- #1
nickthequick
- 53
- 0
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
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