= Two images of a point-mass lens
{title2=$\theta_\pm=(\beta\pm\sqrt{\beta^2+4\theta_E^2})/2$}
Choose a signed angular coordinate along the source-lens line. The <point-mass gravitational lens> becomes
$$
\beta=\theta-\frac{\theta_E^2}{\theta},\qquad\theta\ne0.
$$
For $\beta\ne0$, multiplication by $\theta$ gives the <quadratic equation> $\theta^2-\beta\theta-\theta_E^2=0$. Its positive <discriminant> yields the two displayed roots. Their product is $\theta_+\theta_-=-\theta_E^2$, so they lie on opposite sides of the lens. With $\beta>0$, $\theta_+>\theta_E$ and $-\theta_E<\theta_-<0$: the outer image lies on the source side. For $\beta<0$, the outer image is instead the minus root. These are image directions seen by the observer, not two emitting objects.<Gravitational Lensing by Matthias Bartelmann>[Section 2.2, printed pp. 15-16, equations (100)-(110): point-mass model, two roots, critical circle and magnification]{ref}
For a dimensionless example, set $u=\beta/\theta_E=0.6$ and $x=\theta/\theta_E$. Solving $u=x-1/x$ gives $x_+\simeq1.344$ and $x_-\simeq-0.744$, with separation $\sqrt{u^2+4}\simeq2.088$. The <lensing magnification> follows from <image parity in gravitational lensing>: $\mu_+\simeq1.442$, $\mu_-\simeq-0.442$, and total amplification $\simeq1.884$. Changing $M$ or the distances rescales the angles through the <Einstein angular radius>, while these dimensionless values remain the same.
For $u\gg1$, the <quadratic equation> gives $x_+\simeq u+1/u$ and $x_-\simeq-1/u$. Using $\mu=1/(1-x^{-4})$, the outer image tends to unit <lensing magnification> and the inner image has $|\mu_-|\simeq u^{-4}$. Two mathematical images can therefore look like one at finite sensitivity or <angular resolution>. Furthermore, the inner <impact parameter> eventually becomes too small for the stated <weak-field approximation> or passes through the opaque lens: the ideal formula must then be replaced by an appropriate physical model.
\Image[/point-mass-lens-image-mapping-and-alignment.png]
{title=Signed point-mass image mapping and the aligned Einstein ring}
{description=Left: the horizontal source line u=0.6 intersects the two branches of u=x-1/x at images with opposite parity. Right: exact alignment gives a full circle of image directions, rather than only the two points visible in a one-dimensional cut. This is a theoretical diagram.}
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