## 11 Simplification ¶

### 11.1 Introduction to Simplification ¶

Maxima interacts with the user through a cycle of actions called the read-eval-print loop (REPL). This consists of three steps: reading and parsing, evaluating and simplifying, and outputting. Parsing converts a syntactically valid sequence of typed characters into a internal data structure. Evaluation replaces variable and function names with their values and simplification rewrites expressions to be easier for the user or other programs to understand. Output displays results in a variety of different formats and notations.

Evaluation and simplification sometimes appear to have similar functionality, and Maxima uses simplification in many cases where other systems use evaluation. For example, arithmetic both on numbers and on symbolic expressions is simplification, not evaluation: `2+2` simplifies to `4`, `2+x+x` simplifies to `2+2*x`, and `sqrt(7)^4` simplifies to `49`. Evaluation and simplification are interleaved. For example, `factor(integrate(x+1,x))` first calls the built-in function `integrate`, giving `x+x*x*2^-1`; that simplifies to `x+(1/2)*x^2`; this in turn is passed to the `factor` function, which returns `(x*(x+2))/2`.

Evaluation is what makes Maxima a programming language: it implements functions, subroutines, variables, values, loops, assignments and so on. Evaluation replaces built-in or user-defined function names by their definitions and variables by their values. This is largely the same as activities of a conventional programming language, but extended to work with symbolic mathematical data. The system has various optional "flags" which the user can set to control the details of evaluation. See Functions and Variables for Evaluation.

Simplification maintains the value of an expression while re-formulating its form to be smaller, simpler to understand, or to conform to a particular specification (like expanded). For example, `sin(%pi/2)` to `1`, and `x+x` to `2*x`. There are many flags which control simplification. For example, with `triginverses:true`, `atan(tan(x))` does not simplify to `x`, but with `triginverses:all`, it does.

Simplification can be provided in three ways:

- The internal, built-in automated simplifier,
- User-written pattern-matching transformations, linked to the simplifier by using "tellsimp" or "tellsimpafter" and called automatically,
- User-written simplification routines adding using the `simplifying` subsystem.

The internal simplifier belongs to the heart of Maxima. It is a large and complicated collection of programs, and it has been refined over many years and by thousands of users. Nevertheless, especially if you are trying out novel ideas or unconventional notation, you may find it helpful to make small (or large) changes to the program yourself. For details see for example the paper at the end of https://people.eecs.berkeley.edu/~fateman/papers/intro5.txt.

Maxima internally represents expressions as "trees" with operators or "roots" like `+`, `*` , `=` and operands ("leaves") which are variables like x, y, z, functions or sub-trees, like `x*y`. Each operator has a simplification program associated with it. `+` (which also covers binary `-` since `a-b = a+(-1)*b)` and `*` (which also covers `/` since `a/b = a*b^(-1)`) have rather elaborate simplification programs. These simplification programs (simplus, simptimes, simpexpt, etc.) are called whenever the simplifier encounters the respective arithmetic operators in an expression tree to be analyzed.

The structure of the simplifier dates back to 1965, and many hands have worked on it through the years. It is data-directed, or object-oriented in the sense that it dispatches to the appropriate routine depending on the root of some sub-tree of the expression, recursively. This general approach means that modifications to simplification are generally localized. In many cases it is straightforward to add an operator and its simplification routine without disturbing existing code.

Maxima also provides a variety of transformation routines that can change the form of an expression, including `factor` (polynomial factorization), `horner` (reorganize a polynomial using Horner’s rule), `partfrac` (rewrite a rational function as partial fractions), `trigexpand` (apply the sum formulas for trigonometric functions), and so on.

Users can also write routines that change the form of an expression.

Besides this general simplifier operating on algebraic expression trees, there are several other representations of expressions in Maxima which have separate methods. For example, the `rat` function converts polynomials to vectors of coefficients to assist in rapid manipulation of such forms. Other representations include Taylor series and the (rarely used) Poisson series.

All operators introduced by the user initially have no simplification programs associated with them. Maxima does not know anything about function "f" and so typing `f(a,b)` will result in simplifying a,b, but not `f`. Even some built-in operators have no simplifications. For example, `=` does not "simplify" – it is a place-holder with no simplification semantics other than to simplify its two arguments, in this case referred to as the left and right sides. Other parts of Maxima such as the solve program take special note of equations, that is, trees with `=` as the root. (Note – in Maxima, the assignment operation is `:` . That is, `q: 4` sets the value of the symbol q to `4`. Function definition is done with `:=`. )

The general simplifier returns results with an internal flag indicating the expression and each sub-expression has been simplified. This does not guarantee that it is unique over all possible equivalent expressions. That’s too hard (theoretically, not possible given the generality of what can be expressed in Maxima). However, some aspects of the expression, such as the ordering of terms in a sum or product, are made uniform. This is important for the other programs to work properly.

