Maxima – 164 – Somme, prodotti e serie – 1

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Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.493.

Funzioni e variabili per somme e prodotti

bashindices (expr)
Transforms the expression expr by giving each summation and product a unique index. This gives changevar greater precision when it is working with summations or products. The form of the unique index is jnumber. The quantity number is determined by referring to gensumnum, which can be changed by the user. For example, gensumnum:0$ resets it.

lsum (expr, x, L)
Represents the sum of expr for each element x in L. A noun form 'lsum is returned if the argument L does not evaluate to a list.

(%i1) lsum (x^i, i, [1, 2, 7]);
                                   7    2
(%o1)                             x  + x  + x
(%i2) lsum (i^2, i, rootsof (x^3 - 1, x));
                           ====
                           \      2
(%o2)                       >    i
                           /
                           ====
                                         3
                           i in rootsof(x  - 1, x)

intosum (expr)
Moves multiplicative factors outside a summation to inside. If the index is used in the outside expression, then the function tries to find a reasonable index, the same as it does for sumcontract. This is essentially the reverse idea of the outative property of summations, but note that it does not remove this property, it only bypasses it.

In some cases, a scanmap (multthru, expr) may be necessary before the intosum.

simpproduct
Default value: false.

When simpproduct is true, the result of a product is simplified. This simplification may sometimes be able to produce a closed form. If simpproduct is false or if the quoted form 'product is used, the value is a product noun form which is a representation of the pi notation used in mathematics.

product (expr, i, i_0, i_1)
Represents a product of the values of expr as the index i varies from i_0 to i_1. The noun form 'product is displayed as an uppercase letter pi.

product evaluates expr and lower and upper limits i_0 and i_1, product quotes (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 product.

Otherwise, the range of the index is indefinite. Some rules are applied to simplify the product. When the global variable simpproduct is true, additional rules are applied.

In some cases, simplification yields a result which is not a product; otherwise, the
result is a noun form 'product.

See also nouns and evflag.

(%i3) product (x + i*(i+1)/2, i, 1, 4);
(%o3)                  (x + 1) (x + 3) (x + 6) (x + 10)
(%i4) product (i^2, i, 1, 7);
(%o4)                              25401600
(%i5) product (a[i], i, 1, 7);
(%o5)                        a  a  a  a  a  a  a
                              1  2  3  4  5  6  7
(%i6) product (a(i), i, 1, 7);
(%o6)                 a(1) a(2) a(3) a(4) a(5) a(6) a(7)
(%i7) product (a(i), i, 1, n);
                                    n
                                  /===\
                                   ! !
(%o7)                              ! !  a(i)
                                   ! !
                                  i = 1
(%i8) product (k, k, 1, n);
                                      n
                                    /===\
                                     ! !
(%o8)                                ! !  k
                                     ! !
                                    k = 1
(%i9) product (k, k, 1, n), simpproduct;
(%o9)                                 n!
(%i10) product (integrate (x^k, x, 0, 1), k, 1, n);
                                    n
                                  /===\
                                   ! !    1
(%o10)                             ! !  -----
                                   ! !  k + 1
                                  k = 1
(%i11) product (if k <= 5 then a^k else b^k, k, 1, 10);
                                     15  40
(%o11)                              a   b

simpsum
Default value: false.

When simpsum is true, the result of a sum is simplified. This simplification may sometimes be able to produce a closed form. If simpsum is false or if the quoted form 'sum is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

sum (expr, i, i_0, i_1)
Represents a summation of the values of expr as the index i varies from i_0 to i_1.

The noun form ‘sum is displayed as an uppercase letter sigma.

sum evaluates its summand expr and lower and upper limits i_0 and i_1, sum quotes (does not evaluate) the index i.

If the upper and lower limits differ by an integer, the summand expr is evaluated for each value of the summation index i, and the result is an explicit sum.

Otherwise, the range of the index is indefinite. Some rules are applied to simplify the summation. When the global variable simpsum is true, additional rules are applied.

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

When the evflag (evaluation flag) cauchysum is true, a product of summations is expressed as a Cauchy product, in which the index of the inner summation is a function of the index of the outer one, rather than varying independently.

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 lsum, sumcontract, intosum, bashindices, niceindices, nouns, evflag, and zeilberger-pkg.

(%i12) sum (i^2, i, 1, 7);
(%o12)                                140
(%i13) sum (a[i], i, 1, 7);
(%o13)                 a  + a  + a  + a  + a  + a  + a
                        7    6    5    4    3    2    1
(%i14) sum (a(i), i, 1, 7);
(%o14)          a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
(%i15) sum (a(i), i, 1, n);
                                   n
                                  ====
                                  \
(%o15)                             >    a(i)
                                  /
                                  ====
                                  i = 1
(%i16) sum (2^i + i^2, i, 0, n);
                                 n
                                ====
                                \       i    2
(%o16)                           >    (2  + i )
                                /
                                ====
                                i = 0
(%i17) sum (2^i + i^2, i, 0, n), simpsum;
                                     3      2
                          n + 1   2 n  + 3 n  + n
(%o17)                   2      + --------------- - 1
                                         6
(%i18) sum (1/3^i, i, 1, inf);
                                   inf
                                   ====
                                   \     1
(%o18)                              >    --
                                   /      i
                                   ====  3
                                   i = 1
(%i19) sum (1/3^i, i, 1, inf), simpsum;
                                       1
(%o19)                                 -
                                       2
(%i20) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                                     inf
                                     ====
                                     \     1
(%o20)                            30  >    --
                                     /      2
                                     ====  i
                                     i = 1
(%i21) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                         2
(%o21)                              5 %pi
(%i22) sum (integrate (x^k, x, 0, 1), k, 1, n);
                                   n
                                  ====
                                  \       1
(%o22)                             >    -----
                                  /     k + 1
                                  ====
                                  k = 1
(%i23) sum (if k <= 5 then a^k else b^k, k, 1, 10);
                 10    9    8    7    6    5    4    3    2
(%o23)          b   + b  + b  + b  + b  + a  + a  + a  + a  + a

sumcontract (expr)
Combines all sums of an addition that have upper and lower bounds that differ by constants. The result is an expression containing one summation for each set of such summations added to all appropriate extra terms that had to be extracted to form this sum. sumcontract combines all compatible sums and uses one of the indices from one of the sums if it can, and then try to form a reasonable index if it cannot use any supplied.

It may be necessary to do an intosum (expr) before the sumcontract.

sumexpand
Default value: false.

When sumexpand is true, products of sums and exponentiated sums simplify to nested sums.

See also cauchysum.

(%i24) sumexpand: true$

(%i25) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                            m      n
                           ====   ====
                           \      \
(%o25)                      >      >     f(i1) g(i2)
                           /      /
                           ====   ====
                           i1 = 0 i2 = 0
(%i26) sum (f (i), i, 0, m)^2;
                            m      m
                           ====   ====
                           \      \
(%o26)                      >      >     f(i3) f(i4)
                           /      /
                           ====   ====
                           i3 = 0 i4 = 0

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