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A Framework for an ICT-Based Study of Parametric Integrals

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Thierry Dana-Picard at Jerusalem College of Technology
Thierry Dana-Picard
David G. Zeitoun at Orot College of edeucation
David G. Zeitoun
  • Orot College of edeucation
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Abstract

We describe working session for the study of 1-parameter families of definite integrals in a technology-rich environment. The joint usage of paper-and-pencil work together with a Computer Algebra System and eventually a web based database may lead to closed forms for the integrals, to the derivation of combinatorial identities, and other kinds of output. This joint usage applied to parametric integrals lead to mathematical expressions rather than graphic representation. This permits a new learning process in the teaching of mathematics, physics and engineering. We begin with a short survey of classical cases, using telescopic methods. Then we show a new example where the integrals depend not only on an integer parameter, but also on a real variable. In this last case, the study of the parametric integral involves the study of a recurrence differential equation.
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Math.Comput.Sci. (2017) 11:285–296
DOI 10.1007/s11786-017-0299-z
Mathematics in Computer Science
A Framework for an ICT-Based Study of Parametric Integrals
Thierry Dana-Picard ·David G. Zeitoun
Received: 27 November 2016 / Revised: 20 February 2017 / Accepted: 21 February 2017 / Published online: 25 April 2017
© Springer International Publishing 2017
Abstract We describe working session for the study of 1-parameter families of definite integrals in a technology-
rich environment. The joint usage of paper-and-pencil work together with a Computer Algebra System and eventually
a web based database may lead to closed forms for the integrals, to the derivation of combinatorial identities, and
other kinds of output. This joint usage applied to parametric integrals lead to mathematical expressions rather than
graphic representation. This permits a new learning process in the teaching of mathematics, physics and engineering.
We begin with a short survey of classical cases, using telescopic methods. Then we show a new example where
the integrals depend not only on an integer parameter, but also on a real variable. In this last case, the study of the
parametric integral involves the study of a recurrence differential equation.
Keywords Parametric integrals ·Computer Algebra Systems ·Recurrence differential equation
Mathematics Subject Classification Primary 26A06 ·26A18 ·97U70; Secondary 34A99
1 Introduction
A Computer Algebra System (CAS) is a multi-purpose package which is intended to be an assistant to the mathe-
matical work both of students and of teachers and researchers. The various mathematical objects may have multi-
representations. For example a function of a real variable has traditionally the three following kinds of representation:
•Algebraic representation, given by a closed expression of the form f(x)=..., where xis the variable.
•Graphical representation, i.e. the curve defined by the equation y=f(x). This is a geometrical object.
•A numerical representation, generally presented as a table of values.
Other objects may have more or less registers of representation. For a 1-parameter family of definite integrals, the
work we describe in this paper is performed using first a numerical representation, trying to conjecture a general
T. Dana-Picard (B)
Jerusalem College of Technology, Havaad Haleumi Street 21, 91160 Jerusalem, Israel
e-mail: ndp@jct.ac.il
D. G. Zeitoun
Orot Israel College, Elkana, D.N. Hare Efraim, Israel
e-mail: ed.technologie@gmail.com
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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