  We study a special class of four-point correlation functions of infinitely
heavy half-BPS operators in planar N=4 SYM which admit factorization into a
product of two octagon form factors. We demonstrate that these functions
satisfy a system of nonlinear integro-differential equations which are powerful
enough to fully determine their dependence on the 't Hooft coupling and two
cross ratios. At weak coupling, solution to these equations yields a known
series representation of the octagon in terms of ladder integrals. At strong
coupling, we develop a systematic expansion of the octagon in the inverse
powers of the coupling constant and calculate accompanying expansion
coefficients analytically. We examine the strong coupling expansion of the
correlation function in various kinematical regions and observe a perfect
agreement both with the expected asymptotic behavior dictated by the OPE and
with results of numerical evaluation. We find that, surprisingly enough, the
strong coupling expansion is Borel summable. Applying the Borel-Pade summation
method, we show that the strong coupling expansion correctly describes the
correlation function over a wide region of the 't Hooft coupling.
