Elliptically Symmetric Lenses and Violation of Burke's Theorem
  We show that the outside equation of a bounded elliptically symmetric lens
(ESL) exhibits a pseudo-caustic that arises from a branch cut. A pseudo-caustic
is a curve in the source plane across which the number of images changes by
one. The inside lens equation of a bounded ESL is free of a pseudo-caustic.
Thus the total parity of the images of a point source lensed by a bounded
elliptically symmetric mass is not an invariant in violation of the Burke's
theorem. A smooth mass density function does not guarantee the validity of the
Burke's theorem.
  Pseudo-caustics of various lens equations are discussed. In the Appendix,
Bourassa and Kantowski's deflection angle formula for an elliptically symmetric
lens is reproduced using the Schwarz function of the ellipse for an easy
access; the outside and inside lens equations of an arbitrary set of truncated
circularly or elliptically symmetric lenses, represented as points, sticks, and
disks, are presented as a reasonable approximation of the realistic galaxy or
cluster lenses.
  One may consider smooth density functions that are not bounded but fall
sufficiently fast asymptotically to preserve the total parity invariance. Any
bounded function may be sufficiently closely approximated by an unbounded
smooth function obtained by truncating its Fourier integral at a high frequency
mode. Whether to use a bounded function or an unbounded smooth function for an
ESL lens mass density, whereby whether to observe the total parity invariance
or not, incurs philosophical questions. For example, is it sensible to insist
that the elliptical symmetry of an elliptic lens galaxy be valid in the entire
sky? How a pseudo-caustic close to or intersecting with a caustic must be
withered away during a smoothing process and what it means will be investigated
in a separate work.
