Calculating $partial_t|f(t,z)|^2$ with $tinmathbb R$, $zinmathbb C$ and $f(t,z)inmathbb C$

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Let $f: mathbb Rtimesmathbb Cto mathbb C$ be differentiable. What is the derivative of $|f(t,z)|^2$ w.r.t. $tinmathbb R$, i.e. how do you calculate $partial_t |f(t,z)|^2$?

I've tried to use $|f(t,z)|^2=Re(f(t,z))^2+Im(f(t,z))^2$ and hence $partial_t |f(t,z)|^2=2Re(f(t,z))partial_tf(t,z)+2Im(f(t,z))partial_tf(t,z)$ which doesn't seem to be right since it's complex valued now. Any hints?

calculus real-analysis complex-analysis

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edited Jul 31 at 13:00

asked Jul 31 at 12:44

leonard

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Let $f: mathbb Rtimesmathbb Cto mathbb C$ be differentiable. What is the derivative of $|f(t,z)|^2$ w.r.t. $tinmathbb R$, i.e. how do you calculate $partial_t |f(t,z)|^2$?

I've tried to use $|f(t,z)|^2=Re(f(t,z))^2+Im(f(t,z))^2$ and hence $partial_t |f(t,z)|^2=2Re(f(t,z))partial_tf(t,z)+2Im(f(t,z))partial_tf(t,z)$ which doesn't seem to be right since it's complex valued now. Any hints?

calculus real-analysis complex-analysis

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edited Jul 31 at 13:00

asked Jul 31 at 12:44

leonard

33

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Let $f: mathbb Rtimesmathbb Cto mathbb C$ be differentiable. What is the derivative of $|f(t,z)|^2$ w.r.t. $tinmathbb R$, i.e. how do you calculate $partial_t |f(t,z)|^2$?

I've tried to use $|f(t,z)|^2=Re(f(t,z))^2+Im(f(t,z))^2$ and hence $partial_t |f(t,z)|^2=2Re(f(t,z))partial_tf(t,z)+2Im(f(t,z))partial_tf(t,z)$ which doesn't seem to be right since it's complex valued now. Any hints?

calculus real-analysis complex-analysis

share|cite|improve this question

edited Jul 31 at 13:00

asked Jul 31 at 12:44

leonard

33

Let $f: mathbb Rtimesmathbb Cto mathbb C$ be differentiable. What is the derivative of $|f(t,z)|^2$ w.r.t. $tinmathbb R$, i.e. how do you calculate $partial_t |f(t,z)|^2$?

I've tried to use $|f(t,z)|^2=Re(f(t,z))^2+Im(f(t,z))^2$ and hence $partial_t |f(t,z)|^2=2Re(f(t,z))partial_tf(t,z)+2Im(f(t,z))partial_tf(t,z)$ which doesn't seem to be right since it's complex valued now. Any hints?

calculus real-analysis complex-analysis

share|cite|improve this question

edited Jul 31 at 13:00

asked Jul 31 at 12:44

leonard

33

share|cite|improve this question

share|cite|improve this question

edited Jul 31 at 13:00

edited Jul 31 at 13:00

edited Jul 31 at 13:00

asked Jul 31 at 12:44

leonard

33

asked Jul 31 at 12:44

leonard

33

asked Jul 31 at 12:44

leonard

33

33

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add a comment | 

2 Answers
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We have $|f(t,z)|^2=f(t,z)overlinef(t,z)$.

Hence :

$$partial_t|f(t,z)|^2=overlinef(t,z)partial_tf(t,z)+f(t,z)partial_toverlinef(t,z).$$

But :

$$partial_toverlinef(t,z)=overlinepartial_tf(t,z).$$

Finally :

$$partial_t|f(t,z)|^2= 2Releft( overlinef(t,z)partial_tf(t,z)right).$$

PS : not a very convenient formula, but I doubt one can do better without more assumptions.

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answered Jul 31 at 13:03

nicomezi

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Actually, $partial_tRe(f(t,z))^2=2Re(f(t,z)) cdot partial_t Re(f(t,z)) = 2Re(f(t,z)) cdot Re(partial_t f(t,z))$, so

$$partial_t |f(t,z)|^2=2Re(f(t,z)) cdot Re(partial_t f(t,z))+2Im(f(t,z)) cdot Im(partial_t f(t,z))$$

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answered Jul 31 at 13:06

lisyarus

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote



accepted

We have $|f(t,z)|^2=f(t,z)overlinef(t,z)$.

