  In weak gravitational lensing, the image distortion caused by shear measures
the projected tidal gravitational field of the deflecting mass distribution. To
lowest order, the shear is proportional to the mean image ellipticity. If the
image sizes are not small compared to the scale over which the shear varies,
higher-order distortions occur, called flexion. For ordinary weak lensing, the
observable quantity is not the shear, but the reduced shear, owing to the
mass-sheet degeneracy. Likewise, the flexion itself is unobservable. Rather,
higher-order image distortions measure the reduced flexion, i.e., derivatives
of the reduced shear. We derive the corresponding lens equation in terms of the
reduced flexion and calculate the resulting relation between brightness moments
of source and image. Assuming an isotropic distribution of source orientations,
estimates for the reduced shear and flexion are obtained; these are then tested
with simulations. In particular, the presence of flexion affects the
determination of the reduced shear. The results of these simulations yield the
amount of bias of the estimators, as a function of the shear and flexion. We
point out and quantify a fundamental limitation of the flexion formalism, in
terms of the product of reduced flexion and source size. If this product
increases above the derived threshold, multiple images of the source are formed
locally, and the formalism breaks down. Finally, we show how a general
(reduced) flexion field can be decomposed into its four components: two of them
are due to a shear field, carrying an E- and B-mode in general. The other two
components do not correspond to a shear field; they can also be split up into
corresponding E- and B-modes.
