A NEW LIBRARY PROGRAM FOR GENERATING AUGMENTED JACOBI POLYNOMIALS FOR TEXTURE CALCULATIONS

. A new library program for generating augmented Jacobi polynomials for texture analysis is presented. By using this program, the spatial orientation distribution maps for the three-dimensional texture analysis can be produced.


INTRODUCTION
In the three-dimensional texture analysis, numerical values of polynomials should be calculated by using generalized Legendre polynomials or augmented Jacobi polynomials.
These polynomials denoted pn () by Bunge i and Z mn () by Roe 2 are identical with each other except for their normalization constants and, in some cases, sign.
However, the polynomials Z mn () (we define hereafter, according to the notations used by Roe) have up to now been calculated in the form of Fourier series3-6; the Fourier coefficients in tabular form have already been provided by Morris et al. 3,4 If the numerical values of polynomials expanded in Fourier series are adopted in the computer program for ODF analysis, the following disadvantages occur: (i) A computer with a large memory is required to store resident data.
(2) Errors of numerical values in a table of Fourier coefficients cause errors in the calculation of the polynomials Z mn () To avoid these disadvantages, two methods have been considered feasible to calculate the numerical values of polynomials Z mn () The first one is deducing Z mn () from a 143 recurrence relation. The second is deducing Zmn () from the hypergeometric series directly. Liang et al. suggested in the recent report the method of generating the polynomials Zimn () by the first method. To apply this method, however, initial values for each , m, n must be given previously, which would inevitably lead to the increase in program size, required memory, and processing time.
The purposes of this report are to produce a library program for generating Zmn () based on the second method; and by using this program to work out the orientation distribution maps for the rolled texture of b.c.c, metals, as an example, by use of a small computer (NEAC-3100).

DEFINITION OF Z mn () AND ITS GENERATION
The augmented Jacobi polynomials is defined by Roe 2 follows as 2FI (e,8;y;t) is Gauss' hypergeometric series (see Appendix) The FI is generated easily as shown in Figure

PRODUCTION OF SPATIAL ORIENTATION DISTRIBUTION OF CRYSTALLITES
In setting up our library program by using the coefficients of ODF given by Hu, e the spatial orientation distribution of crystallites in the as-cold-rolled phosphorus steel sheet can be obtained, as shown in Figure 3. The picture at constant ph ( 45 is almost the same as that obtained by Hu. Figure 4 shows the spatial orientation distribution of crystallites in the as-rolled molybdenum TZM-sheet at constant phi ( 45), which is produced by using coefficients of ODF determined from our library of Zmn (). For the spatial orientation distribution map, texture data from X-ray

CONCLUSION
A new library program for generating augmented Jacobi polynomials for three-dimensional texture analysis has been carried out for the purpose of obtaining the spatial orientation distribution of crystallites by the use of a small computer (NEAC-3100) or a personal computer (PC-8001). The details of the program, the required memory capacity and the accuracy in the computer calculation will be presented in a forthcoming paper.

AC KNOWLEDGEMENTS
The authors would like to thank Dr. I. Nishida of the Metal Physics Division of the National Research Institute for Metals for his helpful discussion, and J. Y. Kido of the Electrical Engineering Department of the Salesian Polytechnic for his cordial assistance with the computer program.

APPENDIX
The series 2FI (e,8;y;t) is known as Gauss' hypergeometric series. 9 This series has been generalized by the in-10 troduction of parameters p and q as follows: (i) n (2)n (p) n t n pFq (i;j ;t) (i) (2)  When x is integer, The series pFq is known as Pochhammer's generalized hypergeometric series. This series is terminated by the negative integer as, in which case it is useful in the physical sciences.
In the case of p q i, the series is written as IFI (e;y,t) and called "Kummer's confluent hypergeometric series. i 1 Moreover, Kummer's confluent hypergeometric series leads to Laguerre, associated Laguerre and Hermite polynomials. Also, Gauss' hypergeometric series leads to Legendre, associated Legendre, Jacobi, Gegenbauer and Tchebycheff polynomials. 12,13 It is, therefore, very important to deduce the polynomials pFq.
The series pFq is generated easily according to the algorithm as shown in Figure 5.