A number of option variables control simplification. Indeed, simplification can be turned off entirely using `simp:false`. However, many internal routines will not operate correctly with `simp:false`. (About the only time it seems plausible to turn off the simplifier is in the rare case that you want to over-ride a built-in simplification. In that case you might temporarily disable the simplifier, put in the new transformation via `tellsimp`, and then re-enable the simplifier by `simp:true`.)

It is more plausible for you to associate user-defined symbolic function names or operators with properties (`additive`, `lassociative`, `oddfun`, `antisymmetric`, `linear`, `outative`, `commutative`, `multiplicative`, `rassociative`, `evenfun`, `nary` and `symmetric`). These options steer the simplifier processing in systematic directions.

For example, `declare(f,oddfun)` specifies that `f` is an odd function. Maxima will simplify `f(-x)` to `-f(x)`. In the case of an even function, that is `declare(g,evenfun)`, Maxima will simplify `g(-x)` to `g(x)`. You can also associate a programming function with a name such as `h(x):=x^2+1`. In that case the evaluator will immediately replace `h(3)` by `10`, and `h(a+1)` by `(a+1)^2+1`, so any properties of `h` will be ignored.

In addition to these directly related properties set up by the user, facts and properties from the actual context may have an impact on the simplifier’s behavior, too. See Introduction to Maxima’s Database.

Example: `sin(n*%pi)` is simplified to zero, if n is an integer.

    (%i1) sin(n*%pi);
    (%o1)                      sin(%pi n)

    (%i2) declare(n, integer);
    (%o2)                         done

    (%i3) sin(n*%pi);
    (%o3)                           0

If automated simplification is not sufficient, you can consider a variety of built-in, but explicitly called simplification functions (`ratsimp`, `expand`, `factor`, `radcan` and others). There are also flags that will push simplification into one or another direction. Given `demoivre:true` the simplifier rewrites complex exponentials as trigonometric forms. Given `exponentialize:true` the simplifier tries to do the reverse: rewrite trigonometric forms as complex exponentials.

As everywhere in Maxima, by writing your own functions (be it in the Maxima user language or in the implementation language Lisp) and explicitly calling them at selected places in the program, you can respond to your individual simplification needs. Lisp gives you a handle on all the internal mechanisms, but you rarely need this full generality. "Tellsimp" is designed to generate much of the Lisp internal interface into the simplifier automatically. See See Rules and Patterns.

Over the years (Maxima/Macsyma’s origins date back to about 1966!) users have contributed numerous application packages and tools to extend or alter its functional behavior. Various non-standard and "share" packages exist to modify or extend simplification as well. You are invited to look into this more experimental material where work is still in progress See Package simplification.

The following appended material is optional on a first reading, and reading it is not necessary for productive use of Maxima. It is for the curious user who wants to understand what is going on, or the ambitious programmer who might wish to change the (open-source) code. Experimentation with redefining Maxima Lisp code is easily possible: to change the definition of a Lisp program (say the one that simplifies `cos()`, named `simp%cos`), you simply load into Maxima a text file that will overwrite the `simp%cos` function from the maxima package.

### 11.2 Functions and Variables for Simplification ¶

Property: additive ¶

  If `declare(f,additive)` has been executed, then:

  (1) If `f` is univariate, whenever the simplifier encounters `f` applied to a sum, `f` will be distributed over that sum. I.e. `f(y+x)` will simplify to `f(y)+f(x)`.

  (2) If `f` is a function of 2 or more arguments, additivity is defined as additivity in the first argument to `f`, as in the case of `sum` or `integrate`, i.e. `f(h(x)+g(x),x)` will simplify to `f(h(x),x)+f(g(x),x)`. This simplification does not occur when `f` is applied to expressions of the form `sum(x[i],i,lower-limit,upper-limit)`.

  Example:

      (%i1) F3 (a + b + c);
      (%o1)                     F3(c + b + a)

      (%i2) declare (F3, additive);
      (%o2)                         done

      (%i3) F3 (a + b + c);
      (%o3)                 F3(c) + F3(b) + F3(a)

  Categories: Operators · Declarations and inferences ·

Property: antisymmetric ¶

  If `declare(h,antisymmetric)` is done, this tells the simplifier that `h` is antisymmetric. E.g. `h(x,z,y)` will simplify to `- h(x, y, z)`. That is, it will give (-1)^n times the result given by `symmetric` or `commutative`, where n is the number of interchanges of two arguments necessary to convert it to that form.

