(maxima.info)Functions and Variables for Sums and Products
28.1 Functions and Variables for Sums and Products
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-- Function: 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 'j<number>'. The quantity <number> is determined
by referring to 'gensumnum', which can be changed by the user. For
example, 'gensumnum:0$' resets it.
-- Function: 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.
Examples:
(%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)
-- Function: 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'.
-- Option variable: 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.
-- Function: 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'.
Examples:
(%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);
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
-- Option variable: 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.
-- Function: 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 Note: zeilberger-pkg
Examples:
(%i1) sum (i^2, i, 1, 7);
(%o1) 140
(%i2) sum (a[i], i, 1, 7);
(%o2) a + a + a + a + a + a + a
7 6 5 4 3 2 1
(%i3) sum (a(i), i, 1, 7);
(%o3) a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
(%i4) sum (a(i), i, 1, n);
n
====
\
(%o4) > a(i)
/
====
i = 1
(%i5) sum (2^i + i^2, i, 0, n);
n
====
\ i 2
(%o5) > (2 + i )
/
====
i = 0
(%i6) sum (2^i + i^2, i, 0, n), simpsum;
3 2
n + 1 2 n + 3 n + n
(%o6) 2 + --------------- - 1
6
(%i7) sum (1/3^i, i, 1, inf);
inf
====
\ 1
(%o7) > --
/ i
==== 3
i = 1
(%i8) sum (1/3^i, i, 1, inf), simpsum;
1
(%o8) -
2
(%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
inf
====
\ 1
(%o9) 30 > --
/ 2
==== i
i = 1
(%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
2
(%o10) 5 %pi
(%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
n
====
\ 1
(%o11) > -----
/ k + 1
====
k = 1
(%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
10 9 8 7 6 5 4 3 2
(%o12) b + b + b + b + b + a + a + a + a + a
-- Function: 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'.
-- Option variable: sumexpand
Default value: 'false'
When 'sumexpand' is 'true', products of sums and exponentiated sums
simplify to nested sums.
See also 'cauchysum'.
Examples:
(%i1) sumexpand: true$
(%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
m n
==== ====
\ \
(%o2) > > f(i1) g(i2)
/ /
==== ====
i1 = 0 i2 = 0
(%i3) sum (f (i), i, 0, m)^2;
m m
==== ====
\ \
(%o3) > > f(i3) f(i4)
/ /
==== ====
i3 = 0 i4 = 0
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