RT Conference Proceedings
T1 Advanced techniques to compute improper integrals using a CAS
A1 Aguilera-Venegas, Gabriel
A1 Galán-García, José Luis
A1 Galán-García, María Ángeles
A1 Padilla-Domínguez, Yolanda Carmen
A1 Rodríguez-Cielos, Pedro
A1 Rodríguez-Cielos, Ricardo
K1 Fourier, Transformaciones de
K1 Laplace, Transformación de
AB Let us consider the following types of improper integrals:$$\int_0^\infty f(t)\:{\rm d}t \qquad ; \qquad \int_{-\infty}^0 f(t)\:{\rm d}t \qquad {\rm and} \qquad \int_{-\infty}^\infty f(t)\:{\rm d}t$$\medskipLet $F$ 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.\medskipIn 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.\medskipWe will introduce the file \textbf{\tt ImproperIntegrals.mth}, developed in {\sc Derive} 6, which deals with such computations.\medskipSome {\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.
YR 2014
FD 2014-07-16
LK http://hdl.handle.net/10630/7850
UL http://hdl.handle.net/10630/7850
LA eng
NO Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 24 ene 2026