Jr., I am grateful for help with the sections on radical expressions. My wife, Jane, has not only provided me wlth encouragement and understanding, but she has contributed to this thesis in many material ways. AJ+J_ong other things, she typed the rough drafts and helped with the programming. Mrs. Janet Delaney provided frequent assistance with a sometimes difficult Formula Algol system, and Miss Carol Miller did an expert Job of typing the final draft. I am also most appreciative for financial support received from the National Science Foundation in the form of a graduate fellowship.

No abstract

No abstract

Abstract. In Part of this paper, we give an extension of Liouville's Theorem and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone. Our main result generalizes Liouville's Theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral (but not the dilogorithm or exponential integral) to appear in the integral of an elementary function. The basic conclusion is that these functions, if they appear, appear linearly. We give an algorithm which decides if an elementary function, built up using only exponential functions and rational operations has an integral which can be expressed in terms of elementary functions and error functions.