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\chapter{DEFINT: Definite Integration for REDUCE} \label{DEFINT} \typeout{{DEFINT: Definite Integration for REDUCE}} {\footnotesize \begin{center} Kerry Gaskell and Winfried Neun \\ Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\ Takustra\"se 7 \\ D--14195 Berlin--Dahlem, Germany \\[0.05in] e--mail: neun@zib.de \\[0.10in] Stanley L. Kameny \\ Los Angeles, U.S.A. \end{center} } \ttindex{DEFINT} \REDUCE{}'s definite integration package is able to calculate the definite integrals of many functions, including several special functions. There are a number of parts of this package, including contour integration. The innovative integration process is to represent each function as a Meijer G-function, and then calculating the integral by using the following Meijer G integration formula. \begin{displaymath} \int_{0}^{\infty} x^{\alpha-1} G^{s t}_{u v} \left( \sigma x \ \Bigg\vert \ {( c_u) \atop (d_v)} \right) G^{m n}_{p q} \left( \omega x^{l/k} \ \Bigg\vert \ {(a_p) \atop (b_q)} \right) dx = k G^{i j}_{k l} \left( \xi \ \Bigg\vert \ {(g_k) \atop (h_l)} \right) \hspace{5mm} (1) \end{displaymath} The resulting Meijer G-function is then retransformed, either directly or via a hypergeometric function simplification, to give the answer. The user interface is via a four argument version of the \f{INT}\ttindex{INT} operator, with the lower and upper limits added. \begin{verbatim} load_package defint; int(sin x,x,0,pi/2); 1 \end{verbatim} \newpage \begin{verbatim} int(log(x),x,1,5); 5*log(5) - 4 int(x*e^(-1/2x),x,0,infinity); 4 int(x^2*cos(x)*e^(-2*x),x,0,infinity); 4 ----- 125 int(x^(-1)*besselj(2,sqrt(x)),x,0,infinity); 1 int(si(x),x,0,y); cos(y) + si(y)*y - 1 int(besselj(2,x^(1/4)),x,0,y); 1/4 4*besselj(3,y )*y --------------------- 1/4 y \end{verbatim} The DEFINT package also defines a number of additional transforms, such as the Laplace transform\index{Laplace transform}\footnote{See Chapter~\ref{LAPLACE} for an alternative Laplace transform with inverse Laplace transform}, the Hankel transform\index{Hankel transform}, the Y-transform\index{Y-transform}, the K-transform\index{K-transform}, the StruveH transform\index{StruveH transform}, the Fourier sine transform\index{Fourier sine transform}, and the Fourier cosine transform\index{Fourier cosine transform}. \begin{verbatim} laplace_transform(cosh(a*x),x); - s --------- 2 2 a - s laplace_transform(Heaviside(x-1),x); 1 ------ s e *s hankel_transform(x,x); n + 4 gamma(-------) 2 ------------------- n - 2 2 gamma(-------)*s 2 fourier_sin(e^(-x),x); s -------- 2 s + 1 fourier_cos(x,e^(-1/2*x^2),x); 2 i*s s /2 sqrt( - pi)*erf(---------)*s + e *sqrt(2) sqrt(2) ---------------------------------------------- 2 s /2 e *sqrt(2) \end{verbatim} It is possible to the user to extend the pattern-matching process by which the relevant Meijer G representation for any function is found. Details can be found in the complete documentation. \noindent{\bf Acknowledgement:} This package depends greatly on the pioneering work of Victor Adamchik, to whom thanks are due.