Derivative of the complex conjugate of z with respect to z

[fairy._.queen]

Hi all!

From Wirtinger derivatives, given $z=x+iy$ and indicating as $\overline{z}$ the complex conjugate, I get:
$\frac{\partial\overline{z}}{\partial z}=\frac{1}{2}\left(\frac{\partial (x-iy)}{\partial x}-i\frac{\partial (x-iy)}{\partial y}\right)=0$

This puzzles me, because I cannot see why a number and its complex conjugate could be independent variables.

Thank you in advance!

 

[Ray Vickson]

Hi all!

From Wirtinger derivatives, given $z=x+iy$ and indicating as $\overline{z}$ the complex conjugate, I get:
$\frac{\partial\overline{z}}{\partial z}=\frac{1}{2}\left(\frac{\partial (x-iy)}{\partial x}-i\frac{\partial (x-iy)}{\partial y}\right)=0$

This puzzles me, because I cannot see why a number and its complex conjugate could be independent variables.

Thank you in advance!

If $\Delta z = h + i k$ then $\Delta \bar{z} = h - ik$ ($h,k$ real), so
$$\frac{\Delta \bar{z}}{\Delta z} =\left( \frac{h^2 - k^2}{h^2+k^2}\right) - i \left( \frac{2kh}{h^2+k^2} \right)$$
Since the limit as $h, k \to 0$ depends on the "direction", the function $\bar{z}$ is not differentiable in the complex-analysis sense.

 

[fairy._.queen]

Thank you very much for your reply! Then, what about the following situation:

I have an ordinary differential equation system that depends on a number of variables, including $z$. I have to compute the Jacobian of the function $f$ defining the system, so that I need $\frac{\partial f}{\partial z}$, but $\overline{z}$ is not a variable for which the algorithm solves.

To simplify things, let's assume I have
$f(a) = cz + d\overline{z}$
and the integrator I'm using needs $\frac{\partial f(a)}{\partial z}$.

How can I compute that entry of the Jacobian?

 

[Ray Vickson]

Thank you very much for your reply! Then, what about the following situation:

I have an ordinary differential equation system that depends on a number of variables, including $z$. I have to compute the Jacobian of the function $f$ defining the system, so that I need $\frac{\partial f}{\partial z}$, but $\overline{z}$ is not a variable for which the algorithm solves.

To simplify things, let's assume I have
$f(a) = cz + d\overline{z}$
and the integrator I'm using needs $\frac{\partial f(a)}{\partial z}$.

How can I compute that entry of the Jacobian?

Sorry: I have no idea, and I am not sure the problem even makes sense.

I guess it all depends on whether z is some parameter and you need df/dz is the sense of complex analysis, or just in the sense that using z is a convenient way of encapsulating two real variables and you actually have real derivatives. I just cannot tell from the information you supply.

 

[fairy._.queen]

I have a system of complex ODEs, with complex variables.

Let's assume that it looks like this (it's an oversimplification, but will do):
$z' = w+iz\\ w'=c\overline{z}+dz$

My algorithm requires me to provide the Jacobian manually, so that I need $\frac{\partial w'}{\partial z}$.

What can I do?

 

[Ray Vickson]

I have a system of complex ODEs, with complex variables.

Let's assume that it looks like this (it's an oversimplification, but will do):
$z' = w+iz\\ w'=c\overline{z}+dz$

My algorithm requires me to provide the Jacobian manually, so that I need $\frac{\partial w'}{\partial z}$.

What can I do?

What does the "prime" stand for (in $z^{\prime}$ for example)?

 

[fairy._.queen]

Derivative with respect to a time variable, i.e. $z = z(t)$, for instance. Thanks!

 

[Ray Vickson]

Derivative with respect to a time variable, i.e. $z = z(t)$, for instance. Thanks!

OK, that is sort of what I suspected.

Assuming t is real, that means that using complex quantities is really just a convenient way of encapsulating real quantities; that is, you can re-write the DE system as a system of coupled DEs for real quantities---similar to what we do in electronics/electrical engineering when we use complex voltages, currents, impedances, etc. You are not really doing "complex analysis", so you don't have to worry about Cauchy-Riemann equations and all that. The Jacobians you need are just ordinary multivariate Jacobians for real functions. Perhaps just re-writing everything in purely real terms would be the least confusing way to go---although it may be much longer (just as in electronics we can always re-write everything in purely real terms, but the expressions get much longer in many cases).

 

[fairy._.queen]

Thanks a lot, it's clear now!