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Fixed points: $z = 2i, z = \infty$. Image: The transformation leads to the imaginary axis being mapped onto a circle centered at the origin with radius 2.

Fixed points: $z = 0, z = -i$. Image: There's an expansion and a $90^\circ$ rotation of the complex plane.

Fixed points: $z = -\frac{1}{6}, z = \infty$. Image: The points on the real axis move to form a circle centered on the imaginary axis.

Fixed points: $z = 0, z = -i$. Image: Transformation results in a circular arc in the upper half-plane being mapped onto the line segment on the real axis between the fixed points.

Fixed points: $z = -1/4, z = 3i$. Image: A circle passes through the fixed points, intersecting the real axis at right angles.

Fixed points: $z = 4, z = -4i$. Image: A Möbius transformation where the unit circle would be mapped onto a line segment in the imaginary axis.

Fixed points: $z = i, z = \infty$. Image: The extended complex plane transforms with a rotation and dilation centered about the point $z = i$.

Fixed points: $z = -i, z = \infty$. Image: The unit circle is transformed into itself, but rotated by $90^\circ$ clockwise.

Fixed points: $z = i, z = -\frac{1}{3}i$. Image: Any circle orthogonal to the unit circle remains orthogonal after the transformation.

Fixed points: $z = \frac{i}{7}, z = \infty$. Image: A dilation centered on $z = \frac{i}{7}$ coupled with a $90^\circ$ counterclockwise rotation.

Fixed points: $z = \frac{1}{6}i, z = -7i$. Image: An inversion and a rotation around $z = \frac{1}{6}i$ take place.

Fixed points: $z = -1/2 \pm i/2$. Image: The transformation is an inversion in the unit circle followed by a translation.

Fixed points: $z = 0, z = 1 + i$. Image: The transformation maps the unit circle onto the real axis.

Fixed points: $z = \infty, z = \frac{-1}{3}i$. Image: A transformation that notably rotates and dilates the complex plane around $z = \frac{-1}{3}i$.

Fixed points: $z = 2, z = \infty$. Image: The transformation results in an inversion and a translation in the complex plane.

Fixed points: $z = -i, z = \frac{1}{2}$. Image: The transformation maps the imaginary axis onto a circle that passes through the fixed points.

Fixed points: $z = i/2, z = \infty$. Image: Circle centering at $z = i$ with radius rac{1}{2} is invariant under this transformation.

Fixed points: $z = 3i, z = \infty$. Image: Horizontal lines are mapped onto circles with center at the y-axis.

Fixed points: $z = 1, z = \infty$. Image: The transformation reflects the real axis over the circle centered at the origin with radius 1.

Fixed points: $z = 1, z = \infty$. Image: The extended complex plane is transformed so that the line Im$(z) = 0$ is invariant.

Fixed points: $z = 0, z = 1 + \sqrt{2}, z = 1 - \sqrt{2}$. Image: There is a shift of the whole complex plane by 2 to the right after transformation.

Fixed points: $z = \frac{1}{5}, z = -2i$. Image: Any line through $z = -2i$ is mapped to itself. Other lines are bent into circles.

Fixed points: $z = 1, z = \infty$. Image: An orthogonally intersecting pair of lines through $z = 1$ are mapped to a pair of orthogonal circles.

Fixed points: $z = -2, z = -2i$. Image: The transformation provides a geometric inversion, where a line passing through these points remains invariant.

Fixed points: $z = \frac{3}{7}i, z = \infty$. Image: An eccentric dilation with a shift along the imaginary axis culminates in a rotation about $z = \frac{3}{7}i$.

Fixed points: $z = -5i, z = \frac{1}{2}i$. Image: A line segment on the real axis is mapped onto the imaginary axis between the fixed points.

Fixed points: $z = \frac{i}{2}, z = \infty$. Image: The transformation results in an anticlockwise rotation centered about $z = \frac{i}{2}$.

Fixed points: $z = -\frac{4}{9}, z = \infty$. Image: The transformation maps the vertical line Re$(z) = -\frac{4}{9}$ onto itself.

Fixed points: $z = -i/4, z = -3/2$. Image: This transformation creates a mapping where the imaginary axis gets reflected to a circle through the origins.