Next: Introduction to Fourier series, Previous: Introduction to Series, Up: Sums, Products, and Series [Contents][Index]
Default value: false
When multiplying together sums with inf as their upper limit,
if sumexpand is true and cauchysum is true
then the Cauchy product will be used rather than the usual
product.
In the Cauchy product the index of the inner summation is a
function of the index of the outer one rather than varying
independently.
Example:
(%i1) sumexpand: false$ (%i2) cauchysum: false$
(%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
inf inf
==== ====
\ \
(%o3) ( > f(i)) > g(j)
/ /
==== ====
i = 0 j = 0
(%i4) sumexpand: true$ (%i5) cauchysum: true$
(%i6) expand(s,0,0);
inf i1
==== ====
\ \
(%o6) > > g(i1 - i2) f(i2)
/ /
==== ====
i1 = 0 i2 = 0
For each function f_i of one variable x_i,
deftaylor defines expr_i as the Taylor series about zero.
expr_i is typically a polynomial in x_i or a summation;
more general expressions are accepted by deftaylor without complaint.
powerseries (f_i(x_i), x_i, 0)
returns the series defined by deftaylor.
deftaylor returns a list of the functions f_1, …, f_n.
deftaylor evaluates its arguments.
Example:
(%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
(%o1) [f]
(%i2) powerseries (f(x), x, 0);
inf
==== i1
\ x 2
(%o2) > -------- + x
/ i1 2
==== 2 i1!
i1 = 4
(%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
2 3 4
x 3073 x 12817 x
(%o3)/T/ 1 + x + -- + ------- + -------- + . . .
2 18432 307200
Default value: true
When maxtayorder is true, then during algebraic
manipulation of (truncated) Taylor series, taylor tries to retain
as many terms as are known to be correct.
Renames the indices of sums and products in expr. niceindices
attempts to rename each index to the value of niceindicespref[1], unless
that name appears in the summand or multiplicand, in which case
niceindices tries the succeeding elements of niceindicespref in
turn, until an unused variable is found. If the entire list is exhausted,
additional indices are constructed by appending integers to the value of
niceindicespref[1], e.g., i0, i1, i2, …
niceindices returns an expression.
niceindices evaluates its argument.
Example:
(%i1) niceindicespref;
(%o1) [i, j, k, l, m, n]
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
inf inf
/===\ ====
! ! \
(%o2) ! ! > f(bar i j + foo)
! ! /
bar = 1 ====
foo = 1
(%i3) niceindices (%);
inf inf
/===\ ====
! ! \
(%o3) ! ! > f(i j l + k)
! ! /
l = 1 ====
k = 1
Default value: [i, j, k, l, m, n]
niceindicespref is the list from which niceindices
takes the names of indices for sums and products.
The elements of niceindicespref are typically names of variables,
although that is not enforced by niceindices.
Example:
(%i1) niceindicespref: [p, q, r, s, t, u]$
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
inf inf
/===\ ====
! ! \
(%o2) ! ! > f(bar i j + foo)
! ! /
bar = 1 ====
foo = 1
(%i3) niceindices (%);
inf inf
/===\ ====
! ! \
(%o3) ! ! > f(i j q + p)
! ! /
q = 1 ====
p = 1
Carries out indefinite hypergeometric summation of expr with respect to x using a decision procedure due to R.W. Gosper. expr and the result must be expressible as products of integer powers, factorials, binomials, and rational functions.
The terms "definite"
and "indefinite summation" are used analogously to "definite" and
"indefinite integration".
To sum indefinitely means to give a symbolic result
for the sum over intervals of variable length, not just e.g. 0 to
inf. Thus, since there is no formula for the general partial sum of
the binomial series, nusum can’t do it.
nusum and unsum know a little about sums and differences of
finite products. See also unsum.
