Find the minimum value of the Hermitian polynomial
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Let $ainmathbb{R}$, $binmathbb{C}$, and $c>0$. Find the minimum of the Hermitian polynomial $R$:
$R(t,bar t)=a+bt+bar b bar t+c|t|^2$.
The only progress I have made is rewriting the function as : $(a+c|t|^2)+ 2Re(bt)$. My problem is that, from what I remember about complex differentiation, is that $f(z)=|z|^2$ is only differentiable at the origin and $g(z)=Re(z)$ is not differentiable.
There must be a way to work with the the differential equation involving $R$.
complex-analysis
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
$endgroup$
add a comment |
$begingroup$
Let $ainmathbb{R}$, $binmathbb{C}$, and $c>0$. Find the minimum of the Hermitian polynomial $R$:
$R(t,bar t)=a+bt+bar b bar t+c|t|^2$.
The only progress I have made is rewriting the function as : $(a+c|t|^2)+ 2Re(bt)$. My problem is that, from what I remember about complex differentiation, is that $f(z)=|z|^2$ is only differentiable at the origin and $g(z)=Re(z)$ is not differentiable.
There must be a way to work with the the differential equation involving $R$.
complex-analysis
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
$endgroup$
$begingroup$
Do you want $cdots+overline boverline t+cdots$?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:00
$begingroup$
Yes. Thank you.
$endgroup$
– Jungleshrimp
Jan 15 at 2:00
$begingroup$
Try writing it as $c|t-w|^2+d$ for suitable complex $w$ and real $d$.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:01
$begingroup$
Could you elaborate on that a bit?
$endgroup$
– Jungleshrimp
Jan 15 at 2:10
$begingroup$
I am getting $a+(c-2)|b|^2$
$endgroup$
– Jungleshrimp
Jan 15 at 2:43
add a comment |
$begingroup$
Let $ainmathbb{R}$, $binmathbb{C}$, and $c>0$. Find the minimum of the Hermitian polynomial $R$:
$R(t,bar t)=a+bt+bar b bar t+c|t|^2$.
The only progress I have made is rewriting the function as : $(a+c|t|^2)+ 2Re(bt)$. My problem is that, from what I remember about complex differentiation, is that $f(z)=|z|^2$ is only differentiable at the origin and $g(z)=Re(z)$ is not differentiable.
There must be a way to work with the the differential equation involving $R$.
complex-analysis
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
$endgroup$
Let $ainmathbb{R}$, $binmathbb{C}$, and $c>0$. Find the minimum of the Hermitian polynomial $R$:
$R(t,bar t)=a+bt+bar b bar t+c|t|^2$.
The only progress I have made is rewriting the function as : $(a+c|t|^2)+ 2Re(bt)$. My problem is that, from what I remember about complex differentiation, is that $f(z)=|z|^2$ is only differentiable at the origin and $g(z)=Re(z)$ is not differentiable.
There must be a way to work with the the differential equation involving $R$.
complex-analysis
complex-analysis
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
edited Jan 15 at 2:01
Jungleshrimp
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
asked Jan 15 at 1:52
JungleshrimpJungleshrimp
366112
366112
$begingroup$
Do you want $cdots+overline boverline t+cdots$?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:00
$begingroup$
Yes. Thank you.
$endgroup$
– Jungleshrimp
Jan 15 at 2:00
$begingroup$
Try writing it as $c|t-w|^2+d$ for suitable complex $w$ and real $d$.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:01
$begingroup$
Could you elaborate on that a bit?
$endgroup$
– Jungleshrimp
Jan 15 at 2:10
$begingroup$
I am getting $a+(c-2)|b|^2$
$endgroup$
– Jungleshrimp
Jan 15 at 2:43
add a comment |
$begingroup$
Do you want $cdots+overline boverline t+cdots$?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:00
$begingroup$
Yes. Thank you.
$endgroup$
– Jungleshrimp
Jan 15 at 2:00
$begingroup$
Try writing it as $c|t-w|^2+d$ for suitable complex $w$ and real $d$.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:01
$begingroup$
Could you elaborate on that a bit?
$endgroup$
– Jungleshrimp
Jan 15 at 2:10
$begingroup$
I am getting $a+(c-2)|b|^2$
$endgroup$
– Jungleshrimp
Jan 15 at 2:43
$begingroup$
Do you want $cdots+overline boverline t+cdots$?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:00
$begingroup$
Do you want $cdots+overline boverline t+cdots$?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:00
$begingroup$
Yes. Thank you.
$endgroup$
– Jungleshrimp
Jan 15 at 2:00
$begingroup$
Yes. Thank you.
$endgroup$
– Jungleshrimp
Jan 15 at 2:00
$begingroup$
Try writing it as $c|t-w|^2+d$ for suitable complex $w$ and real $d$.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:01
$begingroup$
Try writing it as $c|t-w|^2+d$ for suitable complex $w$ and real $d$.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:01
$begingroup$
Could you elaborate on that a bit?
$endgroup$
– Jungleshrimp
Jan 15 at 2:10
$begingroup$
Could you elaborate on that a bit?
$endgroup$
– Jungleshrimp
Jan 15 at 2:10
$begingroup$
I am getting $a+(c-2)|b|^2$
$endgroup$
– Jungleshrimp
Jan 15 at 2:43
$begingroup$
I am getting $a+(c-2)|b|^2$
$endgroup$
– Jungleshrimp
Jan 15 at 2:43
add a comment |
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$begingroup$
Do you want $cdots+overline boverline t+cdots$?
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:00
$begingroup$
Yes. Thank you.
$endgroup$
– Jungleshrimp
Jan 15 at 2:00
$begingroup$
Try writing it as $c|t-w|^2+d$ for suitable complex $w$ and real $d$.
$endgroup$
– Lord Shark the Unknown
Jan 15 at 2:01
$begingroup$
Could you elaborate on that a bit?
$endgroup$
– Jungleshrimp
Jan 15 at 2:10
$begingroup$
I am getting $a+(c-2)|b|^2$
$endgroup$
– Jungleshrimp
Jan 15 at 2:43