  Examples:

      (%i1) S (b, a);
      (%o1)                        S(b, a)

      (%i2) declare (S, symmetric);
      (%o2)                         done

      (%i3) S (b, a);
      (%o3)                        S(a, b)

      (%i4) S (a, c, e, d, b);
      (%o4)                   S(a, b, c, d, e)

      (%i5) T (b, a);
      (%o5)                        T(b, a)

      (%i6) declare (T, antisymmetric);
      (%o6)                         done

      (%i7) T (b, a);
      (%o7)                       - T(a, b)

      (%i8) T (a, c, e, d, b);
      (%o8)                   T(a, b, c, d, e)

  Categories: Operators · Declarations and inferences ·

Function: combine (expr) ¶

  Simplifies the sum expr by combining terms with the same denominator into a single term.

  See also: `rncombine`.

  Example:

      (%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a;
                            5 b f   a b c   2 a c
      (%o1)                 ----- + ----- + -----
                              4       5       3

      (%i2) combine (%);
                        75 b f + 4 (3 a b c + 10 a c)
      (%o2)             -----------------------------
                                     60

  Categories: Expressions ·

Property: commutative ¶

  If `declare(h, commutative)` is done, this tells the simplifier that `h` is a commutative function. E.g. `h(x, z, y)` will simplify to `h(x, y, z)`. This is the same as `symmetric`.

  Example:

      (%i1) S (b, a);
      (%o1)                        S(b, a)

      (%i2) S (a, b) + S (b, a);
      (%o2)                   S(b, a) + S(a, b)

      (%i3) declare (S, commutative);
      (%o3)                         done

      (%i4) S (b, a);
      (%o4)                        S(a, b)

      (%i5) S (a, b) + S (b, a);
      (%o5)                       2 S(a, b)

      (%i6) S (a, c, e, d, b);
      (%o6)                   S(a, b, c, d, e)

  Categories: Operators · Declarations and inferences ·

Function: demoivre (expr) ¶
Option variable: demoivre ¶

  The function `demoivre (expr)` converts one expression without setting the global variable `demoivre`.

  When the variable `demoivre` is `true`, complex exponentials are converted into equivalent expressions in terms of circular functions: `exp (a + b*%i)` simplifies to `%e^a * (cos(b) + %i*sin(b))` if `b` is free of `%i`. `a` and `b` are not expanded.

  The default value of `demoivre` is `false`.

  `exponentialize` converts circular and hyperbolic functions to exponential form. `demoivre` and `exponentialize` cannot both be true at the same time.

  Categories: Complex variables · Trigonometric functions · Hyperbolic functions ·

Function: distrib (expr) ¶

  Distributes sums over products. It differs from `expand` in that it works at only the top level of an expression, i.e., it doesn’t recurse and it is faster than `expand`. It differs from `multthru` in that it expands all sums at that level.

  Examples:

      (%i1) distrib ((a+b) * (c+d));
      (%o1)                 b d + a d + b c + a c

      (%i2) multthru ((a+b) * (c+d));
      (%o2)                 (b + a) d + (b + a) c

      (%i3) distrib (1/((a+b) * (c+d)));
                                      1
      (%o3)                    ---------------
                               (b + a) (d + c)

      (%i4) expand (1/((a+b) * (c+d)), 1, 0);
                                      1
      (%o4)                 ---------------------
                            b d + a d + b c + a c

  Categories: Expressions ·

Option variable: distribute_over ¶

  Default value: `true`

  `distribute_over` controls the mapping of functions over bags like lists, matrices, and equations. At this time not all Maxima functions have this property. It is possible to look up this property with the command `properties`..

  The mapping of functions is switched off, when setting `distribute_over` to the value `false`.

  Examples:

  The `sin` function maps over a list:

      (%i1) sin([x,1,1.0]);
      (%o1)         [sin(x), sin(1), 0.8414709848078965]

  `mod` is a function with two arguments which maps over lists. Mapping over nested lists is possible too:

      (%i1) mod([x,11,2*a],10);
      (%o1)             [mod(x, 10), 1, 2 mod(a, 5)]

      (%i2) mod([[x,y,z],11,2*a],10);
      (%o2) [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]

  Mapping of the `floor` function over a matrix and an equation:

      (%i1) floor(matrix([a,b],[c,d]));
                           [ floor(a)  floor(b) ]
      (%o1)                [                    ]
                           [ floor(c)  floor(d) ]

      (%i2) floor(a=b);
      (%o2)                  floor(a) = floor(b)

  Functions with more than one argument map over any of the arguments or all arguments:

      (%i1) expintegral_e([1,2],[x,y]);
      (%o1) [[expintegral_e(1, x), expintegral_e(1, y)],
                            [expintegral_e(2, x), expintegral_e(2, y)]]

  Check if a function has the property distribute_over:

      (%i1) properties(abs);
      (%o1) [limit function, integral, rule, distributes over bags,
                                                noun, gradef, transfun]

  The mapping of functions is switched off, when setting `distribute_over` to the value `false`.

