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Just express $x_0$ and $y_0$ directly in terms of $z$ and $\bar{z}$ (direct subsitution is more effective than "pattern matching" substitution with $x_0+iy_0$):

Zr0=Zr.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
Zi0=Zi.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
f=Zr0+Zi0
show("f : ",f.simplify_full())

$$\verb|f|\phantom{\verb!x!}\verb|:| \left( x_{0}, y_{0}, r \right) \ {\mapsto} \ \frac{2 \, z \overline{z} - \sqrt{-4 \, r^{2} + 4 \, z \overline{z}} \sqrt{z \overline{z}} \sqrt{-\frac{r^{2} - z \overline{z}}{z \overline{z}}}}{2 \, \overline{z}}$$

Just express $x_0$ and $y_0$ directly in terms of $z$ and $\bar{z}$ (direct subsitution is more effective reliable than "pattern matching" substitution with $x_0+iy_0$):$x_0+iy_0$ and $x_0^2+y_0^2$):

Zr0=Zr.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
Zi0=Zi.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
f=Zr0+Zi0
show("f : ",f.simplify_full())

$$\verb|f|\phantom{\verb!x!}\verb|:| \left( x_{0}, y_{0}, r \right) \ {\mapsto} \ \frac{2 \, z \overline{z} - \sqrt{-4 \, r^{2} + 4 \, z \overline{z}} \sqrt{z \overline{z}} \sqrt{-\frac{r^{2} - z \overline{z}}{z \overline{z}}}}{2 \, \overline{z}}$$

Just express $x_0$ and $y_0$ directly in terms of $z$ and $\bar{z}$ (direct subsitution is more reliable than "pattern matching" substitution with $x_0+iy_0$ and $x_0^2+y_0^2$):

Zr0=Zr.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
Zi0=Zi.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
f=Zr0+Zi0
show("f show(LatexExpr("f : ",f.simplify_full())
"), f.simplify_full())

$$\verb|f|\phantom{\verb!x!}\verb|:| $$f : \left( x_{0}, y_{0}, r \right) \ {\mapsto} \ \frac{2 \, z \overline{z} - \sqrt{-4 \, r^{2} + 4 \, z \overline{z}} \sqrt{z \overline{z}} \sqrt{-\frac{r^{2} - z \overline{z}}{z \overline{z}}}}{2 \, \overline{z}}$$

Just express $x_0$ and $y_0$ directly in terms of $z$ and $\bar{z}$ (direct subsitution is more reliable than "pattern matching" substitution with $x_0+iy_0$ and $x_0^2+y_0^2$):

Zr0=Zr.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
Zi0=Zi.subs({x_0: (z+z.conjugate())/2, y_0: (z-z.conjugate())/(2*I)})
Zr0=Zr((z+z.conjugate())/2, (z-z.conjugate())/(2*I), r)
Zi0=Zi((z+z.conjugate())/2, (z-z.conjugate())/(2*I), r)
f=Zr0+Zi0
show(LatexExpr("f show(LatexExpr(r"f : "), f.simplify_full())
(z,\bar{z},r) \mapsto"),f.simplify_full())

$$f : \left( x_{0}, y_{0}, z, \bar{z}, r \right) \ {\mapsto} \ \frac{2 \, z \overline{z} - \sqrt{-4 \, r^{2} + 4 \, z \overline{z}} \sqrt{z \overline{z}} \sqrt{-\frac{r^{2} - z \overline{z}}{z \overline{z}}}}{2 \, \overline{z}}$$