my idea wrong and the function is continuous at $z=0$?
(2) If $f$ is not continuous, which path can be used to prove it?
complex-analysis holomorphic-functions
$endgroup$
1
$begingroup$
If it were holomorphic it would have a power-series expansion around zero. What can you say about the coefficients?
$endgroup$
– uniquesolution
Sep 7 '17 at 7:47
1
$begingroup$
@uniquesolution I suppose that the OP meant to say that it is not differentiable at the origin.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:49
1
$begingroup$
@Jose Carlos Santos But it is differentiable at the origin. In fact, it is differentiable infinitely many times at the origin, but it is not holmorphic there. (holomorphic=analytic). I suppose the OP meant to say exactly what (s)he said.
$endgroup$
– uniquesolution
Sep 7 '17 at 7:53
1
$begingroup$
@uniquesolution No, it is not differentiable at the origin. It is not even continuous there.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:55
1
$begingroup$
Yes you are right -- I assumed - mistakenly - that $z$ is real.
$endgroup$
– uniquesolution
Sep 7 '17 at 8:02
|
show 5 more comments
$begingroup$
Let $$ f(z) = left{ begin{align}
&e^{-frac{1}{z^4}} &hspace{1mm} mbox{if} hspace{1mm} z neq 0 \
&0 &hspace{1mm} mbox{if} hspace{1mm} z = 0 \
end{align} right. $$
I have to show that this function satisfies the Cauchy-Riemann equation at $z=0$, but is not holomorphic.
I have been able to prove that it satisfies Cauchy-Riemann. But I am struggling with proving that it is not holomorphic at $z=0$.
My attempt:
Let $z=x+iy$ where $x,yinmathbb{R}$. Then
$$frac{1}{z}=frac{x-iy}{x^2+y^2}.$$
So
$$frac{1}{z^4}=frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}+ifrac{4xy^3-4x^3y}{(x^2+y^2)^4}.$$
Then
$$e^{-frac{1}{z^4}}=e^{-frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}}left(cosleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)-i sinleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)right)$$
My idea is the following:
If we can show that $f$ is not continuous at $z=0$, then it will follow automatically that $f$ is not holomorphic.
To show that $f$ is not continuous, I am trying to show that $f$ has different limits as $(x,y)to (0,0)$ along different paths. I have tried the paths $y=x$, $y=-x$, $y=x^2$. But all these paths give the limit to be $0$, which is the value of $f$ at $z=0$.
(1) Is my idea wrong and the function is continuous at $z=0$?
(2) If $f$ is not continuous, which path can be used to prove it?
complex-analysis holomorphic-functions
$endgroup$
Let $$ f(z) = left{ begin{align}
&e^{-frac{1}{z^4}} &hspace{1mm} mbox{if} hspace{1mm} z neq 0 \
&0 &hspace{1mm} mbox{if} hspace{1mm} z = 0 \
end{align} right. $$
I have to show that this function satisfies the Cauchy-Riemann equation at $z=0$, but is not holomorphic.
I have been able to prove that it satisfies Cauchy-Riemann. But I am struggling with proving that it is not holomorphic at $z=0$.
My attempt:
Let $z=x+iy$ where $x,yinmathbb{R}$. Then
$$frac{1}{z}=frac{x-iy}{x^2+y^2}.$$
So
$$frac{1}{z^4}=frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}+ifrac{4xy^3-4x^3y}{(x^2+y^2)^4}.$$
Then
$$e^{-frac{1}{z^4}}=e^{-frac{x^4+y^4-6x^2y^2}{(x^2+y^2)^4}}left(cosleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)-i sinleft(frac{4xy^3-4x^3y}{(x^2+y^2)^4} right)right)$$
My idea is the following:
If we can show that $f$ is not continuous at $z=0$, then it will follow automatically that $f$ is not holomorphic.
To show that $f$ is not continuous, I am trying to show that $f$ has different limits as $(x,y)to (0,0)$ along different paths. I have tried the paths $y=x$, $y=-x$, $y=x^2$. But all these paths give the limit to be $0$, which is the value of $f$ at $z=0$.
