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## Main Question or Discussion Point

I'm in truble with a partial differential equation. Actually it is a system of PDE but It would be useful to solve at least one of them.

The most easy one is this one

[tex]

2 \bar{\xi}\left(\bar{s},\bar{t},\bar{u}\right) - 2 \xi\left(s,t,u\right) + \left(s-\bar{s}\right)\left(\bar{\partial}_\bar{s} \bar{\xi} + \partial_s \xi \right) = 0

[/tex]

This equation can be simplified to

[tex]

2 A^*\left(z^*\right) - 2 A\left(z\right) + \left(z-z^*\right)\left(\bar{\partial}_{z^*}A^*+\partial_{z}A\right)= 0

[/tex]

I further developed my computation using [tex] A(z) = u(x,y) + i v(x,y) [/tex] with [tex] u,v \in \mathbb{R}[/tex]

finding (I used Cauchy-Riemann equations)

[tex] v(x,y) = y^2 f(x+y) [/tex]

Here is where I get stucked since I cannot find a suitable form of "f(x+y)" in order to obtain "u" and satisfy Cauchy-Riemann equations...

Any ideas?

The most easy one is this one

[tex]

2 \bar{\xi}\left(\bar{s},\bar{t},\bar{u}\right) - 2 \xi\left(s,t,u\right) + \left(s-\bar{s}\right)\left(\bar{\partial}_\bar{s} \bar{\xi} + \partial_s \xi \right) = 0

[/tex]

This equation can be simplified to

[tex]

2 A^*\left(z^*\right) - 2 A\left(z\right) + \left(z-z^*\right)\left(\bar{\partial}_{z^*}A^*+\partial_{z}A\right)= 0

[/tex]

I further developed my computation using [tex] A(z) = u(x,y) + i v(x,y) [/tex] with [tex] u,v \in \mathbb{R}[/tex]

finding (I used Cauchy-Riemann equations)

[tex] v(x,y) = y^2 f(x+y) [/tex]

Here is where I get stucked since I cannot find a suitable form of "f(x+y)" in order to obtain "u" and satisfy Cauchy-Riemann equations...

Any ideas?