      (%i1) distribute_over;
      (%o1)                         true

      (%i2) sin([x,1,1.0]);
      (%o2)         [sin(x), sin(1), 0.8414709848078965]

      (%i3) distribute_over : not distribute_over;
      (%o3)                         false

      (%i4) sin([x,1,1.0]);
      (%o4)                   sin([x, 1, 1.0])

  Categories: Simplification flags and variables ·

Option variable: domain ¶

  Default value: `real`

  When `domain` is set to `complex`, `sqrt (x^2)` will remain `sqrt (x^2)` instead of returning `abs(x)`.

  Categories: Simplification flags and variables ·

Property: evenfun ¶
Property: oddfun ¶

  `declare(f, evenfun)` or `declare(f, oddfun)` tells Maxima to recognize the function `f` as an even or odd function.

  Examples:

      (%i1) o (- x) + o (x);
      (%o1)                     o(x) + o(- x)

      (%i2) declare (o, oddfun);
      (%o2)                         done

      (%i3) o (- x) + o (x);
      (%o3)                           0

      (%i4) e (- x) - e (x);
      (%o4)                     e(- x) - e(x)

      (%i5) declare (e, evenfun);
      (%o5)                         done

      (%i6) e (- x) - e (x);
      (%o6)                           0

Function: expand
expand (expr)
expand (expr, p, n) ¶

  Expand expression expr. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplication (commutative and non-commutative) are distributed over addition at all levels of expr.

  For polynomials one should usually use `ratexpand` which uses a more efficient algorithm.

  `maxnegex` and `maxposex` control the maximum negative and positive exponents, respectively, which will expand.

  `expand (expr, p, n)` expands expr, using p for `maxposex` and n for `maxnegex`. This is useful in order to expand part but not all of an expression.

  `expon` - the exponent of the largest negative power which is automatically expanded (independent of calls to `expand`). For example if `expon` is 4 then `(x+1)^(-5)` will not be automatically expanded.

  `expop` - the highest positive exponent which is automatically expanded. Thus `(x+1)^3`, when typed, will be automatically expanded only if `expop` is greater than or equal to 3. If it is desired to have `(x+1)^n` expanded where `n` is greater than `expop` then executing `expand ((x+1)^n)` will work only if `maxposex` is not less than `n`.

  `expand(expr, 0, 0)` causes a resimplification of `expr`. `expr` is not reevaluated. In distinction from `ev(expr, noeval)` a special representation (e. g. a CRE form) is removed. See also `resimplify` and `ev`.

  The `expand` flag used with `ev` causes expansion.

  The file `share/simplification/facexp.mac` contains several related functions (in particular `facsum`, `factorfacsum` and `collectterms`, which are autoloaded) and variables (`nextlayerfactor` and `facsum_combine`) that provide the user with the ability to structure expressions by controlled expansion. Brief function descriptions are available in `simplification/facexp.usg`. A demo is available by doing `demo("facexp")`.

  Examples:

      (%i1) expr:(x+1)^2*(y+1)^3;
                                     2        3
      (%o1)                   (x + 1)  (y + 1)

      (%i2) expand(expr);
             2  3        3    3      2  2        2      2      2
      (%o2) x  y  + 2 x y  + y  + 3 x  y  + 6 x y  + 3 y  + 3 x  y
                                                            2
                                           + 6 x y + 3 y + x  + 2 x + 1

      (%i3) expand(expr,2);
                     2        3              3          3
      (%o3)         x  (y + 1)  + 2 x (y + 1)  + (y + 1)

      (%i4) expr:(x+1)^-2*(y+1)^3;
                                         3
                                  (y + 1)
      (%o4)                       --------
                                         2
                                  (x + 1)

      (%i5) expand(expr);
                  3               2
                 y             3 y            3 y             1
      (%o5) ------------ + ------------ + ------------ + ------------
             2              2              2              2
            x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1

      (%i6) expand(expr,2,2);
                                         3
                                  (y + 1)
      (%o6)                     ------------
                                 2
                                x  + 2 x + 1

