## COMPLEX-VARIABLE SOLUTIONS IN MATHEMATICA

Copies of the files referred to in package format (.m) are given here. Although Mathematica can handle complex numbers and variables, I have not found a good way to utilize this resource directly in the solution of complex-variable problems. Instead, I have had some success by defining the complex variable and its conjugate as separate variables z1=x+I*y, z2=x-I*y, with solution x=(z1+z2)/2, y=I*(z2-z1)/2.

If f(z1,z2) is a function of z1 and z2, its conjugate fb can then be defined by the operation

fb:=subs({z1=z2,z2=z1,I=-I}f);

This notation and technique is used in the file rtoch.nb that expresses a real harmonic function of x and y in the form

f+fb

where f is a function of z1 and fb is its conjugate. The file rtocb.nb uses a similar technique to convert a real biharmonic function to the form

g1+g1b+z2*g2+z1*g2b,

where g1(z1), g2(z1) are functions of z1 only and g1b, g2b are their conjugates. The file Airytocomplex.nb uses this technique to convert a real Airy stress function to the complex potentials chi and theta, using results from Section 19.4. One consequence of this conversion is that the displacement can then be found (apart from a rigid-body displacement), using equation (19.44). The reverse operation (converting complex potentials to an equivalent Airy stress function) is performed by the file complextoAiry.nb.

## THREE-DIMENSIONAL PROBLEMS

The complex version of the Papkovich-Neuber solution of Section 21.3 is contained in the file omegaPNcomp.nb. For problems of the prismatic bar (Chapter 28), omega can be set to zero. We therefore also include a version of the P-N solution in which the omega terms have been omitted as compPN.nb.

The file PrandtltoPN.nb converts a given real Prandtl stress function phi (as defined in Chapters 15-17) to complex P-N form. If the Prandtl function is used to define a shear problem, as in Chapter 17, it will not be harmonic, but the constants A,B,C (here denoted by AA,BB,CC) can be determined by substituting into equation (17.10) and equating coefficients of x,y and constants. This file calls the file GH.nb (see below).

The file intz.nb uses the procedure of Section 24.8 to generate partial integrals with respect to z of three-dimensionally harmonic functions of z1,z2,z. This program is then used as the basis for determining complex versions of the bounded spherical harmonics [equations (24.66) et seq. ] in the connected files chi.nb, chim.nb. Starting from z1^m, chim.nb proceeds iteratively to generate the functions chi[n,m] for n=0,7. The file chi.nb calls chim and uses it to develop explicit expressions for chi[n,m] for m,n=0,6.

The functions G[m,n], H[m,n] of equations (28.27) are generated by the file GH.nb, which also calls chi and hence chim. Also, the functions Q[m,n],V[m,n] of equations (28.42) are generated by the file QV.nb.