Advanced techniques to compute improper integrals using a CAS

Abstract

Let us consider the following types of improper integrals:

$$ \int_0^\infty f(t)\:{\rm d}t \qquad ; \qquad \int_{-\infty}^0 f(t)\:{\rm d}t \qquad {\rm and} \qquad \int_{-\infty}^\infty f(t)\:{\rm d}t $$

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Let $F$ be an antiderivative of $f$. The basic approach to compute such integrals involves the following computations:

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\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*}

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\noindent But, what happens if an antiderivative $F$ for $f$ or the above limits do not exist?

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\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.

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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.

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We will introduce the file \textbf{\tt ImproperIntegrals.mth}, developed in {\sc Derive} 6, which deals with such computations.

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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.