  Resimplify an expression without expansion:

      (%i1) expr:(1+x)^2*sin(x);
                                      2
      (%o1)                    (x + 1)  sin(x)

      (%i2) exponentialize:true;
      (%o2)                         true

      (%i3) expand(expr,0,0);
                           %i x     - %i x            2
                        (%e     - %e      ) %i (x + 1)
      (%o3)           - -------------------------------
                                       2

  Categories: Expressions ·

Function: expandwrt (expr, x_1, …, x_n) ¶

  Expands expression `expr` with respect to the variables x_1, …, x_n. All products involving the variables appear explicitly. The form returned will be free of products of sums of expressions that are not free of the variables. x_1, …, x_n may be variables, operators, or expressions.

  By default, denominators are not expanded, but this can be controlled by means of the switch `expandwrt_denom`.

  This function is autoloaded from `simplification/stopex.mac`.

  Categories: Expressions ·

Option variable: expandwrt_denom ¶

  Default value: `false`

  `expandwrt_denom` controls the treatment of rational expressions by `expandwrt`. If `true`, then both the numerator and denominator of the expression will be expanded according to the arguments of `expandwrt`, but if `expandwrt_denom` is `false`, then only the numerator will be expanded in that way.

  Categories: Expressions ·

Function: expandwrt_factored (expr, x_1, …, x_n) ¶

  is similar to `expandwrt`, but treats expressions that are products somewhat differently. `expandwrt_factored` expands only on those factors of `expr` that contain the variables x_1, …, x_n.

  This function is autoloaded from `simplification/stopex.mac`.

  Categories: Expressions ·

Option variable: expon ¶

  Default value: 0

  `expon` is the exponent of the largest negative power which is automatically expanded (independent of calls to `expand`). For example, if `expon` is 4 then `(x+1)^(-5)` will not be automatically expanded.

  Categories: Expressions ·

Function: exponentialize (expr) ¶
Option variable: exponentialize ¶

  The function `exponentialize (expr)` converts circular and hyperbolic functions in expr to exponentials, without setting the global variable `exponentialize`.

  When the variable `exponentialize` is `true`, all circular and hyperbolic functions are converted to exponential form. The default value is `false`.

  `demoivre` converts complex exponentials into circular functions. `exponentialize` and `demoivre` cannot both be true at the same time.

  Categories: Complex variables · Trigonometric functions · Hyperbolic functions ·

Option variable: expop ¶

  Default value: 0

  `expop` is the highest positive exponent which is automatically expanded. Thus `(x + 1)^3`, when typed, will be automatically expanded only if `expop` is greater than or equal to 3. If it is desired to have `(x + 1)^n` expanded where `n` is greater than `expop` then executing `expand ((x + 1)^n)` will work only if `maxposex` is not less than n.

  Categories: Expressions ·

Property: lassociative ¶

  `declare (g, lassociative)` tells the Maxima simplifier that `g` is left-associative. E.g., `g (g (a, b), g (c, d))` will simplify to `g (g (g (a, b), c), d)`.

  See also `rassociative`.

  Categories: Declarations and inferences · Operators · Simplification ·

Property: linear ¶

  One of Maxima’s operator properties. For univariate `f` so declared, "expansion" `f(x + y)` yields `f(x) + f(y)`, `f(a*x)` yields `a*f(x)` takes place where `a` is a "constant". For functions of two or more arguments, "linearity" is defined to be as in the case of `sum` or `integrate`, i.e., `f (a*x + b, x)` yields `a*f(x,x) + b*f(1,x)` for `a` and `b` free of `x`.

  Example:

      (%i1) declare (f, linear);
      (%o1)                         done

      (%i2) f(x+y);
      (%o2)                      f(y) + f(x)

      (%i3) declare (a, constant);
      (%o3)                         done

      (%i4) f(a*x);
      (%o4)                        a f(x)

  `linear` is equivalent to `additive` and `outative`. See also `opproperties`.

  Example:

      (%i1) 'sum (F(k) + G(k), k, 1, inf);
                             inf
                             ____
                             \
      (%o1)                   >    (G(k) + F(k))
                             /
                             ----
                             k = 1

      (%i2) declare (nounify (sum), linear);
      (%o2)                         done

      (%i3) 'sum (F(k) + G(k), k, 1, inf);
                           inf          inf
                           ____         ____
                           \            \
      (%o3)                 >    G(k) +  >    F(k)
                           /            /
                           ----         ----
                           k = 1        k = 1

  Categories: Declarations and inferences · Operators · Simplification ·

Option variable: maxnegex ¶

  Default value: 1000

  `maxnegex` is the largest negative exponent which will be expanded by the `expand` command, see also `maxposex`.