(1) Is my idea wrong and the function is continuous at $z=0$?
(2) If $f$ is not continuous, which path can be used to prove it?
complex-analysis holomorphic-functions
complex-analysis holomorphic-functions
edited Sep 7 '17 at 8:06
José Carlos Santos
163k22131234
163k22131234
asked Sep 7 '17 at 7:43
learning_mathlearning_math
1,4531417
1,4531417
1
$begingroup$
If it were holomorphic it would have a power-series expansion around zero. What can you say about the coefficients?
$endgroup$
– uniquesolution
Sep 7 '17 at 7:47
1
$begingroup$
@uniquesolution I suppose that the OP meant to say that it is not differentiable at the origin.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:49
1
$begingroup$
@Jose Carlos Santos But it is differentiable at the origin. In fact, it is differentiable infinitely many times at the origin, but it is not holmorphic there. (holomorphic=analytic). I suppose the OP meant to say exactly what (s)he said.
$endgroup$
– uniquesolution
Sep 7 '17 at 7:53
1
$begingroup$
@uniquesolution No, it is not differentiable at the origin. It is not even continuous there.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:55
1
$begingroup$
Yes you are right -- I assumed - mistakenly - that $z$ is real.
$endgroup$
– uniquesolution
Sep 7 '17 at 8:02
|
show 5 more comments
1
$begingroup$
If it were holomorphic it would have a power-series expansion around zero. What can you say about the coefficients?
$endgroup$
– uniquesolution
Sep 7 '17 at 7:47
1
$begingroup$
@uniquesolution I suppose that the OP meant to say that it is not differentiable at the origin.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:49
1
$begingroup$
@Jose Carlos Santos But it is differentiable at the origin. In fact, it is differentiable infinitely many times at the origin, but it is not holmorphic there. (holomorphic=analytic). I suppose the OP meant to say exactly what (s)he said.
$endgroup$
– uniquesolution
Sep 7 '17 at 7:53
1
$begingroup$
@uniquesolution No, it is not differentiable at the origin. It is not even continuous there.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:55
1
$begingroup$
Yes you are right -- I assumed - mistakenly - that $z$ is real.
$endgroup$
– uniquesolution
Sep 7 '17 at 8:02
1
1
$begingroup$
If it were holomorphic it would have a power-series expansion around zero. What can you say about the coefficients?
$endgroup$
– uniquesolution
Sep 7 '17 at 7:47
$begingroup$
If it were holomorphic it would have a power-series expansion around zero. What can you say about the coefficients?
$endgroup$
– uniquesolution
Sep 7 '17 at 7:47
1
1
$begingroup$
@uniquesolution I suppose that the OP meant to say that it is not differentiable at the origin.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:49
$begingroup$
@uniquesolution I suppose that the OP meant to say that it is not differentiable at the origin.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:49
1
1
$begingroup$
@Jose Carlos Santos But it is differentiable at the origin. In fact, it is differentiable infinitely many times at the origin, but it is not holmorphic there. (holomorphic=analytic). I suppose the OP meant to say exactly what (s)he said.
$endgroup$
– uniquesolution
Sep 7 '17 at 7:53
$begingroup$
@Jose Carlos Santos But it is differentiable at the origin. In fact, it is differentiable infinitely many times at the origin, but it is not holmorphic there. (holomorphic=analytic). I suppose the OP meant to say exactly what (s)he said.
$endgroup$
– uniquesolution
Sep 7 '17 at 7:53
1
1
$begingroup$
@uniquesolution No, it is not differentiable at the origin. It is not even continuous there.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:55
$begingroup$
@uniquesolution No, it is not differentiable at the origin. It is not even continuous there.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:55
1
1
$begingroup$
Yes you are right -- I assumed - mistakenly - that $z$ is real.
$endgroup$
– uniquesolution
Sep 7 '17 at 8:02
$begingroup$
Yes you are right -- I assumed - mistakenly - that $z$ is real.
