Maple Share Library:  Analysis (Calculus)
=========================================

BesselH         > readshare(hankel, program);
                SEE ALSO: hankel.ms

                Implementation of Hankel Functions in terms of BesselJ and
                BesselY.  Contains BesselH, evalf/BesselH, evalc/BesselH,
                expand/BesselH, simplify/BesselH, diff/BesselH.
                AUTHOR: Douglas B. Meade, meade@math.scarolina.edu

billiard.ms     FILE: geometry/billiard.ms
                A Maple worksheet showing how to solve the billiard problem
                for a circular billiard table.  This is an example of solving
                a difficult equation, first analytically, then numerically.
                The numerical solution shows the use of automatic
                differentiation to differentiate a program.
                AUTHOR: Dominik Gruntz, gruntz@inf.ethz.ch

education       FILES:  calculus/area.ms, calculus/deriv.ms,
                        calculus/diffeq.ms, calculus/fundtheo.ms,
                        calculus/limit.ms, calculus/maxima.ms,

                These six worksheets show how Maple can be applied to
                solve simple problems in calculus at the educational level.
   area.ms      1: Finding areas by integration
   deriv.ms     2: Finding an equation for the tangent line of a graph
   diffeq.ms    3: Solving a DE by the method of separation of variables.
   fundtheo.ms  4: The fundamental theorem of calculus
   limit.ms     5: An introduction to limits.
   maxima.ms    6: Solving two problems of Constrained maxima.
                AUTHOR: Stan Devitt, jsdevitt@maplesoft.on.ca 

fjeforms        > readshare(fjeforms, fjeforms); (54K)
                SEE ALSO (under fjeforms): fjeforms.tex
                WORKSHEETS (under fjeforms):  basis.ms, kerr.ms, np.ms, ts.ms
                INPUT FILES (under fjeforms): basis.in, kerr.in, np.in, ts.in

                The Maple V package fjeforms permits the user to manipulate
                differential forms in a way that has been found useful in
                connection with the general theory of relativity.
                The 4 worksheets show examples of using the package
   basis.ms     1: Calculates the Rici rotation matrix of differential forms
   kerr.ms      2: Verification of the Kerr solution of the Einstein equations
                   and evaluation of its curvature quantities.
   np.ms        3: Derivation of the Newman-Penrose form of the Einstein
                   equations and the Bianchi identities 
   ts.ms        4: Verification of the simplest Tomimatsu-Sato solution of the
                   Ernst equation
                AUTHOR: Fred Ernst, ernst@sun.mcs.clarkson.edu

fourier.ms      FILE: engineer/fourier.ms
                A worksheet which animates how a fourier series
                converges to a function on [0,1]
                AUTHOR: Ocie_Mitchell@hmc.edu

FPS             > readshare(FPS, calculus); (58K)
                The FormalPowerSeries (FPS) function tries to find a formal
                power series expansion for a function in terms of a formula
                for the coefficients.  For example

                  > FormalPowerSeries(exp(x)*sin(x),x);

                       infinity
                        -----     k 1/2                k
                         \      (2 )    sin(1/4 k Pi) x
                          )     ------------------------
                         /                 k!
                        -----
                        k = 0
   
                  > FPS(exp(x^2)*(1-erf(x)),x=infinity,left);
   
                      infinity
                       -----           (- k)     (- k)      (2 k + 1)
                        \      (2 k)! 4      (-1)      (1/x)
                         )     --------------------------------------
                        /                        k!
                       -----
                       k = 0
                      -----------------------------------------------
                                             1/2
                                           Pi

                AUTHOR: Dominik Gruntz, gruntz@inf.ethz.ch

fullparfrac     > readshare(fparfrac, calculus);
                Full partial fraction decomposition of a rational function
                in Q(x) over the algebraic closure of Q without factorization
                AUTHOR: Bruno.Salvy@inria.fr

gdev            > readshare(gdev, calculus); (155K)
                SEE ALSO: calculus/gdev.tex

                A facility for more general series expansions and limits.
                Uses a different model for asymptotic series expansions
                than Maple's asympt and series commands.
                See ?gdev and ?glimit
                AUTHOR: Bruno.Salvy@inria.fr

gfun            > readshare(gfun, calculus); (76K)
                SEE ALSO: calculus/gfun.tex  (46K)

                A package for computing with generating functions.
                Converts implicit equations into differential equations,
                differential equations into recurrences and vice-versa,
                ordinary into exponential recurrences.
                Also recovers linear recurrences or differential equations
                from finite lists of coefficients.
                E.g. given the first few Chebyshev polynomials T(n,x) where
                T(0,x) = 1, T(1,x) = x, T(n,x) = 2*x*T(n-1,x) - T(n-2,x),
                it can recover the generating function for T(n,x)

                      infinity
                       -----
                        \               n       1 - x z
                         )     T(n, x) z  = --------------
                        /                    2
                       -----                z  - 2 x z + 1
                       n = 0