  Categories: Expressions ·

Option variable: maxposex ¶

  Default value: 1000

  `maxposex` is the largest exponent which will be expanded with the `expand` command, see also `maxnegex`.

  Categories: Expressions ·

Property: multiplicative ¶

  `declare(f, multiplicative)` tells the Maxima simplifier that `f` is multiplicative.

  1. If `f` is univariate, whenever the simplifier encounters `f` applied to a product, `f` distributes over that product. E.g., `f(x*y)` simplifies to `f(x)*f(y)`. This simplification is not applied to expressions of the form `f('product(...))`.
  2. If `f` is a function of 2 or more arguments, multiplicativity is defined as multiplicativity in the first argument to `f`, e.g., `f (g(x) * h(x), x)` simplifies to `f (g(x) ,x) * f (h(x), x)`.

  `declare(nounify(product), multiplicative)` tells Maxima to simplify symbolic products.

  Example:

      (%i1) F2 (a * b * c);
      (%o1)                       F2(a b c)

      (%i2) declare (F2, multiplicative);
      (%o2)                         done

      (%i3) F2 (a * b * c);
      (%o3)                   F2(a) F2(b) F2(c)

  `declare(nounify(product), multiplicative)` tells Maxima to simplify symbolic products.

      (%i1) product (a[i] * b[i], i, 1, n);
                                   n
                                 _____
                                 |   |
      (%o1)                      |   | a  b
                                 |   |  i  i
                                 i = 1

      (%i2) declare (nounify (product), multiplicative);
      (%o2)                         done

      (%i3) product (a[i] * b[i], i, 1, n);
                                n         n
                              _____     _____
                              |   |     |   |
      (%o3)                  (|   | a ) |   | b
                              |   |  i  |   |  i
                              i = 1     i = 1

  Categories: Declarations and inferences · Expressions · Simplification ·

Function: multthru
multthru (expr)
multthru (expr_1, expr_2) ¶

  Multiplies a factor (which should be a sum) of expr by the other factors of expr. That is, expr is `f_1 f_2 ... f_n` where at least one factor, say f_i, is a sum of terms. Each term in that sum is multiplied by the other factors in the product. (Namely all the factors except f_i). `multthru` does not expand exponentiated sums. This function is the fastest way to distribute products (commutative or noncommutative) over sums. Since quotients are represented as products `multthru` can be used to divide sums by products as well.

  `multthru (expr_1, expr_2)` multiplies each term in expr_2 (which should be a sum or an equation) by expr_1. If expr_1 is not itself a sum then this form is equivalent to `multthru (expr_1*expr_2)`.

      (%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
                            1        x         f(x)
      (%o1)             - ----- + -------- - --------
                          x - y          2          3
                                  (x - y)    (x - y)

      (%i2) multthru ((x-y)^3, %);
                                            2
      (%o2)              x (x - y) - (x - y)  - f(x)

      (%i3) ratexpand (%);
                                 2
      (%o3)                   - y  + x y - f(x)

      (%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
                              10  2              2  2
                       (b + a)   s  + 2 a b s + a  b
      (%o4)            ------------------------------
                                        2
                                   a b s

      (%i5) multthru (%);  /* note that this does not expand (b+a)^10 */
                                              10
                             2   a b   (b + a)
      (%o5)                  - + --- + ---------
                             s    2       a b
                                 s
      (%o6)            a . f + a . c . (e + d) + a . b

      (%i7) multthru (a.(b+c.(d+e)+f));
      (%o7)         a . f + a . c . e + a . c . d + a . b

      (%i8) expand (a.(b+c.(d+e)+f));

  Categories: Expressions ·

Property: nary ¶

  `declare(f, nary)` tells Maxima to recognize the function `f` as an n-ary function.

  The `nary` declaration is not the same as calling the `nary` function. The sole effect of `declare(f, nary)` is to instruct the Maxima simplifier to flatten nested expressions, for example, to simplify `foo(x, foo(y, z))` to `foo(x, y, z)`. See also `declare`.

  Example:

      (%i1) H (H (a, b), H (c, H (d, e)));
      (%o1)               H(H(a, b), H(c, H(d, e)))

      (%i2) declare (H, nary);
      (%o2)                         done

      (%i3) H (H (a, b), H (c, H (d, e)));
      (%o3)                   H(a, b, c, d, e)

Option variable: negdistrib ¶

  Default value: `true`

  When `negdistrib` is `true`, -1 distributes over an expression. E.g., `-(x + y)` becomes `- y - x`. Setting it to `false` will allow `- (x + y)` to be displayed like that. This is sometimes useful but be very careful: like the `simp` flag, this is one flag you do not want to set to `false` as a matter of course or necessarily for other than local use in your Maxima.

