Maxima - руководство на русском языке

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

2

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

(%i6) (%i6) multthru (a.(b+c.(d+e)+f));

(%o6) a . f + a . c . (e + d) + a . b

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

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

Функция: nounify (f)

Returns the noun-существительную form of the function name f. This is needed if one

wishes to refer to the name of a verb-глагол function as if it were a noun-существительным.

Note-заметьте that some verb functions will return their noun forms if they can't be evaluated

for certain-определённых arguments. This is also the form returned if a function call is

preceded-предшествуется by a quote-кавычкой.

Функция: nterms (expr)

Returns-возвращает the number-число of terms-термов that expr would have if it were

fully expanded out and no cancellations or combination of terms occurred. Note-заметьте that

expressions like sin (expr), sqrt (expr), exp (expr), etc-и_т.д. count as just one term

regardless-независимо of how many terms expr has (if it is a sum).

Примеры (в КОНСОЛИ):

(%i1) nterms(x^2+y^3);

(%o1) 2

(%i2) nterms(x^2+y^3-sin(x^2-y^5));

(%o2) 3

Функция: op (expr)

Returns the main (который последний работает) operator of the expression expr.

Функция op(expr) is equivalent to part(expr,0).

Функция op returns a string if the main operator is a built-in or user-defined prefix, binary

or n-ary infix, postfix, matchfix, or nofix operator. Otherwise-иначе, if expr is a subscripted-

индексированная function expression, op returns the subscripted-индексированную function;

61

in this case the return value is not an atom. Otherwise-иначе, expr is an array-массив function

or ordinary function expression, and op returns a symbol.

Функция op observes-отслеживает the value of the global flag inflag.

Функция op evaluates it argument.

See also ordergreat and mainvar.

(%i1) example(orderless);

(%i2) b*x+y^2;

2

(%o2) y + b x

(%i3) orderless(y);

(%o3) done

(%i4) b*x+y^2;

2

(%o4) b x + y

(%i5) %-%th(3);

2 2

(%o5) y - y

(%i6) unorder();

(%o6) [y]

(%o6) done

Функция: orderlessp (expr_1, expr_2)

Returns true if expr_1 precedes-предшествует expr_2 in the ordering set up by the

orderless command.

Примеры (в КОНСОЛИ):

(%i1) orderlessp(x2,x1);

(%o1) false

(%i2) orderlessp(x1,x2);

(%o2) true

Функция: part (expr, n_1, ..., n_k)

Returns parts of the displayed form of expr. It obtains-получает the part of expr as

specified by the indices n_1, ..., n_k. First part n_1 of expr is obtained, then part n_2 of that,

etc. The result is part n_k of ... part n_2 of part n_1 of expr.

part can be used to obtain an element of a list, a row of a matrix, etc.

If the last argument to a part function is a list of indices then several subexpressions are

picked out, each one corresponding to an index of the list. Thus part (x + y + z, [1,

3]) is z+x.

piece holds the last expression selected when using the part functions. It is set during

the execution of the function and thus may be referred to in the function itself as shown below.

If partswitch is set to true then end is returned when a selected part of an expression

doesn't exist, otherwise an error message is given.

Example: part (z+2*y, 2, 1) yields 2.

example (part) displays additional examples.

64

Функция: partition (expr, x)

Returns a list-список of two expressions-выражений. They are (1) the factors-множители

of expr (if it is a product-произведение), the terms-термы of expr (if it is a sum-сумма), or the

list-список (if it is a list) which don't contain-содержит var and, (2) the factors-множители,

terms-термы, or list-список which do.

Примеры (в КОНСОЛИ):

(%i1) partition (2*a*x*f(x), x);

(%o1) [2 a, x f(x)]

(%i2) partition (a+b, x);

(%o2) [b + a, 0]

(%i3) partition ([a, b, f(a), c], a);

(%o3) [[b, c], [a, f(a)]]

Option variable: partswitch

Default value: false

When partswitch is true, end is returned when a selected part-часть of an expression

doesn't exist-существует, otherwise-иначе an error message is given.

Функция: pickapart (expr, n)

Assigns intermediate-промежуточное expression labels to subexpressions of expr at

depth n, an integer. Subexpressions at greater or lesser depths are not assigned labels.

pickapart returns an expression in terms of intermediate expressions equivalent to the original

expression expr.

See also part, dpart, lpart, inpart, and reveal.