                AUTHORS: Bruno.Salvy@inria.fr, Paul.Zimmermann@loria.fr

guesss          > readshare(guesss, calculus);
                A routine to guess the next values in a sequence.
                This routine searches for more general differential equations
                than the gfun package does.  It uses a generalized continued
                fraction method to find the differential equation.
                AUTHOR: Harm Derksen, hderksen@sci.kun.nl

hint            > readshare(hint, calculus);
hyperint        A routine to compute indefinite hyperelliptic integrals.
                These are integrals of the form
                  /
                 |        p(x)                 p(x),q(x),a(x) are in K[x]
                 |  --------------- dx  where  a(x) has odd degree
                 |  q(x) sqrt(a(x))            q(x) is square-free
                /
                This specialized routine is much faster than the general
                algorithm used by Maple.  Note: implementation for K=Q only.
                AUTHOR: Laurent Bertrand, lbertran@cict.fr

Hurwitz         > readshare(Hurwitz, calculus);
                Test if a polynomial is of Hurwitz type, i.e. the real part
                of it's roots is < 0.  Optionally returns the conditions under
                which the polynomial will be of Hurwitz type -- for use
                with polynomials whose coefficients involve parameters.
                Note: this implementation handles more general cases.
                AUTHOR: Robert Corless, rcorless@uwovax.uwo.ca

IF              > readshare(IF);
                SEE ALSO: IF.tex
                Package of routines for computing with real algebraic
                numbers, signs of the roots of polynomials, solving
                systems of equations, and computing a cylindrical algebraic
                decomposition of a plane algebraic curve.
                AUTHOR: Felipe Cucker, cucker@lsi.upc.es

IntSolve        > readshare(IntSolve, linalg);
                Version 1 of an integral equation solver.
                Note: this package requires the routines in the file
                linalg/Echelon.
                AUTHORS: Honglin Ye & Robert M. Corless, rcorless@uwovax.uwo.ca

LambertW.ps     FILE: calculus/LambertW.ps (249K)
                A PostScript file documenting Lambert's W function
                (the W function in Maple).  Describes known properties of
                the function, integration formulae and techniques,
                asymptotics, and gives many nice applications in various
                areas of mathematics, in particular solutions to ODE's.
                AUTHOR: R. M. Corless, rcorless@uwovax.uwo.ca
                AUTHOR: D. J. Jeffrey, djj@uwo.ca
                AUTHOR: G. H. Gonnet, gonnet@inf.ethz.ch
                AUTHOR: D. E. G. Hare, deghare@daisy.uwaterloo.ca

logmap.ms       FILE: calculus/logmap.ms
                This worksheet explores the period-doubling bifurcation
                sequence of the discrete logistic map.  Although numerical
                tools can be used, we believe it is useful to examine the
                polynomial algebra here in some detail.
                The worksheet makes use of polynomial factorization,
                discriminants, and numerical root finding.
                AUTHOR: R. M. Corless, rcorless@uwovax.uwo.ca

parfrac.ms      FILE: (calculus/parfrac.ms)
                Shows how to use Maple to perform each step of a partial
                fraction decomposition for rational function integration.
                A good example of using Maple as a tool in teaching.
                AUTHOR: Michael Monagan, monagan@inf.ethz.ch

pendulum.ms     FILE: science/pendulum.ms
                Models the motion of a pendulum in a fluid as a second order
                ODE, solves the ODE analytically using Maple, and then plots
                the motion for different fluid resistances.
                AUTHOR: David Harper, D.Harper@qmw.ac.uk
                
PS              > readshare(PS, calculus); (32K)
                The power series package PS is an improved version of the
                library package powseries for manipulating formal power series.
                Improvements are
                1: a power series may be created from a given expression
                   (the coefficients don't have to be specified by equations)
                2: it is no longer necessary to name each intermediate series
                3: the trigonometric functions are implemented
                4: series are automatically displayed to order Order
                   (using the print formatting facility in Maple V)
                AUTHOR: Dominik Gruntz, gruntz@inf.ethz.ch

puiseux         > readshare(IntBasis, algebra);  (24K)
                SEE ALSO: algebra/IntBasis.tex   (30K)
                Computes a Puiseux series expansion of an algebraic function.
                Especially intended for expansions about singular points where
                the result contains more than one branch.
                SEE ALSO: genus and integral_basis in ?share,calculus.
                AUTHOR: Mark van Hoeij, hoeij@sci.kun.nl

ratlode         > readshare(ratlode, calculus);
                Finds the rational solutions to a linear n'th order ODE in y(x)
                with rational coefficients, i.e. an equation of the form

                        sum( a[i](x)*diff(y(x),x$i), i=0..n ) = b(x)

                where a[i](x) are rational functions in x over a field.
                Because of limitations in Maple, the implementation only works
                for coefficients which are transcendental extensions of Q.
                AUTHOR: Manuel Bronstein, bronstei@inf.ethz.ch

taylor.ms       FILE: engineer/taylor.ms
                A worksheet which animates how a Taylor series
                converges to a function on a given interval.
                AUTHOR: Ocie_Mitchell@hmc.edu

