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From: Robert Dodier <robert_dodier@us...>  20050821 16:25:48

Update of /cvsroot/maxima/maxima/doc/intromax In directory sc8prcvs1.sourceforge.net:/tmp/cvsserv32346 Modified Files: intromax.ltx Log Message: Change function names from uppercase to lowercase (except in examples, still need to do those) and change C/D labels to i/o labels. No changes of content. Index: intromax.ltx =================================================================== RCS file: /cvsroot/maxima/maxima/doc/intromax/intromax.ltx,v retrieving revision 1.2 retrieving revision 1.3 diff u d r1.2 r1.3  intromax.ltx 11 Feb 2003 11:17:12 0000 1.2 +++ intromax.ltx 21 Aug 2005 16:25:37 0000 1.3 @@ 17,7 +17,7 @@ \title{Introduction to MACSYMA% +\title{Introduction to Maxima% \thanks{Adapted from ``Perturbation Methods, Bifurcation Theory and Computer Algebra'' by Rand and Armbruster, Springer, 1987} } @@ 33,7 +33,7 @@ \section{Introduction \label{sec:introduction}} To invoke MAXIMA in Linux, type +To invoke Maxima in Linux, type \begin{verbatim} maxima<ret> \end{verbatim} @@ 49,17 +49,17 @@ (C1) \end{verbatim} The {\tt (C1)} is a ``label''. Each input or output line is labelled and can be referred to by its own label for the rest of the session. {\tt C} labels denote your commands and {\tt D} labels denote Displays of the machine's response. \emph{Never use variable names like {\tt C1} or {\tt D5}, as these will be confused with the lines so labeled}. +The {\tt (\%i1)} is a ``label''. Each input or output line is labelled and can be referred to by its +own label for the rest of the session. {\tt i} labels denote your commands and {\tt o} labels +denote Displays of the machine's response. \emph{Never use variable names like {\tt \%i1} or {\tt +\%o5}, as these will be confused with the lines so labeled}. MAXIMA is pragmatic about lower and upper case: regardless of your typing {\tt sin(x)} or {\tt SIN(x)}, +Maxima is pragmatic about lower and upper case: regardless of your typing {\tt sin(x)} or {\tt +sin(x)}, {\tt \verb+%e^x+} or {\tt \verb+%E^x+}, it will understand that you mean the sine and exponential functions, and will echo in standard uppercase {\tt SIN} and {\tt \verb+%E+}. +functions, and will echo in standard uppercase {\tt sin} and {\tt \verb+%e+}. \emph{That doesn't apply to user variables, though: {\tt x} and {\tt X} are \emph{different} variables for MAXIMA ! (Try it.)} +variables for Maxima ! (Try it.)} @@ 68,7 +68,7 @@ \section{Special keys and symbols \label{sec:keys}} \begin{enumerate} \item To end a MAXIMA session, type {\tt quit();}. If you type \verb+^C+, here is what happens: +\item To end a Maxima session, type {\tt quit();}. If you type \verb+^C+, here is what happens: \begin{verbatim} Correctable error: Console interrupt. Signalled by MACSYMATOPLEVEL. @@ 79,14 +79,16 @@ (C1) \end{verbatim} Notice that typing {\tt :q} or {\tt :t} (for \emph{top level}) after the {\tt MAXIMA>>} prompt gets you back to the MAXIMA level. +you back to the Maxima level. \verb+^Y+, on the other hand, won't have any effect but being echoed on the screen; finally \verb+^Z+ will have the same effect as {\tt quit();}. (Here \verb+^+ stands for the control key, so that \verb+^C+ means first press the key marked control and hold it down while pressing the C key.) \item To abort a computation without leaving MAXIMA, type \verb+^C+. It is important for you to +\item To abort a computation without leaving Maxima, type \verb+^C+. It is important for you to know how to do this in case, for example, you begin a computation which is taking too long. Remember to type {\tt :q} at the {\tt MAXIMA>>} prompt to return to MAXIMA. For example: +Remember to type {\tt :q} at the {\tt MAXIMA>>} prompt to return to Maxima. For example: +% sum (1/x^2, x, 1, 10000); +% ^C \begin{verbatim} (C1) sum(1/x^2,x,1,1000); @@ 100,30 +102,30 @@ (C2) \end{verbatim} \item In order to tell MAXIMA that you have finished your command, use the semicolon ({\tt ;}), +\item In order to tell Maxima that you have finished your command, use the semicolon ({\tt ;}), followed by a return. Note that the return key alone does not signal that you are done with your input. \item An alternative input terminator to the semicolon ({\tt ;}) is the dollar sign ({\tt \$}), which, however, supresses the display of MAXIMA's computation. This is useful if you are computing +which, however, supresses the display of Maxima's computation. This is useful if you are computing some long intermediate result, and you don't want to waste time having it displayed on the screen. %\item If you want to completely delete the current input line (and start this line fresh from the %beginning), type a double question mark ({\tt ??