## [Maxima-commits] CVS: maxima/doc/info Series.texi,1.7,1.8

 [Maxima-commits] CVS: maxima/doc/info Series.texi,1.7,1.8 From: Robert Dodier - 2005-02-27 21:38:27 ```Update of /cvsroot/maxima/maxima/doc/info In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv2846 Modified Files: Series.texi Log Message: Edits for greater clarity; reworked some examples. Still needs work. Index: Series.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/Series.texi,v retrieving revision 1.7 retrieving revision 1.8 diff -u -d -r1.7 -r1.8 --- Series.texi 27 Feb 2005 19:03:38 -0000 1.7 +++ Series.texi 27 Feb 2005 21:38:18 -0000 1.8 @@ -293,23 +293,31 @@ @end defun @defvar powerdisp - default: @code{false} - if @code{true} will cause sums to be displayed -with their terms in the reverse order. Thus polynomials would display -as truncated power series, i.e., with the lowest power first. +Default value: @code{false} + +When @code{powerdisp} is @code{true}, +a sum is displayed with its terms in order of increasing power. +Thus a polynomial is displayed as a truncated power series, +with the constant term first and the highest power last. + +By default, terms of a sum are displayed in order of decreasing power. +@c NEED AN EXAMPLE HERE @end defvar -@... powerseries (exp, var, pt) -generates the general form of the power -series expansion for exp in the variable var about the point pt (which -may be @code{inf} for infinity). If @code{powerseries} is unable to expand exp, the -@...{taylor} function may give the first several terms of the series. -@...{verbose} - if @code{true} will cause comments about the progress of -@...{powerseries} to be printed as the execution of it proceeds. +@defun powerseries (@var{expr}, @var{x}, @var{x_0}) +Returns the general form of the power series expansion for @var{expr} +in the variable @var{x} about the point @var{x_0} (which may be @code{inf} for infinity). + +If @code{powerseries} is unable to expand @var{expr}, +@code{taylor} may give the first several terms of the series. + +When @code{verbose} is @code{true}, +@code{powerseries} prints progress messages. @example -(%i1) verbose:true\$ -(%i2) powerseries(log(sin(x)/x),x,0); +(%i1) verbose: true\$ +(%i2) powerseries (log(sin(x)/x), x, 0); can't expand log(sin(x)) so we'll try again after applying the rule: @@ -339,67 +347,115 @@ @end defun @defvar psexpand - default: @code{false} - if @code{true} will cause extended rational -function expressions to display fully expanded. (@code{ratexpand} will also -cause this.) If @code{false}, multivariate expressions will be displayed -just as in the rational function package. If @code{psexpand:multi}, then -terms with the same total degree in the variables are grouped -together. +Default value: @code{false} + +When @code{psexpand} is @code{true}, +an extended rational function expression is displayed fully expanded. +The switch @code{ratexpand} has the same effect. + +@c WE NEED TO BE EXPLICIT HERE +When @code{psexpand} is @code{false}, +a multivariate expression is displayed just as in the rational function package. + +@c TERMS OF WHAT ?? +When @code{psexpand} is @code{multi}, +then terms with the same total degree in the variables are grouped together. @end defvar -@... revert (expression,variable) -Does reversion of Taylor Series. -"Variable" is the variable the original Taylor expansion is in. Do -@...{load(revert)} to access this function. Try +@defun revert (@var{expr}, @var{x}) +@defunx revert2 (@var{expr}, @var{x}, @var{n}) +These functions return the reversion of @var{expr}, a Taylor series about zero in the variable @var{x}. +@code{revert} returns a polynomial of degree equal to the highest power in @var{expr}. +@code{revert2} returns a polynomial of degree @var{n}, +which may be greater than, equal to, or less than the degree of @var{expr}. + +@code{load ("revert")} loads these functions. + +Examples: @example -revert2(expression,variable,hipower) +(%i1) load ("revert")\$ +(%i2) t: taylor (exp(x) - 1, x, 0, 6); + 2 3 4 5 6 + x x x x x +(%o2)/T/ x + -- + -- + -- + --- + --- + . . . + 2 6 24 120 720 +(%i3) revert (t, x); + 6 5 4 3 2 + 10 x - 12 x + 15 x - 20 x + 30 x - 60 x +(%o3)/R/ - -------------------------------------------- + 60 +(%i4) ratexpand (%); + 6 5 4 3 2 + x x x x x +(%o4) - -- + -- - -- + -- - -- + x + 6 5 4 3 2 +(%i5) taylor (log(x+1), x, 0, 6); + 2 3 4 5 6 + x x x x x +(%o5)/T/ x - -- + -- - -- + -- - -- + . . . + 2 3 4 5 6 +(%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6)); +(%o6) 0 +(%i7) revert2 (t, x, 4); + 4 3 2 + x x x +(%o7) - -- + -- - -- + x + 4 3 2 @end example -also. @code{revert} only works on -expansions around 0. - @end defun -@... srrat (exp) -this command has been renamed to @code{taytorat}. +@c OBSOLETE AS ADVERTISED; NOT CALLED FROM ANY USER-LEVEL FILE (.mac OR .dem) +@c CUT THIS OUT ON NEXT PASS +@c @defun srrat (exp) +@c this command has been renamed to @code{taytorat}. +@c +@c @end defun -@... defun +@defun taylor (@var{expr}, @var{x}, @var{x_0}, @var{n}) +Expands the expression @var{expr} in a truncated +Taylor series (or Laurent series, if required) in the variable @var{x} +around the point @var{x_0}, +containing terms through @code{(@var{x} - @var{x_0})^@var{n}}. -@... taylor (exp, var, pt, pow) -expands the expression exp in a truncated -Taylor series (or Laurent series, if required) in the variable var -around the point pt. The terms through (var-pt)**pow are generated. -If exp is of the form f(var)/g(var) and g(var) has no terms up to +If @var{expr} is of the form f(x)/g(x) and g(var) has no terms up to degree pow then @code{taylor} will try to expand g(var) up to degree 2*pow. If there are still no non-zero terms @code{taylor} will keep doubling the degree of the expansion of g(var) until reaching pow*2**n where n is -the value of the variable @code{taylordepth} [3]. If @code{maxtayorder} [FALSE] is -set to @code{true}, then during algebraic manipulation of (truncated) Taylor -series, @code{taylor} will try to retain as many terms as are certain to be -correct. Do @code{example(taylor)} for examples. - -@...{taylor(exp,[var1,pt1,ord1],[var2,pt2,ord2],...)} returns a truncated -power series in the variables vari about the points pti, truncated at -ordi. +the value of the variable @code{taylordepth}. -@...{psexpand} [FALSE] if @code{true} will cause extended rational function -expressions to display fully expanded. (@code{ratexpand} will also cause -this.) If @code{false}, multivariate expressions will be displayed just as in -the rational function package. If @code{psexpand:multi}, then terms with the -same total degree in the variables are grouped together. +@code{taylor (@var{expr}, [@var{x_1}, @var{x_@{0,1@}}, @var{n_1}], [@var{x_2}, @var{x_@{0,2@}}, @var{n_2}], ...)} +returns a truncated power series in the variables @var{x_1}, @var{x_2}, ... +about the points @var{x_@{0,1@}}, @var{x_@{0,2@}}, ..., +truncated at @var{n_1}, @var{n_2}, .... -@...{taylor(exp, [var1, var2, . . .], pt, ord)} where each of pt and ord +@code{taylor (@var{expr}, [@var{x_1}, @var{x_2}, ...], @var{x_0}, @var{n})} +where each of pt and ord may be replaced by a list which will correspond to the list of -variables. that is, the nth items on each of the lists will be +variables. That is, the nth items on each of the lists will be associated together. -@...{taylor(exp, [x,pt,ord,ASYMP])} will give an expansion of exp in +@code{taylor (@var{expr}, [x, pt, ord, 'asymp])} will give an expansion of exp in negative powers of (x-pt). The highest order term will be (x-pt)^(-ord). The @code{asymp} is a syntactic device and not to be assigned to. See also the @code{taylor_logexpand} switch for controlling expansion. +When @code{maxtayorder} is @code{true}, then during algebraic +manipulation of (truncated) Taylor series, @code{taylor} tries to retain +as many terms as are known to be correct. + +When @code{psexpand} is @code{true}, +an extended rational function expression is displayed fully expanded. +The switch @code{ratexpand} has the same effect. +When @code{psexpand} is @code{false}, +a multivariate expression is displayed just as in the rational function package. +When @code{psexpand} is @code{multi}, +then terms with the same total degree in the variables are grouped together. + +Do @code{example(taylor)} for examples. + @end defun @defvar taylordepth ```