Примеры (в КОНСОЛИ):

(%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));

2

sin(x ) b + a

(%o1) - log(sqrt(x + 1) + 1) + ------- + -----

3 2

(%i2) pickapart (expr, 0);

2

sin(x ) b + a

(%t2) - log(sqrt(x + 1) + 1) + ------- + -----

3 2

(%o2) %t2

(%i3) pickapart (expr, 1);

(%t3) - log(sqrt(x + 1) + 1)

2

sin(x )

(%t4) -------

3

b + a

(%t5) -----

2

(%o5) %t5 + %t4 + %t3

(%i6) pickapart (expr, 2);

(%t6) log(sqrt(x + 1) + 1)

65

2

(%t7) sin(x )

(%t8) b + a

%t8 %t7

(%o8) --- + --- - %t6

2 3

(%i9) pickapart (expr, 3);

(%t9) sqrt(x + 1) + 1

2

(%t10) x

b + a sin(%t10)

(%o10) ----- - log(%t9) + ---------

2 3

(%i11) pickapart (expr, 4);

(%t11) sqrt(x + 1)

2

sin(x ) b + a

(%o11) ------- + ----- - log(%t11 + 1)

3 2

(%i12) pickapart (expr, 5);

(%t12) x + 1

2

sin(x ) b + a

(%o12) ------- + ----- - log(sqrt(%t12) + 1)

3 2

(%i13) pickapart (expr, 6);

2

sin(x ) b + a

(%o13) ------- + ----- - log(sqrt(x + 1) + 1)

3 2

System variable: piece

Holds-удерживает the last expression selected when using the part functions. It is set

during-в_течение the execution-выполнения of the function and thus-так may be referred-

связано to in the function itself.

Функция: polarform (expr)

Returns an expression r %e^(%i theta) equivalent to expr, such that r and theta

are purely-чисто real-действительное (число).

Примеры (в КОНСОЛИ):

(%i1) example(polarform);

(%i2) rectform(sin(x+2*%i));

(%o2) cosh(2) sin(x) + %i sinh(2) cos(x)

(%i3) polarform(%);

2 2 2 2

(%o3) sqrt(cosh (2) sin (x) + sinh (2) cos (x))

%i atan2(sinh(2) cos(x), cosh(2) sin(x))

%e

(%i4) rectform(log(4*%i+3));

66

4

(%o4) log(5) + %i atan(-)

3

(%i5) polarform(%);

atan(4/3)

%i atan(---------)

2 2 4 log(5)

(%o5) sqrt(log (5) + atan (-)) %e

3

(%i6) ev(rectform((3.5*%i+2)^0.25),numer);

(%o6) 0.36825879181669 %i + 1.368266274774369

(%i7) polarform(%);

(%o7) 1.416956998795383 (0.25989411967319 %i +

0.96563711950157)

(%o7) done

Функция: powers (expr, x)

Gives the powers-степени of x occuring in expr.

load (powers) loads this function.

Функция: product (expr, i, i_0, i_1)

Represents a product-произведение of the values of expr as the index i varies-

изменяющийся from i_0 to i_1. The noun-существительная form 'product is displayed as

an uppercase-заглавная letter pi.

product evaluates expr and lower and upper limits i_0 and i_1, product quotes-

кавычки (does not evaluate) the index i.

If the upper and lower limits differ by an integer, expr is evaluated for each value of the

index i, and the result is an explicit-явное product-произведение.

Otherwise-иначе, the range-область of the index is indefinite-неопределенная. Some

rules-правила are applied- примененяются to simplify-упростить the product. When the global

variable simpproduct is true, additional rules are applied. In some cases-причинам,

simplification yields a result which is not a product; otherwise-иначе, the result is a noun-

существительная form 'product.

See also nouns and evflag.

Примеры (в КОНСОЛИ):

(%i1) product (x + i*(i+1)/2, i, 1, 4);

(%o1) (x + 1) (x + 3) (x + 6) (x + 10)

(%i2) product (i^2, i, 1, 7);

(%o2) 25401600

(%i3) product (a[i], i, 1, 7);

(%o3) a a a a a a a

1 2 3 4 5 6 7

(%i4) product (a(i), i, 1, 7);

(%o4) a(1) a(2) a(3) a(4) a(5) a(6) a(7)

(%i5) product (a(i), i, 1, n);

67

n

/===\

! !

(%o5) ! ! a(i)

! !

i = 1

(%i6) product (k, k, 1, n);

n

/===\

! !

(%o6) ! ! k

! !

k = 1

(%i7) product (k, k, 1, n), simpproduct;

(%o7) n!

