Let H be a hyperexponential function of x over a field K of characteristic 0. The MultiplicativeDecomposition[i](H,x) calling sequence constructs the ith multiplicative decomposition for H, .

If the MultiplicativeDecomposition command is called without an index, the first multiplicative decomposition is constructed.

A multiplicative decomposition of H is a pair of rational functions  such that . Let R be the rational certificate of H, i.e., . Let  be a differential rational normal form of R. Then  is a multiplicative decomposition of H. Hence, each differential rational normal form  of the certificate R of H is also a multiplicative decomposition of H.

The construction of MultiplicativeDecomposition[i](H,x) is based on , for .

The output is of the form  where V and F are rational function of x over K.

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press, (2004): 183-190.