  Example:

      (%i1) negdistrib;
      (%o1)                         true

      (%i2) -(x+y);
      (%o2)                        - y - x

      (%i3) negdistrib : not negdistrib ;
      (%o3)                         false

      (%i4) -(x+y);
      (%o4)                       - (y + x)

  Categories: Simplification flags and variables ·

System variable: opproperties ¶

  `opproperties` is the list of the special operator properties recognized by the Maxima simplifier.

  Items are added to the `opproperties` list by the function `define_opproperty`.

  Example:

      (%i1) opproperties;
      (%o1) [linear, additive, multiplicative, outative, evenfun,
      oddfun, commutative, symmetric, antisymmetric, nary,
      lassociative, rassociative]

  Categories: Global variables · Operators · Simplification ·

Function: define_opproperty (property_name, simplifier_fn) ¶

  Declares the symbol property_name to be an operator property, which is simplified by simplifier_fn, which may be the name of a Maxima or Lisp function or a lambda expression. After `define_opproperty` is called, functions and operators may be declared to have the property_name property, and simplifier_fn is called to simplify them.

  simplifier_fn must be a function of one argument, which is an expression in which the main operator is declared to have the property_name property.

  simplifier_fn is called with the global flag `simp` disabled. Therefore simplifier_fn must be able to carry out its simplification without making use of the general simplifier.

  `define_opproperty` appends property_name to the global list `opproperties`.

  `define_opproperty` returns `done`.

  Example:

  Declare a new property, `identity`, which is simplified by `simplify_identity`. Declare that `f` and `g` have the new property.

      (%i1) define_opproperty (identity, simplify_identity);
      (%o1)                         done

      (%i2) simplify_identity(e) := first(e);
      (%o2)           simplify_identity(e) := first(e)

      (%i3) declare ([f, g], identity);
      (%o3)                         done

      (%i4) f(10 + t);
      (%o4)                        t + 10

      (%i5) g(3*u) - f(2*u);
      (%o5)                           u

  Categories: Operators · Simplification ·

Property: outative ¶

  `declare(f, outative)` tells the Maxima simplifier that constant factors in the argument of `f` can be pulled out.

  1. If `f` is univariate, whenever the simplifier encounters `f` applied to a product, that product will be partitioned into factors that are constant and factors that are not and the constant factors will be pulled out. E.g., `f(a*x)` will simplify to `a*f(x)` where `a` is a constant. Non-atomic constant factors will not be pulled out.
  2. If `f` is a function of 2 or more arguments, outativity is defined as in the case of `sum` or `integrate`, i.e., `f (a*g(x), x)` will simplify to `a * f(g(x), x)` for `a` free of `x`.

  `sum`, `integrate`, and `limit` are all `outative`.

  Example:

      (%i1) F1 (100 * x);
      (%o1)                       F1(100 x)

      (%i2) declare (F1, outative);
      (%o2)                         done

      (%i3) F1 (100 * x);
      (%o3)                       100 F1(x)

      (%i4) declare (zz, constant);
      (%o4)                         done

      (%i5) F1 (zz * y);
      (%o5)                       zz F1(y)

  Categories: Declarations and inferences · Operators ·

Function: radcan (expr) ¶

  Simplifies expr, which can contain logs, exponentials, and radicals, by converting it into a form which is canonical over a large class of expressions and a given ordering of variables; that is, all functionally equivalent forms are mapped into a unique form. For a somewhat larger class of expressions, `radcan` produces a regular form. Two equivalent expressions in this class do not necessarily have the same appearance, but their difference can be simplified by `radcan` to zero.

  For some expressions `radcan` is quite time consuming. This is the cost of exploring certain relationships among the components of the expression for simplifications based on factoring and partial-fraction expansions of exponents.

  Examples:

      (%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
                                          a/2
      (%o1)                     log(x + 1)

      (%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
      (%o2)                           2

      (%i3) radcan((%e^x-1)/(1+%e^(x/2)));
                                    x/2
      (%o3)                       %e    - 1

  Categories: Simplification functions ·

Option variable: radexpand ¶

  Default value: `true`

  `radexpand` controls some simplifications of radicals.

  When `radexpand` is `all`, causes nth roots of factors of a product which are powers of n to be pulled outside of the radical. E.g. if `radexpand` is `all`, `sqrt (16*x^2)` simplifies to `4*x`.