 [Maxima-commits] CVS: maxima/doc/info Series.texi,1.7,1.8 From: Robert Dodier - 2005-02-27 21:38:27 ```Update of /cvsroot/maxima/maxima/doc/info In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv2846 Modified Files: Series.texi Log Message: Edits for greater clarity; reworked some examples. Still needs work. Index: Series.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/Series.texi,v retrieving revision 1.7 retrieving revision 1.8 diff -u -d -r1.7 -r1.8 --- Series.texi 27 Feb 2005 19:03:38 -0000 1.7 +++ Series.texi 27 Feb 2005 21:38:18 -0000 1.8 @@ -293,23 +293,31 @@ @end defun @defvar powerdisp - default: @code{false} - if @code{true} will cause sums to be displayed -with their terms in the reverse order. Thus polynomials would display -as truncated power series, i.e., with the lowest power first. +Default value: @code{false} + +When @code{powerdisp} is @code{true}, +a sum is displayed with its terms in order of increasing power. +Thus a polynomial is displayed as a truncated power series, +with the constant term first and the highest power last. + +By default, terms of a sum are displayed in order of decreasing power. +@c NEED AN EXAMPLE HERE @end defvar -@... powerseries (exp, var, pt) -generates the general form of the power -series expansion for exp in the variable var about the point pt (which -may be @code{inf} for infinity). If @code{powerseries} is unable to expand exp, the -@...{taylor} function may give the first several terms of the series. -@...{verbose} - if @code{true} will cause comments about the progress of -@...{powerseries} to be printed as the execution of it proceeds. +@defun powerseries (@var{expr}, @var{x}, @var{x_0}) +Returns the general form of the power series expansion for @var{expr} +in the variable @var{x} about the point @var{x_0} (which may be @code{inf} for infinity). + +If @code{powerseries} is unable to expand @var{expr}, +@code{taylor} may give the first several terms of the series. + +When @code{verbose} is @code{true}, +@code{powerseries} prints progress messages. @example -(%i1) verbose:true\$ -(%i2) powerseries(log(sin(x)/x),x,0); +(%i1) verbose: true\$ +(%i2) powerseries (log(sin(x)/x), x, 0); can't expand log(sin(x)) so we'll try again after applying the rule: @@ -339,67 +347,115 @@ @end defun @defvar psexpand - default: @code{false} - if @code{true} will cause extended rational -function expressions to display fully expanded. (@code{ratexpand} will also -cause this.) If @code{false}, multivariate expressions will be displayed -just as in the rational function package. If @code{psexpand:multi}, then -terms with the same total degree in the variables are grouped -together. +Default value: @code{false} + +When @code{psexpand} is @code{true}, +an extended rational function expression is displayed fully expanded. +The switch @code{ratexpand} has the same effect. + +@c WE NEED TO BE EXPLICIT HERE +When @code{psexpand} is @code{false}, +a multivariate expression is displayed just as in the rational function package. + +@c TERMS OF WHAT ?? +When @code{psexpand} is @code{multi}, +then terms with the same total degree in the variables are grouped together. @end defvar -@... revert (expression,variable) -Does reversion of Taylor Series. -"Variable" is the variable the original Taylor expansion is in. Do -@...{load(revert)} to access this function. Try +@defun revert (@var{expr}, @var{x}) +@defunx revert2 (@var{expr}, @var{x}, @var{n}) +These functions return the reversion of @var{expr}, a Taylor series about zero in the variable @var{x}. +@code{revert} returns a polynomial of degree equal to the highest power in @var{expr}. +@code{revert2} returns a polynomial of degree @var{n}, +which may be greater than, equal to, or less than the degree of @var{expr}. + +@code{load ("revert")} loads these functions. + +Examples: @example -revert2(expression,variable,hipower) +(%i1) load ("revert")\$ +(%i2) t: taylor (exp(x) - 1, x, 0, 6); + 2 3 4 5 6 + x x x x x +(%o2)/T/ x + -- + -- + -- + --- + --- + . . . + 2 6 24 120 720 +(%i3) revert (t, x); + 6 5 4 3 2 + 10 x - 12 x + 15 x - 20 x + 30 x - 60 x +(%o3)/R/ - -------------------------------------------- + 60 +(%i4) ratexpand (%); + 6 5 4 3 2 + x x x x x +(%o4) - -- + -- - -- + -- - -- + x + 6 5 4 3 2 +(%i5) taylor (log(x+1), x, 0, 6); + 2 3 4 5 6 + x x x x x +(%o5)/T/ x - -- + -- - -- + -- - -- + . . . + 2 3 4 5 6 +(%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6)); +(%o6) 0 +(%i7) revert2 (t, x, 4); + 4 3 2 + x x x +(%o7) - -- + -- - -- + x + 4 3 2 @end example -also. @code{revert} only works on -expansions around 0. - @end defun -@... srrat (exp) -this command has been renamed to @code{taytorat}. +@c OBSOLETE AS ADVERTISED; NOT CALLED FROM ANY USER-LEVEL FILE (.mac OR .dem) +@c CUT THIS OUT ON NEXT PASS +@c @defun srrat (exp) +@c this command has been renamed to @code{taytorat}. +@c +@c @end defun -@... defun +@defun taylor (@var{expr}, @var{x}, @var{x_0}, @var{n}) +Expands the expression @var{expr} in a truncated +Taylor series (or Laurent series, if required) in the variable @var{x} +around the point @var{x_0}, +containing terms through @code{(@var{x} - @var{x_0})^@var{n}}. -@... taylor (exp, var, pt, pow) -expands the expression exp in a truncated -Taylor series (or Laurent series, if required) in the variable var -around the point pt. The terms through (var-pt)**pow are generated. -If exp is of the form f(var)/g(var) and g(var) has no terms up to +If @var{expr} is of the form f(x)/g(x) and g(var) has no terms up to degree pow then @code{taylor} will try to expand g(var) up to degree 2*pow. If there are still no non-zero terms @code{taylor} will keep doubling the degree of the expansion of g(var) until reaching pow*2**n where n is -the value of the variable @code{taylordepth} [3]. If @code{maxtayorder} [FALSE] is -set to @code{true}, then during algebraic manipulation of (truncated) Taylor -series, @code{taylor} will try to retain as many terms as are certain to be -correct. Do @code{example(taylor)} for examples. - -@...{taylor(exp,[var1,pt1,ord1],[var2,pt2,ord2],...)} returns a truncated -power series in the variables vari about the points pti, truncated at -ordi. +the value of the variable @code{taylordepth}. -@...{psexpand} [FALSE] if @code{true} will cause extended rational function -expressions to display fully expanded. (@code{ratexpand} will also cause -this.) If @code{false}, multivariate expressions will be displayed just as in -the rational function package. If @code{psexpand:multi}, then terms with the -same total degree in the variables are grouped together. +@code{taylor (@var{expr}, [@var{x_1}, @var{x_@{0,1@}}, @var{n_1}], [@var{x_2}, @var{x_@{0,2@}}, @var{n_2}], ...)} +returns a truncated power series in the variables @var{x_1}, @var{x_2}, ... +about the points @var{x_@{0,1@}}, @var{x_@{0,2@}}, ..., +truncated at @var{n_1}, @var{n_2}, .... -@...{taylor(exp, [var1, var2, . . .], pt, ord)} where each of pt and ord +@code{taylor (@var{expr}, [@var{x_1}, @var{x_2}, ...], @var{x_0}, @var{n})} +where each of pt and ord may be replaced by a list which will correspond to the list of -variables. that is, the nth items on each of the lists will be +variables. That is, the nth items on each of the lists will be associated together. -@...{taylor(exp, [x,pt,ord,ASYMP])} will give an expansion of exp in +@code{taylor (@var{expr}, [x, pt, ord, 'asymp])} will give an expansion of exp in negative powers of (x-pt). The highest order term will be (x-pt)^(-ord). The @code{asymp} is a syntactic device and not to be assigned to. See also the @code{taylor_logexpand} switch for controlling expansion. +When @code{maxtayorder} is @code{true}, then during algebraic +manipulation of (truncated) Taylor series, @code{taylor} tries to retain +as many terms as are known to be correct. + +When @code{psexpand} is @code{true}, +an extended rational function expression is displayed fully expanded. +The switch @code{ratexpand} has the same effect. +When @code{psexpand} is @code{false}, +a multivariate expression is displayed just as in the rational function package. +When @code{psexpand} is @code{multi}, +then terms with the same total degree in the variables are grouped together. + +Do @code{example(taylor)} for examples. + @end defun @defvar taylordepth ```