}). \item If you wish to repeat a command which you have already given, say on line {\tt (C5)}, you may +\item If you wish to repeat a command which you have already given, say on line {\tt (\%i5)}, you may do so without typing it over again by preceding its label with two single quotes ({\tt ''}), i.e., {\tt ''C5}. (Note that simply inputing {\tt C5} will not do the job  try it.) +''\%i5}. (Note that simply inputing {\tt \%i5} will not do the job  try it.) \item If you want to refer to the immediately preceding result computed my MAXIMA, you can either use its {\tt D} label, or you can use the special symbol percent ({\tt \%}). +\item If you want to refer to the immediately preceding result computed my Maxima, you can either +use its {\tt o} label, or you can use the special symbol percent ({\tt \%}). \item The standard quantities $e$ (natural log base), $i$ (square root of $1$) and $\pi$ ($3.14159\ldots$) are respectively referred to as \verb+%e+ (or \verb+%E+), \verb+%i+ (or \verb+%I+), and \verb+%pi+ (or \verb+%PI+). Note that the use of {\tt \%} here as a prefix +(or \verb+%i+), and \verb+%pi+ (or \verb+%pi+). Note that the use of {\tt \%} here as a prefix is completely unrelated to the use of {\tt \%} to refer to the preceding result computed. \item In order to assign a value to a variable, MAXIMA uses the colon ({\tt :}), not the equal +\item In order to assign a value to a variable, Maxima uses the colon ({\tt :}), not the equal sign. The equal sign is used for representing equations. \end{enumerate} @@ 139,7 +141,8 @@ \item [{\tt .}] matrix multiplication \item [{\tt sqrt(x)}] square root of {\tt x}. \end{description} MAXIMA's output is characterized by exact (rational) arithmetic. E.g., +Maxima's output is characterized by exact (rational) arithmetic. E.g., +% 1/100 + 1/101; \begin{verbatim} (C1) 1/100+1/101; @@ 148,6 +151,8 @@ 10100 \end{verbatim} If irrational numbers are involved in a computation, they are kept in symbolic form: +% (1 + sqrt(2))^5; +% expand (%); \begin{verbatim} (C2) (1+sqrt(2))^5; @@ 159,27 +164,32 @@ \end{verbatim} However, it is often useful to express a result in decimal notation. This may be accomplished by following the expression you want expanded by ``{\tt ,numer}'': +% %, numer; \begin{verbatim} (C4) %,numer; (D4) 82.01219330881976 \end{verbatim} Note the use here of \verb+%+ to refer to the previous result. In this version of MAXIMA, {\tt numer} gives 16 significant figures, of which the last is often unreliable. However, MAXIMA can offer \emph{arbitrarily high +to refer to the previous result. In this version of Maxima, {\tt numer} gives 16 significant +figures, of which the last is often unreliable. However, Maxima can offer \emph{arbitrarily high precision} by using the {\tt bfloat} function: +% bfloat (%o3); \begin{verbatim} (C5) bfloat(d3); (D5) 8.201219330881976B1 \end{verbatim} The number of significant figures displayed is controlled by the MAXIMA variable {\tt FPPREC}, which +The number of significant figures displayed is controlled by the Maxima variable {\tt fpprec}, which has the default value of 16: +% fpprec; \begin{verbatim} (C6) fpprec; (D6) 16 \end{verbatim} Here we reset {\tt FPPREC} to yield 100 digits: +Here we reset {\tt fpprec} to yield 100 digits: +% fpprec: 100; +% ''%o5; \begin{verbatim} (C7) fpprec:100; @@ 190,8 +200,9 @@ 7540110412666123849550160561B1 \end{verbatim} Note the use of two single quotes ({\tt ''}) in {\tt (C8)} to repeat command {\tt (C5)}. MAXIMA can +Note the use of two single quotes ({\tt ''}) in {\tt (\%i8)} to repeat command {\tt (\%i5)}. Maxima can handle very large numbers without approximation: +% 100!; \begin{verbatim} (C9) 100!; @@ 203,8 +214,10 @@ \section{Algebra \label{sec:algebra}} MAXIMA's importance as a computer tool to facilitate analytical calculations becomes more evident +Maxima's importance as a computer tool to facilitate analytical calculations becomes more evident when we see how easily it does algebra for us. Here's an example in which a polynomial is expanded: +% (x + 3*y + x^2*y)^3; +% expand (%); \begin{verbatim} (C1) (x+3*y+x^2*y)^3; @@ 221,6 +234,7 @@ Now suppose we wanted to substitute {\tt 5/z} for {\tt x} in the above expression: \noindent\begin{minipage}{\textwidth} +% %o2, x=5/z; \begin{verbatim} (C3) d2,x=5/z; @@ 237,7 +251,8 @@ z \end{verbatim} \end{minipage} The MAXIMA function RATSIMP will place this over a common denominator: +The Maxima function {\tt ratsimp} will place this over a common denominator: +% ratsimp (%); \begin{verbatim} (C4) ratsimp(%); @@ 248,6 +263,7 @@ + (5625 y + 1875 y) z + 9375 y z + 15625 y )/z \end{verbatim} Expressions may also be {\tt factor}ed: +% factor (%); \begin{verbatim} (C5) factor(%); @@ 257,8 +273,12 @@ 6 z \end{verbatim} MAXIMA can obtain exact solutions to systems of nonlinear algebraic equations. In this example we {\tt solve} three equations in the three unknowns {\tt A}, {\tt B}, {\tt C}: +Maxima can obtain exact solutions to systems of nonlinear algebraic equations. In this example we +{\tt solve} three equations in the three unknowns {\tt a}, {\tt b}, {\tt c}: +% a + b*c = 1; +% b  a*c = 0; +% a + b = 5; +% solve ([%o6, %o7, %o8], [a, b, c]); \begin{verbatim} (C6) a + b*c=1; @@ 283,8 +303,10 @@ {\tt [ \ldots ]}, which itself contains two lists. Each of the latter contain a distinct solution to the simultaneous equations. Trigonometric identities are easy to manipulate in MAXIMA. The function {\tt trigexpand} uses the +Trigonometric identities are easy to manipulate in Maxima. The function {\tt trigexpand} uses the sumofangles formulas to make the argument inside each trig function as simple as possible: +% sin(u + v) * cos(u)^3; +% trigexpand (%); \begin{verbatim} (C10) sin(u+v)*cos(u)^3; @@ 297,6 +319,7 @@ \end{verbatim} The function {\tt trigreduce}, on the other hand, converts an expression into a form which is a sum of terms, each of which contains only a single {\tt sin} or {\tt cos}: +% trigreduce (%o10); \begin{verbatim} (C12) trigreduce(d10); @@ 307,6 +330,9 @@ \end{verbatim} The functions {\tt realpart} and {\tt imagpart} will return the real and imaginary parts of a complex expression: +% w: 3 + k*%i; +% w^2 * %e^w; +% realpart (w); \begin{verbatim} (C13) w:3+k*%i; @@ 324,9 +350,10 @@ \section{Calculus \label{sec:calculus}} MAXIMA can compute derivatives and integrals, expand in Taylor series, take limits, and obtain exact +Maxima can compute derivatives and integrals, expand in Taylor series, take limits, and obtain exact solutions to ordinary differential equations. We begin by defining the symbol {\tt f} to be the following function of {\tt x}: +% f: x^3 * %e^(k*x) * sin(w*x); \begin{verbatim} (C1) f:x^3*%E^(k*x)*sin(w*x); @@ 334,6 +361,7 @@ (D1) x %E SIN(w x) \end{verbatim} We compute the derivative of {\tt f} with respect to {\tt x}: +% diff (f, x); \begin{verbatim} (C2) diff(f,x); @@ 341,6 +369,7 @@ (D2) k x %E SIN(w x) + 3 x %E SIN(w x) + w x %E COS(w x) \end{verbatim} Now we find the indefinite integral of {\tt f} with respect to {\tt x}: +% integrate (f, x); \begin{verbatim} (C3) integrate(f,x); @@ 363,6 +392,8 @@ /(w + 4 k w + 6 k w + 4 k w + k ) \end{verbatim} A slight change in syntax gives definite integrals: +% integrate (1/x^2, x, 1, inf); +% integrate (1/x, x, 0, inf); \begin{verbatim} (C4) integrate(1/x^2,x,1,inf); @@ 372,11 +403,13 @@ Integral is divergent  an error. Quitting. To debug this try DEBUGMODE(TRUE);) \end{verbatim} Next we define the simbol {\tt g} in terms of {\tt f} (previously defined in {\tt C1}) and the +Next we define the simbol {\tt g} in terms of {\tt f} (previously defined in {\tt \%i1}) and the hyperbolic sine function, and find its Taylor series expansion (up to, say, order 3 terms) about the point {\tt x = 0}: \noindent\begin{minipage}{\textwidth} +% g: f / sinh(k*x)^4; +% taylor (g, x, 0, 3); \begin{verbatim} (C6) g:f/sinh(k*x)^4; @@ 395,6 +428,7 @@ \end{verbatim} \end{minipage} The limit of {\tt g} as {\tt x} goes to 0 is computed as follows: +% limit (g, x, 0); \begin{verbatim} (C8) limit(g,x,0); @@ 402,7 +436,8 @@ (D8)  4 k \end{verbatim} MAXIMA also permits derivatives to be represented in unevaluated form (note the quote): +Maxima also permits derivatives to be represented in unevaluated form (note the quote): +% 'diff (y, x); \begin{verbatim} (C9) 'diff(y,x); @@ 410,14 +445,16 @@ (D9)  dx \end{verbatim} The quote operator in {\tt (C9)} means ``do not evaluate''. Without it, MAXIMA would have obtained +The quote operator in {\tt (\%i9)} means ``do not evaluate''. Without it, Maxima would have obtained 0: +% diff (y, x); \begin{verbatim} (C10) diff(y,x); (D10) 0 \end{verbatim} Using the quote operator we can write differential equations: +% 'diff (y, x, 2) + 'diff (y, x) + y; \begin{verbatim} (C11) 'diff(y,x,2) + 'diff(y,x) + y; @@ 427,7 +464,8 @@ 2 dx dx \end{verbatim} MAXIMA's {\tt ODE2} function can solve first and second order ODE's: +Maxima's {\tt ode2} function can solve first and second order ODE's: +% ode2 (%o11, y, x); \begin{verbatim} (C12) ode2(d11,y,x); @@ 440,9 +478,10 @@ \newpage \section{Matrix calculations \label{sec:matrix}} MAXIMA can compute the determinant, inverse and eigenvalues and eigenvectors of matrices which have +Maxima can compute the determinant, inverse and eigenvalues and eigenvectors of matrices which have symbolic elements (i.e., elements which involve algebraic variables.) We begin by entering a matrix {\tt m} element by element: +% m: entermatrix (3, 3); \begin{verbatim} (C1) m:entermatrix(3,3); @@ 467,6 +506,9 @@ [ 1 1 0 ] \end{verbatim} Next we find its transpose, determinant and inverse: +% transpose (m); +% determinant (m); +% invert (m), detout; \begin{verbatim} (C2) transpose(m); @@ 488,8 +530,11 @@ (D4)  a + 1 \end{verbatim} In {\tt (C4)}, the modifier {\tt DETOUT} keeps the determinant outside the inverse. As a check, we +In {\tt (\%i4)}, the modifier {\tt detout} keeps the determinant outside the inverse. As a check, we multiply {\tt m} by its inverse (note the use of the period to represent matrix multiplication): +% m . %o4; +% expand (%); +% factor (%); \begin{verbatim} (C5) m.d4; @@ 523,9 +568,10 @@ [ 0 0 1 ] \end{verbatim} In order to find the eigenvalues and eigenvectors of {\tt m}, we use the function {\tt EIGENVECTORS}: +eigenvectors}: \noindent\begin{minipage}{\textwidth} +% eigenvectors (m); \begin{verbatim} (C8) eigenvectors(m); @@ 543,10 +589,11 @@ [1, , ], [1,  1, 0]] 2 a + 2 2 a + 2 \end{verbatim} In {\tt D8}, the first triple gives the eigenvalues of {\tt m} and the next gives their respective +In {\tt \%o8}, the first triple gives the eigenvalues of {\tt m} and the next gives their respective multiplicities (here each is unrepeated). The next three triples give the corresponding eigenvectors of {\tt m}. In order to extract from this expression one of these eigenvectors, we may use the {\tt PART} function: +use the {\tt part} function: +% part (%, 2); \begin{verbatim} (C9) part(%,2); @@ 556,15 +603,15 @@ \end{verbatim} \end{minipage} \section{Programming in MAXIMA \label{sec:programming}} +\section{Programming in Maxima \label{sec:programming}} So far, we have used MAXIMA in the interactive mode, rather like a calculator. However, for +So far, we have used Maxima in the interactive mode, rather like a calculator. However, for computations which involve a repetitive sequence of commands, it is better to execute a program. Here we present a short sample program to calculate the critical points of a function {\tt f} of two variables {\tt x} and {\tt y}. The program cues the user to enter the function {\tt f}, then it computes the partial derivatives $\mathtt{f_x}$ and $\mathtt{f_y}$, and then it uses the MAXIMA command {\tt SOLVE} to obtain solutions to $\mathtt{f_x = f_y = 0}$. The program is written outside of MAXIMA with a text editor, and then loaded into MAXIMA with the {\tt BATCH} command. Here is the +computes the partial derivatives $\mathtt{f_x}$ and $\mathtt{f_y}$, and then it uses the Maxima +command {\tt solve} to obtain solutions to $\mathtt{f_x = f_y = 0}$. The program is written outside +of Maxima with a text editor, and then loaded into Maxima with the {\tt batch} command. Here