  The full low energy effective action of N=4 SYM is believed to be self-dual.
Starting with the first two leading terms in a momentum expansion of this
effective action, we perform a duality transformation and find the conditions
for self-duality. These determine some of the higher order terms. We compare
the effective action of N=4 SYM with the probe-source description of type II_B
D3-branes in the AdS_5 \times S_5 background. We find agreement up to six
derivative terms if we identify the separation of the 3-branes with a
redefinition of the gauge scalar that involves the gauge field strength.
