In this note, we present an answer to Exercise 9.3 in the Book Symbolic Integration I (second edition) by M. Bronstein, under an additional assumption that the real elementary extension in the exercise is purely transcendental. Our answer is based on a rather technical lemma derived from a naive attempt to do the exercise inductively.

has no solutions which can be written using elementary functions, or anti-derivatives of elementary functions, or exponentials of such anti-derivatives, or anti-derivative of those, etc. We should note that Equation 1 can be solved using power series, integrals which depend on a parameter, or Bessel functions of order 1/3. However, as we will see, none of these methods of solution are “algebrai...

Communication research requires data that are rich enough to contain explicit evidence about processes of communication. The paper fonnally distinguishes among and discusses three kinds of data: aggregational data (such as accumulated in much of psychological inquiries, small group experiments and survey research) and network data (representing observable patterns of interest to relational biol...

We present a closed-form expression for the irradiance at a point on a surface due to an arbitrary polygonal Lambertian luminaire with linearly-varying radiant exitance. The solution consists of elementary functions and a single wellbehaved special function that can be either approximated directly or computed exactly in terms of classical special functions such as Clausen’s integral or the clos...

Michel has attempted to dismiss the concept of a vacuum electromagnetic zero-point Ðeld (ZPF) as a ““ computational trick.ÏÏ He criticizes the work of several researchers, and speciÐcally the Rueda, Haisch, & Cole proposal that, ““ the ZPF may have something to do with cosmic voids ÏÏ and their reference to ““ various other publications suggesting still other physical applications.ÏÏ We interpr...

Euler’s work on elliptic integrals is a milestone in the history of algebraic geometry. The founders of calculus understood that some algebraic functions could be integrated using elementary functions (logarithms and inverse trigonometric functions). Euler realized that integrating other algebraic functions leads to genuinely different functions, elliptic integrals. These functions are not some...

Despite the huge production of all sorts of multimedia instructions, educational research has yielded surprisingly few general design principles for instructions in which verbal and visual information are combined. Instructional designers seem to base their design choices more on intuitive ideas than on sound research results. There are however some interesting theories that give guidelines for...

Today, many of the computations in signal processing and wireless communication applications are linked with complex analysis of several functions. These complex functions are combination of sine and cosine terms that generally spread in the channel. Most of these functions can be split into elementary functions. Today user’s desire every gadget must be smaller in size and simple to operate (“K...

We consider the problem of finding the density of ones in a configuration obtained by n iterations of a given cellular automaton (CA) rule, starting from a disordered initial condition. While this problem is intractable in full generality for a general CA rule, we argue that for some sufficiently simple classes of rules it is possible to express the density in terms of elementary functions. Ele...

The matrix Whittaker kernel has been introduced by A. Borodin in Part IV of the present series of papers. This kernel describes a point process — a probability measure on a space of countable point configurations. The kernel is expressed in terms of the Whittaker confluent hypergeometric functions. It depends on two parameters and determines a J-symmetric operator K in L(R+)⊕ L(R+). It turns ou...