Advanced Calculus fi..

The function U is Airy's stress function; its second derivativesChapter 8 Functions of a Complex Variable 61 5give components of the stress tensor, which describes the forces acting on an arbitrary"$plane cross section of the solid being studied.Solution of (8.112) in a domain D is equivalent to the solution of two equations:The solutions P of the second equation are harmonic functions. Furthermore, ifU,, U2 satisfy the first equation for given P, thenhence the solutions of the first equation are of formUl + w,where U1 is a particular solution and W is harmonic. Now let harmonic functions u-*and v be chosen, if possible, so thatThenHenceauaxa vayv2(xu + yv) = xv2u + 2- + yv2v + 2- = 4P1Ul = -(xu + yv)4is the desired particular solution, provided u and v can be found. If D is simplyconnected, one can choose Q so that P + i Q = F(z) is analytic in D; thendefines (up to additive constants) harmonic functions u and v such that aulax =av/ay = P and U1 can be written as follows:1 101 = ;(xu + yv) = -Re [i f (z)].4Finally, the solutions U of (8.1 12) can be written thus: