By M.E. Munroe
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Extra resources for Introduction to measure and integration
Example text
4, Then M is holonomic outside the hypersurface defined by q = 0. This implies that f is holonomic in the same way as in the proof of the preceding proposition. 6. For two polynomials fi(z),fi(z)in C l z ~ ,. ,z,], put f(x) = exp(fl(x)/fZ(z)). The system of differential equations M above is not holonomic in general (consider, for example, exp(l/(zf - zzx,"))). A holonomic system for f(x) can be found by the method in [14]. Let f be a holonomic function. By definition, it is a multi-valued analytic function defined on C n \ S.
Hence, there exists a submodule J such that D$';'/J N M y ; J is a system of equations for u(z',0), (d,u)(x',0), . . , (a$u)(z',0), where x' = (21,. . ,~ ~ - 1 ) . An algorithm of finding generators of J from those of I is given in [12]. By 82. 21, we can find a system of equations for u(z’,O) from J . Take an integer m such that 0 I m < n. Let Z be the algebraic set { ( X I , . . , 2,) I x,+1 = . . = z, = 0) and M a left D,-module DL/I where I is a left submodule of DL. The restriction of M to Z is defined by M/(x,+lM ...
W. Schafer. Discrete-Time Signal Processing. Prentice Hall, 1989. 17. M. Petkovsek, H. S. Wilf, D. Zeilberger. A = B. , 1996. 18. W. Press, S. Teukolsky, W. Vetterling, B. Flannery. Numerical Recipes in C++. Cambridge University Press, 1988. 19. M. Saito, B. Sturmfels, N. Takayama. Grobner Deformations of Hypergeometric Differential Equations. Springer, 2000. 20. N. Takayama. An algorithm of constructing the integral of a module, Proceedings of International Symposium on Symbolic and Algebraic Computation (1990), 206-211.
Introduction to measure and integration by M.E. Munroe
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