Hence :

$$partial_t|f(t,z)|^2=overlinef(t,z)partial_tf(t,z)+f(t,z)partial_toverlinef(t,z).$$

But :

$$partial_toverlinef(t,z)=overlinepartial_tf(t,z).$$

Finally :

$$partial_t|f(t,z)|^2= 2Releft( overlinef(t,z)partial_tf(t,z)right).$$

PS : not a very convenient formula, but I doubt one can do better without more assumptions.

share|cite|improve this answer

answered Jul 31 at 13:03

nicomezi

3,3871818

add a comment | 

up vote
0
down vote



accepted

We have $|f(t,z)|^2=f(t,z)overlinef(t,z)$.

Hence :

$$partial_t|f(t,z)|^2=overlinef(t,z)partial_tf(t,z)+f(t,z)partial_toverlinef(t,z).$$

But :

$$partial_toverlinef(t,z)=overlinepartial_tf(t,z).$$

Finally :

$$partial_t|f(t,z)|^2= 2Releft( overlinef(t,z)partial_tf(t,z)right).$$

PS : not a very convenient formula, but I doubt one can do better without more assumptions.

share|cite|improve this answer

answered Jul 31 at 13:03

nicomezi

3,3871818

add a comment | 

up vote
0
down vote



accepted

up vote
0
down vote



accepted

We have $|f(t,z)|^2=f(t,z)overlinef(t,z)$.

Hence :

$$partial_t|f(t,z)|^2=overlinef(t,z)partial_tf(t,z)+f(t,z)partial_toverlinef(t,z).$$

But :

$$partial_toverlinef(t,z)=overlinepartial_tf(t,z).$$

Finally :

$$partial_t|f(t,z)|^2= 2Releft( overlinef(t,z)partial_tf(t,z)right).$$

PS : not a very convenient formula, but I doubt one can do better without more assumptions.

share|cite|improve this answer

answered Jul 31 at 13:03

nicomezi

3,3871818

We have $|f(t,z)|^2=f(t,z)overlinef(t,z)$.

Hence :

$$partial_t|f(t,z)|^2=overlinef(t,z)partial_tf(t,z)+f(t,z)partial_toverlinef(t,z).$$

But :

$$partial_toverlinef(t,z)=overlinepartial_tf(t,z).$$

Finally :

$$partial_t|f(t,z)|^2= 2Releft( overlinef(t,z)partial_tf(t,z)right).$$

PS : not a very convenient formula, but I doubt one can do better without more assumptions.

share|cite|improve this answer

answered Jul 31 at 13:03

nicomezi

3,3871818

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 13:03

nicomezi

3,3871818

answered Jul 31 at 13:03

nicomezi

3,3871818

answered Jul 31 at 13:03

nicomezi

3,3871818

3,3871818

add a comment | 

add a comment | 

up vote
0
down vote

Actually, $partial_tRe(f(t,z))^2=2Re(f(t,z)) cdot partial_t Re(f(t,z)) = 2Re(f(t,z)) cdot Re(partial_t f(t,z))$, so

$$partial_t |f(t,z)|^2=2Re(f(t,z)) cdot Re(partial_t f(t,z))+2Im(f(t,z)) cdot Im(partial_t f(t,z))$$

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answered Jul 31 at 13:06

lisyarus

9,89221433

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up vote
0
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Actually, $partial_tRe(f(t,z))^2=2Re(f(t,z)) cdot partial_t Re(f(t,z)) = 2Re(f(t,z)) cdot Re(partial_t f(t,z))$, so

$$partial_t |f(t,z)|^2=2Re(f(t,z)) cdot Re(partial_t f(t,z))+2Im(f(t,z)) cdot Im(partial_t f(t,z))$$

share|cite|improve this answer

answered Jul 31 at 13:06

lisyarus

9,89221433

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up vote
0
down vote

up vote
0
down vote

Actually, $partial_tRe(f(t,z))^2=2Re(f(t,z)) cdot partial_t Re(f(t,z)) = 2Re(f(t,z)) cdot Re(partial_t f(t,z))$, so

$$partial_t |f(t,z)|^2=2Re(f(t,z)) cdot Re(partial_t f(t,z))+2Im(f(t,z)) cdot Im(partial_t f(t,z))$$

share|cite|improve this answer

answered Jul 31 at 13:06

lisyarus

9,89221433

Actually, $partial_tRe(f(t,z))^2=2Re(f(t,z)) cdot partial_t Re(f(t,z)) = 2Re(f(t,z)) cdot Re(partial_t f(t,z))$, so

$$partial_t |f(t,z)|^2=2Re(f(t,z)) cdot Re(partial_t f(t,z))+2Im(f(t,z)) cdot Im(partial_t f(t,z))$$

share|cite|improve this answer

answered Jul 31 at 13:06

lisyarus

9,89221433

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 13:06

lisyarus

9,89221433

answered Jul 31 at 13:06

lisyarus

9,89221433

answered Jul 31 at 13:06

lisyarus

9,89221433

9,89221433

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add a comment | 

 

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