$endgroup$
– uniquesolution
Sep 7 '17 at 8:02
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The function $f$ is not even continuous at $0$. If $z=rholeft(cosleft(fracpi8right)+sinleft(fracpi8right)iright)$ with $rho>0$, then $z^4=rho^4i$ and therefore$$bigl|f(z)bigr|=leftlvertexpleft(-frac1{z^4}right)rightrvert=leftlvertexpleft(frac i{rho^4}right)rightrvert=1.$$So, $f(z)$ doesn't even get near $0$ when $z$ belongs to the ray with origin at $0$ and passing through $cosleft(fracpi8right)+sinleft(fracpi8right)i$.
$endgroup$
add a comment |
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$begingroup$
The function $f$ is not even continuous at $0$. If $z=rholeft(cosleft(fracpi8right)+sinleft(fracpi8right)iright)$ with $rho>0$, then $z^4=rho^4i$ and therefore$$bigl|f(z)bigr|=leftlvertexpleft(-frac1{z^4}right)rightrvert=leftlvertexpleft(frac i{rho^4}right)rightrvert=1.$$So, $f(z)$ doesn't even get near $0$ when $z$ belongs to the ray with origin at $0$ and passing through $cosleft(fracpi8right)+sinleft(fracpi8right)i$.
$endgroup$
add a comment |
$begingroup$
The function $f$ is not even continuous at $0$. If $z=rholeft(cosleft(fracpi8right)+sinleft(fracpi8right)iright)$ with $rho>0$, then $z^4=rho^4i$ and therefore$$bigl|f(z)bigr|=leftlvertexpleft(-frac1{z^4}right)rightrvert=leftlvertexpleft(frac i{rho^4}right)rightrvert=1.$$So, $f(z)$ doesn't even get near $0$ when $z$ belongs to the ray with origin at $0$ and passing through $cosleft(fracpi8right)+sinleft(fracpi8right)i$.
$endgroup$
add a comment |
$begingroup$
The function $f$ is not even continuous at $0$. If $z=rholeft(cosleft(fracpi8right)+sinleft(fracpi8right)iright)$ with $rho>0$, then $z^4=rho^4i$ and therefore$$bigl|f(z)bigr|=leftlvertexpleft(-frac1{z^4}right)rightrvert=leftlvertexpleft(frac i{rho^4}right)rightrvert=1.$$So, $f(z)$ doesn't even get near $0$ when $z$ belongs to the ray with origin at $0$ and passing through $cosleft(fracpi8right)+sinleft(fracpi8right)i$.
$endgroup$
The function $f$ is not even continuous at $0$. If $z=rholeft(cosleft(fracpi8right)+sinleft(fracpi8right)iright)$ with $rho>0$, then $z^4=rho^4i$ and therefore$$bigl|f(z)bigr|=leftlvertexpleft(-frac1{z^4}right)rightrvert=leftlvertexpleft(frac i{rho^4}right)rightrvert=1.$$So, $f(z)$ doesn't even get near $0$ when $z$ belongs to the ray with origin at $0$ and passing through $cosleft(fracpi8right)+sinleft(fracpi8right)i$.
edited Jan 8 at 19:10
answered Sep 7 '17 at 7:55
José Carlos SantosJosé Carlos Santos
163k22131234
163k22131234
add a comment |
add a comment |
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$begingroup$
If it were holomorphic it would have a power-series expansion around zero. What can you say about the coefficients?
$endgroup$
– uniquesolution
Sep 7 '17 at 7:47
1
$begingroup$
@uniquesolution I suppose that the OP meant to say that it is not differentiable at the origin.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:49
1
$begingroup$
@Jose Carlos Santos But it is differentiable at the origin. In fact, it is differentiable infinitely many times at the origin, but it is not holmorphic there. (holomorphic=analytic). I suppose the OP meant to say exactly what (s)he said.
$endgroup$
– uniquesolution
Sep 7 '17 at 7:53
1
$begingroup$
@uniquesolution No, it is not differentiable at the origin. It is not even continuous there.
$endgroup$
– José Carlos Santos
Sep 7 '17 at 7:55
1
$begingroup$
Yes you are right -- I assumed - mistakenly - that $z$ is real.
$endgroup$
– uniquesolution
Sep 7 '17 at 8:02