Harmonic Functions, conjugates and the Hilbert Transform

[nickthequick]

Hi,

I am currently confused about something I've run across in the literature.

Given that
$\nabla^2\phi = \phi_{xx}+\phi_{zz} = 0$ for $z\in (-\infty, 0]$

and

$\phi_z = \frac{\partial}{\partial x} |A|^2$ at z=0.

for $A= a(x)e^{i \theta(x)}$.

The author claims that

$\phi_x = A_xA^*-AA^*_x$ at z=0

and where A* represents the complex conjugate.

The author then claims a more general formula for $\phi_x$ can be found in terms of the Hilbert Transform.

I do not understand how the author finds the expression for $\left.\phi_x\right|_{z=0}$. Also, although I'm vaguely aware that Hilbert Transforms can be used to find Harmonic conjugates, I don't see how that can be exploited in this case.

Any suggestions are appreciated!

Thanks,

Nick

 

[yus310]

Hi,

I am currently confused about something I've run across in the literature.

Given that
$\nabla^2\phi = \phi_{xx}+\phi_{zz} = 0$ for $z\in (-\infty, 0]$

and

$\phi_z = \frac{\partial}{\partial x} |A|^2$ at z=0.

for $A= a(x)e^{i \theta(x)}$.

The author claims that

$\phi_x = A_xA^*-AA^*_x$ at z=0

and where A* represents the complex conjugate.

The author then claims a more general formula for $\phi_x$ can be found in terms of the Hilbert Transform.

I do not understand how the author finds the expression for $\left.\phi_x\right|_{z=0}$. Also, although I'm vaguely aware that Hilbert Transforms can be used to find Harmonic conjugates, I don't see how that can be exploited in this case.

Any suggestions are appreciated!

Thanks,

Nick

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