  More particularly, consider `sqrt (x^2)`.

  - If `radexpand` is `all` or `assume (x > 0)` has been executed, `sqrt(x^2)` simplifies to `x`.
  - If `radexpand` is `true` and `domain` is `real` (its default), `sqrt(x^2)` simplifies to `abs(x)`.
  - If `radexpand` is `false`, or `radexpand` is `true` and `domain` is `complex`, `sqrt(x^2)` is not simplified.

  Note that `domain` only matters when `radexpand` is `true`.

  Categories: Simplification flags and variables ·

Property: rassociative ¶

  `declare (g, rassociative)` tells the Maxima simplifier that `g` is right-associative. E.g., `g(g(a, b), g(c, d))` simplifies to `g(a, g(b, g(c, d)))`.

  See also `lassociative`.

  Categories: Declarations and inferences · Operators ·

Function: resimplify (expr) ¶

  Resimplifies the expression expr based on the current environment. This function is useful when the fact database, option variables, or tellsimp rules have changed since the expression was last simplified.

  Example:

      (%i1) expr : sin(x)^2 + cos(x)^2;
                                 2         2
      (%o1)                   sin (x) + cos (x)

      (%i2) exponentialize : true;
      (%o2)                         true

      (%i3) expr;
                                 2         2
      (%o3)                   sin (x) + cos (x)

      (%i4) resimplify(%);
                    %i x     - %i x 2      %i x     - %i x 2
                 (%e     + %e      )    (%e     - %e      )
      (%o4)      -------------------- - --------------------
                          4                      4

      (%i5) ratsimp(%);
      (%o5)                           1

  Categories: Simplification functions ·

Function: scsimp (expr, rule_1, …, rule_n) ¶

  Sequential Comparative Simplification (method due to Stoute). `scsimp` attempts to simplify expr according to the rules rule_1, …, rule_n. If a smaller expression is obtained, the process repeats. Otherwise after all simplifications are tried, it returns the original answer.

  `example (scsimp)` displays some examples.

  Categories: Simplification functions ·

Option variable: simp ¶

  Default value: `true`

  `simp` enables simplification. This is the default. `simp` is also an `evflag`, which is recognized by the function `ev`. See `ev`.

  When `simp` is used as an `evflag` with a value `false`, the simplification is suppressed only during the evaluation phase of an expression. The flag does not suppress the simplification which follows the evaluation phase.

  Many Maxima functions and operations require simplification to be enabled to work normally. When simplification is disabled, many results will be incomplete, and in addition there may be incorrect results or program errors.

  Examples:

  The simplification is switched off globally. The expression `sin(1.0)` is not simplified to its numerical value. The `simp`-flag switches the simplification on.

      (%i1) simp:false;
      (%o1)                         false

      (%i2) sin(1.0);
      (%o2)                       sin(1.0)

      (%i3) sin(1.0),simp;
      (%o3)                  0.8414709848078965

  The simplification is switched on again. The `simp`-flag cannot suppress the simplification completely. The output shows a simplified expression, but the variable `x` has an unsimplified expression as a value, because the assignment has occurred during the evaluation phase of the expression.

      (%i1) simp:true;
      (%o1)                         true

      (%i2) x:sin(1.0),simp:false;
      (%o2)                  0.8414709848078965

      (%i3) :lisp $x
      ((%SIN) 1.0)

  Categories: Evaluation flags ·

Property: symmetric ¶

  `declare (h, symmetric)` tells the Maxima simplifier that `h` is a symmetric function. E.g., `h (x, z, y)` simplifies to `h (x, y, z)`.

  `commutative` is synonymous with `symmetric`.

  Categories: Declarations and inferences · Operators ·

Function: xthru (expr) ¶

  Combines all terms of expr (which should be a sum) over a common denominator without expanding products and exponentiated sums as `ratsimp` does. `xthru` cancels common factors in the numerator and denominator of rational expressions but only if the factors are explicit.

  Sometimes it is better to use `xthru` before `ratsimp`ing an expression in order to cause explicit factors of the gcd of the numerator and denominator to be canceled thus simplifying the expression to be `ratsimp`ed.

  Examples:

      (%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
                                      20
                       1       (x + 2)   - 2 y       x
      (%o1)        --------- + --------------- - ---------
                          19             20             20
                   (y + x)        (y + x)        (y + x)

      (%i2) xthru (%);
                                       20
                                (x + 2)   - y
      (%o2)                     -------------
                                         20
                                  (y + x)

  Categories: Expressions ·