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dc.contributor.authorAguilera-Venegas, Gabriel 
dc.contributor.authorGalán-García, José Luis 
dc.contributor.authorGalán-García, María Ángeles 
dc.contributor.authorPadilla-Domínguez, Yolanda
dc.contributor.authorRodriguez-Cielos, Pedro 
dc.contributor.authorRodríguez-Cielos, Ricardo
dc.date.accessioned2014-07-16T10:47:39Z
dc.date.available2014-07-16T10:47:39Z
dc.date.created2014-07-01
dc.date.issued2014-07-16
dc.identifier.urihttp://hdl.handle.net/10630/7850
dc.description.abstractLet us consider the following types of improper integrals: $$ \int_0^\infty f(t)\:{\rm d}t \qquad ; \qquad \int_{-\infty}^0 f(t)\:{\rm d}t \qquad {\rm and} \qquad \int_{-\infty}^\infty f(t)\:{\rm d}t $$ \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.es_ES
dc.description.sponsorshipUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.es_ES
dc.language.isoenges_ES
dc.rightsinfo:eu-repo/semantics/openAccess
dc.subjectFourier, Transformaciones dees_ES
dc.subjectLaplace, Transformación dees_ES
dc.subject.otherImproper integralses_ES
dc.subject.otherCASes_ES
dc.subject.otherLaplace Transformes_ES
dc.subject.otherFourier Transformes_ES
dc.subject.otherResidue Theoremes_ES
dc.titleAdvanced techniques to compute improper integrals using a CASes_ES
dc.typeinfo:eu-repo/semantics/conferenceObjectes_ES
dc.relation.eventtitleTechnology and its Integration in Mathematics Education TIME 2014es_ES
dc.relation.eventplaceKrems, Austriaes_ES
dc.relation.eventdate1-5/07/2014es_ES


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