dc.contributor.author Aguilera-Venegas, Gabriel dc.contributor.author Galán-García, José Luis dc.contributor.author Galán-García, María Ángeles dc.contributor.author Padilla-Domínguez, Yolanda dc.contributor.author Rodriguez-Cielos, Pedro dc.contributor.author Rodríguez-Cielos, Ricardo dc.date.accessioned 2014-07-16T10:47:39Z dc.date.available 2014-07-16T10:47:39Z dc.date.created 2014-07-01 dc.date.issued 2014-07-16 dc.identifier.uri http://hdl.handle.net/10630/7850 dc.description.abstract Let us consider the following types of improper integrals: es_ES $$\int_0^\infty f(t)\:{\rm d}t \qquad ; \qquad \int_{-\infty}^0 f(t)\:{\rm d}t \qquad {\rm and} \qquad \int_{-\infty}^\infty f(t)\:{\rm d}t$$ \medskip Let $F$ be an antiderivative of $f$. The basic approach to compute such integrals involves the following computations: \medskip \begin{eqnarray*} \int_0^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_0^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(0)\big) \\ \\ \int_{-\infty}^0 f(t)\:{\rm d}t & = & \lim_{m\to-\infty} \int_m^0 f(t)\:{\rm d}t = \lim_{m\to-\infty} \big(F(0)-F(m)\big) \\ \\ \int_{-\infty}^\infty f(t)\:{\rm d}t & = & \int_{-\infty}^0 f(t)\:{\rm d}t + \int_0^\infty f(t)\:{\rm d}t \qquad \mbox{or, in case of convergence,} \\ \\ \int_{-\infty}^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_{-m}^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(-m)\big) \qquad \mbox{(Cauchy principal value)} \end{eqnarray*} \medskip \noindent But, what happens if an antiderivative $F$ for $f$ or the above limits do not exist? \medskip \noindent For example, for \quad $\displaystyle\int_0^\infty\frac{{\rm sin}(at)}{t}\:{\rm d}t$ \quad ; \quad $\displaystyle\int_0^\infty\frac{{\rm cos}(at)-{\rm cos}(bt)}{t}\:{\rm d}t$ \quad {\rm or} \quad $\displaystyle\int_{-\infty}^\infty\frac{{\rm cos}(at)}{t^2+1}\:{\rm d}t$ \qquad the antiderivatives can not be computed. Hence, the above procedures cannot be used for these examples. \medskip In this work we will deal with advance techniques to compute this kind of improper integrals using a {\sc Cas}. Laplace and Fourier transforms or Residue Theorem in Complex Analysis are some advance techniques which can be used for this matter. \medskip We will introduce the file \textbf{\tt ImproperIntegrals.mth}, developed in {\sc Derive} 6, which deals with such computations. \medskip Some {\sc Cas} use different rules for computing integrations. For example {\sc Rubi} system, a {\bf ru}le-{\bf b}ased {\bf i}ntegrator developed by Albert Rich (see {\tt http://www.apmaths.uwo.ca/\~{ }arich/}), is a very powerful system for computing integrals using rules. We will be able to develop new rules schemes for some improper integrals using {\tt ImproperIntegrals.mth}. These new rules can extend the types of improper integrals that a {\sc Cas} can compute. dc.description.sponsorship Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. es_ES dc.language.iso eng es_ES dc.rights info:eu-repo/semantics/openAccess dc.subject Fourier, Transformaciones de es_ES dc.subject Laplace, Transformación de es_ES dc.subject.other Improper integrals es_ES dc.subject.other CAS es_ES dc.subject.other Laplace Transform es_ES dc.subject.other Fourier Transform es_ES dc.subject.other Residue Theorem es_ES dc.title Advanced techniques to compute improper integrals using a CAS es_ES dc.type info:eu-repo/semantics/conferenceObject es_ES dc.relation.eventtitle Technology and its Integration in Mathematics Education TIME 2014 es_ES dc.relation.eventplace Krems, Austria es_ES dc.relation.eventdate 1-5/07/2014 es_ES
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