The issue appears to be that Mathematica is handling the assumptions concerning `x`

incorrectly. If you explicitly tell it that $x < -1$, $|x| leq 1$, or $x > 1$, it produces three slightly different functional forms:

```
ulow(x_) = Assuming(x < -1 && (Eta) > 0, ... )
umid(x_) = Assuming(1 > x > -1 && (Eta) > 0, ... )
uhigh(x_) = Assuming(1 < x && (Eta) > 0, ... )
(* (4 I ((Pi) + ArcSin(x)))/(x Sqrt(-1 + x^2)) *)
(* -((4 ArcCos(Sqrt(1 - x^2)))/Sqrt(x^2 - x^4)) *)
(* (4 I ArcSin(x))/(x Sqrt(-1 + x^2)) *)
```

So if desired, we can assemble these into a `Piecewise`

function that works for all `x`

:

```
u(x_) := Piecewise({{ulow(x), x< -1}, {umid(x), -1 <= x <= 1}, {uhigh(x), x > 1}})
Plot(Evaluate(ReIm(u(x))), {x, -Pi, Pi})
```

This does appear to have the property that $Re(u(x))$ is even in $x$ and $Im(u(x))$ is odd in $x$.