is the program listing: \begin{verbatim} /*  @@ 600,9 +647,11 @@ )$ \end{verbatim} The program (which is actually a function with no argument) is called {\tt critpts}. Each line is a valid MAXIMA command which could be executed from the keyboard, and which is separated by the next +valid Maxima command which could be executed from the keyboard, and which is separated by the next command by a comma. The partial derivatives are stored in a variable named {\tt eqs}, and the unknowns are stored in {\tt unk}. Here is a sample run: +% batch ("critpts.max"); +% critpts (); \begin{verbatim} (C1) batch("critpts.max"); @@ 639,117 +688,117 @@ y = 0.86972626928141 %I + 0.4559120701117]] \end{verbatim} \section{A partial list of MAXIMA functions} +\section{A partial list of Maxima functions} See the MAXIMA Manual in the {\tt maxima5.4/info/} directory in texinfo or html format. From MAXIMA itself, you can use {\tt DESCRIBE(\textit{function name})}. +See the Maxima reference manual {\tt doc/html/maxima_toc.html} (under the main Maxima installation directory). +From Maxima itself, you can use {\tt describe(\textit{function name})}. \begin{description}  \item[{\tt ALLROOTS(A)}] Finds all the (generally complex) roots of the polynomial equation {\tt  A}, and lists them in {\tt NUMER}ical format (i.e. to 16 significant figures).  \item[{\tt APPEND(A,B)}] Appends the list {\tt B} to the list {\tt A}, resulting in a single + \item[{\tt allroots(a)}] Finds all the (generally complex) roots of the polynomial equation {\tt + A}, and lists them in {\tt numer}ical format (i.e. to 16 significant figures). + \item[{\tt append(a,b)}] Appends the list {\tt b} to the list {\tt a}, resulting in a single list.  \item[{\tt BATCH(A)}] Loads and runs a BATCH program with filename {\tt A}.  \item[{\tt COEFF(A,B,C)}] Gives the coefficient of {\tt B} raised to the power {\tt C} in  expression {\tt A}.  \item[{\tt CONCAT(A,B)}] Creates the symbol {\tt AB}.  \item[{\tt CONS(A,B)}] Adds {\tt A} to the list {\tt B} as its first element.  \item[{\tt DEMOIVRE(A)}] Transforms all complex exponentials in {\tt A} to their trigonometric + \item[{\tt batch(a)}] Loads and runs a program with filename {\tt a}. + \item[{\tt coeff(a,b,c)}] Gives the coefficient of {\tt b} raised to the power {\tt c} in + expression {\tt a}. + \item[{\tt concat(a,b)}] Creates the symbol {\tt ab}. + \item[{\tt cons(a,b)}] Adds {\tt a} to the list {\tt b} as its first element. + \item[{\tt demoivre(a)}] Transforms all complex exponentials in {\tt a} to their trigonometric equivalents.  \item[{\tt DENOM(A)}] Gives the denominator of {\tt A}.  \item[{\tt DEPENDS(A,B)}] Declares {\tt A} to be a function of {\tt B}. This is useful for + \item[{\tt denom(a)}] Gives the denominator of {\tt a}. + \item[{\tt depends(a,b)}] Declares {\tt a} to be a function of {\tt b}. This is useful for writing unevaluated derivatives, as in specifying differential equations.  \item[{\tt DESOLVE(A,B)}] Attempts to solve a linear system {\tt A} of ODE's for unknowns {\tt B} + \item[{\tt desolve(a,b)}] Attempts to solve a linear system {\tt a} of ODE's for unknowns {\tt b} using Laplace transforms.  \item[{\tt DETERMINANT(A)}] Returns the determinant of the square matrix {\tt A}.  \item[{\tt DIFF(A,B1,C1,B2,C2,\ldots,Bn,Cn)}] Gives the mixed partial derivative of {\tt A} with  respect to each {\tt Bi}, {\tt Ci} times. For brevity, {\tt DIFF(A,B,1)} may be represented by  {\tt DIFF(A,B)}. {\tt 'DIFF(\ldots)} represents the unevaluated derivative, useful in specifying + \item[{\tt determinant(a)}] Returns the determinant of the square matrix {\tt a}. + \item[{\tt diff(a,b1,c1,b2,c2,\ldots,bn,cn)}] Gives the mixed partial derivative of {\tt a} with + respect to each {\tt bi}, {\tt ci} times. For brevity, {\tt diff(a,b,1)} may be represented by + {\tt diff(a,b)}. {\tt 'diff(\ldots)} represents the unevaluated derivative, useful in specifying a differential equation.  \item[{\tt EIGENVALUES(A)}] Returns two lists, the first being the eigenvalues of the square  matrix {\tt A}, and the second being their respective multiplicities.  \item[{\tt EIGENVECTORS(A)}] Does everything that {\tt EIGENVALUES} does, and adds a list of the  eigenvectors of {\tt A}.  \item[{\tt ENTERMATRIX(A,B)}] Cues the user to enter an $\mathtt{A} \times\, \mathtt{B}$ matrix, + \item[{\tt eigenvalues(a)}] Returns two lists, the first being the eigenvalues of the square + matrix {\tt a}, and the second being their respective multiplicities. + \item[{\tt eigenvectors(a)}] Does everything that {\tt eigenvalues} does, and adds a list of the + eigenvectors of {\tt a}. + \item[{\tt entermatrix(a,b)}] Cues the user to enter an $\mathtt{a} \times\, \mathtt{b}$ matrix, element by element.  \item[{\tt EV(A,B1,B2,\ldots,Bn)}] Evaluates {\tt A} subject to the conditions {\tt Bi}. In  particular the {\tt Bi} may be equations, lists of equations (such as that returned by {\tt  SOLVE}), or assignments, in which cases {\tt EV} ``plugs'' the {\tt Bi} into {\tt A}. The {\tt  Bi} may also be words such as {\tt NUMER} (in which case the result is returned in numerical  format), {\tt DETOUT} (in which case any matrix inverses in {\tt A} are performed with the  determinant factored out), or {\tt DIFF} (in which case all differentiations in {\tt A} are  evaluated, i.e., {\tt 'DIFF} in {\tt A} is replaced by {\tt DIFF}). For brevity in a manual  command (i.e., not inside a userdefined function), the {\tt EV} may be dropped, shortening the  syntax to {\tt A,B1,B2,\ldots,Bn}.  \item[{\tt EXPAND(A)}] Algebraically expands {\tt A}. In particular multiplication is + \item[{\tt ev(a,b1,b2,\ldots,bn)}] Evaluates {\tt a} subject to the conditions {\tt bi}. In + particular the {\tt bi} may be equations, lists of equations (such as that returned by {\tt + solve}), or assignments, in which cases {\tt ev} ``plugs'' the {\tt bi} into {\tt a}. The {\tt + Bi} may also be words such as {\tt numer} (in which case the result is returned in numerical + format), {\tt detout} (in which case any matrix inverses in {\tt a} are performed with the + determinant factored out), or {\tt diff} (in which case all differentiations in {\tt a} are + evaluated, i.e., {\tt 'diff} in {\tt a} is replaced by {\tt diff}). For brevity in a manual + command (i.e., not inside a userdefined function), the {\tt ev} may be dropped, shortening the + syntax to {\tt a,b1,b2,\ldots,bn}. + \item[{\tt expand(a)}] Algebraically expands {\tt a}. In particular multiplication is distributed over addition.  \item[{\tt EXPONENTIALIZE(A)}] Transforms all trigonometric functions in {\tt A} to their complex + \item[{\tt exponentialize(a)}] Transforms all trigonometric functions in {\tt a} to their complex exponential equivalents.  \item[{\tt FACTOR(A)}] Factors {\tt A}.  \item[{\tt FREEOF(A,B)}] Is true if the variable {\tt A} is not part of the expression {\tt B}.  \item[{\tt GRIND(A)}] Displays a variable or function {\tt A} in a compact format. When used  with {\tt WRITEFILE} and an editor outside of MAXIMA, it offers a scheme for producing {\tt  BATCH} files which include MAXIMAgenerated expressions.  \item[{\tt IDENT(A)}] Returns an $\mathtt{A} \times\, \mathtt{A}$ identity matrix.  \item[{\tt IMAGPART(A)}] Returns the imaginary part of {\tt A}.  \item[{\tt INTEGRATE(A,B)}] Attempts to find the indefinite integral of {\tt A} with respect to  {\tt B}.  \item[{\tt INTEGRATE(A,B,C,D)}] Attempts to find the indefinite integral of {\tt A} with respect to  {\tt B}. taken from $\mathtt{B=C}$ to $\mathtt{B=D}$. The limits of integration {\tt C} and {\tt  D} may be taken is {\tt INF} (positive infinity) of {\tt MINF} (negative infinity).  \item[{\tt INVERT(A)}] Computes the inverse of the square matrix {\tt A}.  \item[{\tt KILL(A)}] Removes the variable {\tt A} with all its assignments and properties from  the current MAXIMA environment.  \item[{\tt LIMIT(A,B,C)}] Gives the limit of expression {\tt A} as variable {\tt B} approaches  the value {\tt C}. The latter may be taken as {\tt INF} of {\tt MINF} as in {\tt INTEGRATE}.  \item[{\tt LHS(A)}] Gives the lefthand side of the equation {\tt A}.  \item[{\tt LOADFILE(A)}] Loads a disk file with filename {\tt A} from the current default  directory. The disk file must be in the proper format (i.e. created by a {\tt SAVE} command).  \item[{\tt MAKELIST(A,B,C,D)}] Creates a list of {\tt A}'s (each of which presumably depends on  {\tt B}), concatenated from $\mathtt{B=C}$ to $\mathtt{B=D}$  \item[{\tt MAP(A,B)}] Maps the function {\tt A} onto the subexpressions of {\tt B}.  \item[{\tt MATRIX(A1,A2,\ldots,An)}] Creates a matrix consisting of the rows {\tt Ai}, where each  row {\tt Ai} is a list of {\tt m} elements, {\tt [B1, B2, \ldots, Bm]}.  \item[{\tt NUM(A)}] Gives the numerator of {\tt A}.  \item[{\tt ODE2(A,B,C)}] Attempts to solve the first or secondorder ordinary differential  equation {\tt A} for {\tt B} as a function of {\tt C}.  \item[{\tt PART(A,B1,B2,\ldots,Bn)}] First takes the {\tt B1}th part of {\tt A}, then the {\tt + \item[{\tt factor(a)}] Factors {\tt a}. + \item[{\tt freeof(a,b)}] Is true if the variable {\tt a} is not part of the expression {\tt b}. + \item[{\tt grind(a)}] Displays a variable or function {\tt a} in a compact format. When used + with {\tt writefile} and an editor outside of Maxima, it offers a scheme for producing {\tt + batch} files which include Maximagenerated expressions. + \item[{\tt ident(a)}] Returns an $\mathtt{a} \times\, \mathtt{a}$ identity matrix. + \item[{\tt imagpart(a)}] Returns the imaginary part of {\tt a}. + \item[{\tt integrate(a,b)}] Attempts to find the indefinite integral of {\tt a} with respect to + {\tt b}. + \item[{\tt integrate(a,b,c,d)}] Attempts to find the indefinite integral of {\tt a} with respect to + {\tt b}. taken from $\mathtt{b=c}$ to $\mathtt{b=d}$. The limits of integration {\tt c} and {\tt + D} may be taken is {\tt inf} (positive infinity) of {\tt minf} (negative infinity). + \item[{\tt invert(a)}] Computes the inverse of the square matrix {\tt a}. + \item[{\tt kill(a)}] Removes the variable {\tt a} with all its assignments and properties from + the current Maxima environment. + \item[{\tt limit(a,b,c)}] Gives the limit of expression {\tt a} as variable {\tt b} approaches + the value {\tt c}. The latter may be taken as {\tt inf} of {\tt minf} as in {\tt integrate}. + \item[{\tt lhs(a)}] Gives the lefthand side of the equation {\tt a}. + \item[{\tt loadfile(a)}] Loads a disk file with filename {\tt a} from the current default + directory. The disk file must be in the proper format (i.e. created by a {\tt save} command). + \item[{\tt makelist(a,b,c,d)}] Creates a list of {\tt a}'s (each of which presumably depends on + {\tt b}), concatenated from $\mathtt{b=c}$ to $\mathtt{b=d}$ + \item[{\tt map(a,b)}] Maps the function {\tt a} onto the subexpressions of {\tt b}. + \item[{\tt matrix(a1,a2,\ldots,an)}] Creates a matrix consisting of the rows {\tt ai}, where each + row {\tt ai} is a list of {\tt m} elements, {\tt [b1, b2, \ldots, bm]}. + \item[{\tt num(a)}] Gives the numerator of {\tt a}. + \item[{\tt ode2(a,b,c)}] Attempts to solve the first or secondorder ordinary differential + equation {\tt a} for {\tt b} as a function of {\tt c}. + \item[{\tt part(a,b1,b2,\ldots,bn)}] First takes the {\tt b1}th part of {\tt a}, then the {\tt B2}th part of that, and so on.  \item[{\tt PLAYBACK(A)}] Displays the last {\tt A} (an integer) labels and their associated  expressions. If {\tt A} is omitted, all lines are played back. See the Manual for other + \item[{\tt playback(a)}] Displays the last {\tt a} (an integer) labels and their associated + expressions. If {\tt a} is omitted, all lines are played back. See the Manual for other options.  \item[{\tt RATSIMP(A)}] Simplifies {\tt A} and returns a quotient of two polynomials.  \item[{\tt REALPART(A)}] Returns the real part of {\tt A}.  \item[{\tt RHS(A)}] Gives the righthand side of the equation {\tt A}.  \item[{\tt SAVE(A,B1,B2,\ldots, Bn)}] Creates a disk file with filename {\tt A} in the current  default directory, of variables, functions, or arrays {\tt Bi}. The format of the file permits  it to be reloaded into MAXIMA using the {\tt LOADFILE} command. Everything (including labels)  may be {\tt SAVE}d by taking {\tt B1} equal to {\tt ALL}.  \item[{\tt SOLVE(A,B)}] Attempts to solve the algebraic equation {\tt A} for the unknown {\tt B}. A  list of solution equations is returned. For brevity, if {\tt A} is an equation of the form  $\mathtt{C = 0}$, it may be abbreviated simply by the expression {\tt C}.  \item[{\tt STRING(A)}] Converts {\tt A} to MACSYMA's linear notation (similar to FORTRAN's) just as if  it had been typed in and puts {\tt A} into the  buffer for possible editing. The STRING'ed expression should not be used in a computation.  \item[{\tt STRINGOUT(A,B1,B2,\ldots,Bn)}] Creates a disk file with filename {\tt A} in the current  default directory, of variables (e.g. labels) {\tt Bi}. The file is in a text format and is not  reloadable into MAXIMA. However the strungout expressions can be incorporated into a FORTRAN,  BASIC or C program with a minimum of editing.  \item[{\tt SUBST(A,B,C)}] Substitutes {\tt A} for {\tt B} in {\tt C}.  \item[{\tt TAYLOR(A,B,C,D)}] Expands {\tt A} in a Taylor series in {\tt B} about $\mathtt{B=C}$,  up to and including the term $\mathtt{(BC)^D}$. MAXIMA also supports Taylor expansions in more + \item[{\tt ratsimp(a)}] Simplifies {\tt a} and returns a quotient of two polynomials. + \item[{\tt realpart(a)}] Returns the real part of {\tt a}. + \item[{\tt rhs(a)}] Gives the righthand side of the equation {\tt a}. + \item[{\tt save(a,b1,b2,\ldots, bn)}] Creates a disk file with filename {\tt a} in the current + default directory, of variables, functions, or arrays {\tt bi}. The format of the file permits + it to be reloaded into Maxima using the {\tt loadfile} command. Everything (including labels) + may be {\tt save}d by taking {\tt b1} equal to {\tt all}. + \item[{\tt solve(a,b)}] Attempts to solve the algebraic equation {\tt a} for the unknown {\tt b}. A + list of solution equations is returned. For brevity, if {\tt a} is an equation of the form + $\mathtt{c = 0}$, it may be abbreviated simply by the expression {\tt c}. + \item[{\tt string(a)}] Converts {\tt a} to Maxima's linear notation (similar to Fortran's) just as if + it had been typed in and puts {\tt a} into the + buffer for possible editing. The {\tt string}'ed expression should not be used in a computation. + \item[{\tt stringout(a,b1,b2,\ldots,bn)}] Creates a disk file with filename {\tt a} in the current + default directory, of variables (e.g. labels) {\tt bi}. The file is in a text format and is not + reloadable into Maxima. However the strungout expressions can be incorporated into a Fortran, + Basic or C program with a minimum of editing. + \item[{\tt subst(a,b,c)}] Substitutes {\tt a} for {\tt b} in {\tt c}. + \item[{\tt taylor(a,b,c,d)}] Expands {\tt a} in a Taylor series in {\tt b} about $\mathtt{b=c}$, + up to and including the term $\mathtt{(bc)^d}$. Maxima also supports Taylor expansions in more than one independent variable; see the Manual for details.  \item[{\tt TRANSPOSE(A)}] Gives the transpose of the matrix {\tt A}.  \item[{\tt TRIGEXPAND(A)}] Is a trig simplification function which uses the sumofangles  formulas to simplify the arguments of individual {\tt SIN}'s or {\tt COS}'s. For example,  {\tt trigexpand(sin(x+y))} gives {\tt COS(x) SIN(y) + SIN(x) COS(y)}.  \item[{\tt TRIGREDUCE(A)}] Is a trig simplification function which uses trig identities to  convert products and powers of {\tt SIN} and {\tt COS} into a sum of terms, each of which  contains only a single {\tt SIN} or {\tt COS}. For example, \verb+trigreduce(sin(x)^2)+ gives  {\tt (1  COS(2x))/2}.  \item[{\tt TRIGSIMP(A)}] Is a trig simplification function which replaces {\tt TAN}, {\tt SEC},  etc., by their {\tt SIN} and {\tt COS} equivalents. It also uses the identity  $\mathtt{SIN()^2 + COS()^2 = 1}$. + \item[{\tt transpose(a)}] Gives the transpose of the matrix {\tt a}. + \item[{\tt trigexpand(a)}] Is a trig simplification function which uses the sumofangles + formulas to simplify the arguments of individual {\tt sin}'s or {\tt cos}'s. For example, + {\tt trigexpand(sin(x+y))} gives {\tt cos(x) sin(y) + sin(x) cos(y)}. + \item[{\tt trigreduce(a)}] Is a trig simplification function which uses trig identities to + convert products and powers of {\tt sin} and {\tt cos} into a sum of terms, each of which + contains only a single {\tt sin} or {\tt cos}. For example, \verb+trigreduce(sin(x)^2)+ gives + {\tt (1  cos(2x))/2}. + \item[{\tt trigsimp(a)}] Is a trig simplification function which replaces {\tt tan}, {\tt sec}, + etc., by their {\tt sin} and {\tt cos} equivalents. It also uses the identity + $\mathtt{sin()^2 + cos()^2 = 1}$. \end{description} 
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