(%i8) product (integrate (x^k, x, 0, 1), k, 1, n);

n

/===\

! ! 1

(%o8) ! ! -----

! ! k + 1

k = 1

(%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);

15 40

(%o9) a b

Функция: realpart (expr)

Returns the real-действительную part-часть of expr. Функция realpart and

imagpart will work on expressions involving trigonometic and hyperbolic functions, as well as

square root, logarithm, and exponentiation.

Пример (в КОНСОЛИ):

(%i1) realpart(23+12*%i);

(%o1) 23

Функция: rectform (expr)

Returns an expression a + b %i equivalent to expr, such that a and b are purely-

чисто real-действительные (числа).

Функция: rembox (expr, unlabelled)

Функция: rembox (expr, label)

Функция: rembox (expr)

Removes-убирает boxes-рамки from expr.

rembox (expr, unlabelled) removes all unlabelled boxes from expr.

rembox (expr, label) removes only boxes bearing-отношения label-метки.

rembox (expr) removes all boxes, labelled and unlabelled.

Boxes are drawn by the box, dpart, and lpart functions.

Примеры (в КОНСОЛИ) — но они почему-то НЕ работают:

(%i1) expr:a*d-b*c)/h^2+sin(%pi*x);

68

a d - b c

(%o1) sin(%pi x) + ---------

2

h

(%i2) dpart (dpart (expr, 1, 1), 2, 2);

""""""" a d - b c

(%o2) sin("%pi x") + ---------

""""""" """"

" 2"

"h "

""""

(%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);

FOO""""""""""" BAR""""""""

" """"""" " "a d - b c"

(%o3) "sin("%pi x")" + "---------"

" """"""" " " """" "

"""""""""""""" " " 2" "

" "h " "

" """" "

"""""""""""

(%i4) rembox (expr2, unlabelled);

BAR""""""""

FOO""""""""" "a d - b c"

(%o4) "sin(%pi x)" + "---------"

"""""""""""" " 2 "

" h "

"""""""""""

(%i5) rembox (expr2, FOO);

BAR""""""""

""""""" "a d - b c"

(%o5) sin("%pi x") + "---------"

""""""" " """" "

" " 2" "

" "h " "

" """" "

"""""""""""

(%i6) rembox (expr2, BAR);

FOO"""""""""""

" """"""" " a d - b c

(%o6) "sin("%pi x")" + ---------

" """"""" " """"

"""""""""""""" " 2"

"h "

""""

(%i7) rembox (expr2);

a d - b c

(%o7) sin(%pi x) + ---------

2

h

69

Функция: sum (expr, i, i_0, i_1)

Represents a summation-сумму of the values of expr as the index i varies from i_0 to

i_1. The noun-существительная form 'sum is displayed as an uppercase-заглавная letter

sigma-сигма.

sum evaluates its summand expr and lower and upper limits i_0 and i_1, sum quotes-

кавычки (does not evaluate) the index i.

If the upper and lower limits differ-отличен by-от an integer-целого, the summand expr

is evaluated for each value of the summation index i, and the result is an explicit-явная sum-

сумма.

Otherwise-иначе, the range-область of the index is indefinite-неопределённая. Some

rules-правила are applied-применяются to simplify-упростить the summation-сумму. When

the global variable simpsum is true, additional rules are applied. In some cases-причинам,

simplification yields a result which is not a summation; otherwise, the result is a noun form 'sum.

When the evflag (evaluation flag) cauchysum-сумма_Коши is true, a product-

произведение of summations is expressed-выражается as a Cauchy-Коши product-

произведение, in which the index of the inner-внутреннего summation-суммирования is a

function of the index of the outer-внешний one, rather-скорее than varying-варьирование

independently-независимо.

The global variable genindex is the alphabetic prefix used to generate the next index of

summation, when an automatically generated index is needed.

gensumnum is the numeric suffix used to generate the next index of summation, when an

automatically generated index is needed. When gensumnum is false, an automatically-

generated index is only genindex with no numeric suffix.

See also sumcontract, intosum, bashindices, niceindices, nouns, evflag, and

zeilberger.

Примеры (в КОНСОЛИ):

(%i1) sum (i^2, i, 1, 7);

(%o1) 140

(%i2) 'sum (i^2, i, 1, 7);

7

====

\ 2

(%o2) > i

/

====

i = 1

(%i3) sum (a[i], i, 1, 7);

(%o3) a + a + a + a + a + a + a

7 6 5 4 3 2 1

(%i4) 'sum (a[i], i, 1, 7);

7

====

\

(%o4) > a

/ i

====

i = 1

(%i5) sum (a(i), i, 1, 7);

70

Maxima - руководство на русском языке

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