Harmonic Functions, conjugates and the Hilbert Transform

Hi,

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

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

and

[itex] \phi_z = \frac{\partial}{\partial x} |A|^2 [/itex] at z=0.

for [itex] A= a(x)e^{i \theta(x)} [/itex].

The author claims that

[itex] \phi_x = A_xA^*-AA^*_x [/itex] at z=0

and where A* represents the complex conjugate.

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

I do not understand how the author finds the expression for [itex] \left.\phi_x\right|_{z=0} [/itex]. 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
 
81
1
Hi,

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

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

and

[itex] \phi_z = \frac{\partial}{\partial x} |A|^2 [/itex] at z=0.

for [itex] A= a(x)e^{i \theta(x)} [/itex].

The author claims that

[itex] \phi_x = A_xA^*-AA^*_x [/itex] at z=0

and where A* represents the complex conjugate.

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

I do not understand how the author finds the expression for [itex] \left.\phi_x\right|_{z=0} [/itex]. 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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