Examples:
(%i1) nusum (n*n!, n, 0, n);
Dependent equations eliminated: (1)
(%o1) (n + 1)! - 1
(%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
4 3 2 n
2 (n + 1) (63 n + 112 n + 18 n - 22 n + 3) 4 2
(%o2) ------------------------------------------------ - ------
693 binomial(2 n, n) 3 11 7
(%i3) unsum (%, n);
4 n
n 4
(%o3) ----------------
binomial(2 n, n)
(%i4) unsum (prod (i^2, i, 1, n), n);
n - 1
/===\
! ! 2
(%o4) ( ! ! i ) (n - 1) (n + 1)
! !
i = 1
(%i5) nusum (%, n, 1, n);
Dependent equations eliminated: (2 3)
n
/===\
! ! 2
(%o5) ! ! i - 1
! !
i = 1
Returns a list of all rational functions which have the given Taylor series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e. are "best" approximants, and which additionally satisfy the specified degree bounds.
taylor_series is an univariate Taylor series. numer_deg_bound and denom_deg_bound are positive integers specifying degree bounds on the numerator and denominator.
taylor_series can also be a Laurent series, and the degree
bounds can be inf which causes all rational functions whose total
degree is less than or equal to the length of the power series to be
returned. Total degree is defined as numer_deg_bound +
denom_deg_bound.
Length of a power series is defined as
"truncation level" + 1 - min(0, "order of series").
(%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
2 3
(%o1)/T/ 1 + x + x + x + . . .
(%i2) pade (%, 1, 1);
1
(%o2) [- -----]
x - 1
(%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
+ 387072*x^7 + 86016*x^6 - 1507328*x^5
+ 1966080*x^4 + 4194304*x^3 - 25165824*x^2
+ 67108864*x - 134217728)
/134217728, x, 0, 10);
2 3 4 5 6 7
x 3 x x 15 x 23 x 21 x 189 x
(%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
2 16 32 1024 2048 32768 65536
8 9 10
5853 x 2847 x 83787 x
+ ------- + ------- - --------- + . . .
4194304 8388608 134217728
(%i4) pade (t, 4, 4);
(%o4) []
There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.
(%i5) pade (t, 5, 5);
5 4 3
(%o5) [- (520256329 x - 96719020632 x - 489651410240 x
2
- 1619100813312 x - 2176885157888 x - 2386516803584)
5 4 3
/(47041365435 x + 381702613848 x + 1360678489152 x
2
+ 2856700692480 x + 3370143559680 x + 2386516803584)]
Returns the general form of the power series expansion for expr in the
variable x about the point a (which may be inf for infinity):
inf
====
\ n
> b (x - a)
/ n
====
n = 0
If powerseries is unable to expand expr,
taylor may give the first several terms of the series.
When verbose is true,
powerseries prints progress messages.
(%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:
d
/ -- (sin(x))
[ dx
log(sin(x)) = i ----------- dx
] sin(x)
/
in the first simplification we have returned:
/
[
i cot(x) dx - log(x)
]
/
inf
==== i1 2 i1 2 i1
\ (- 1) 2 bern(2 i1) x
> ------------------------------
/ i1 (2 i1)!
====
i1 = 1
(%o2) -------------------------------------
2
Default value: false
When psexpand is true,
an extended rational function expression is displayed fully expanded.
The switch ratexpand has the same effect.
When psexpand is false,
a multivariate expression is displayed just as in the rational function package.
When psexpand is multi,
then terms with the same total degree in the variables are grouped together.
These functions return the reversion of expr, a Taylor series about zero
in the variable x. revert returns a polynomial of degree equal to
the highest power in expr. revert2 returns a polynomial of degree
n, which may be greater than, equal to, or less than the degree of
expr.
load ("revert") loads these functions.
Examples:
(%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
taylor (expr, x, a, n) expands the expression
expr in a truncated Taylor or Laurent series in the variable x
around the point a,
containing terms through (x - a)^n.
If expr is of the form f(x)/g(x) and
g(x) has no terms up to degree n then taylor
attempts to expand g(x) up to degree 2 n.
If there are still no nonzero terms, taylor doubles the degree of the
expansion of g(x) so long as the degree of the expansion is
less than or equal to n 2^taylordepth.
taylor (expr, [x_1, x_2, ...], a, n)
returns a truncated power series
of degree n in all variables x_1, x_2, …
about the point (a, a, ...).
taylor (expr, [x_1, a_1, n_1], [x_2,
a_2, n_2], ...) returns a truncated power series in the variables
x_1, x_2, … about the point
(a_1, a_2, ...), truncated at n_1, n_2, …
taylor (expr, [x_1, x_2, ...], [a_1,
a_2, ...], [n_1, n_2, ...]) returns a truncated power series
in the variables x_1, x_2, … about the point
(a_1, a_2, ...), truncated at n_1, n_2, …
taylor (expr, [x, a, n, 'asymp]) returns an
expansion of expr in negative powers of x - a.
The highest order term is (x - a)^-n.
When maxtayorder is true, then during algebraic
manipulation of (truncated) Taylor series, taylor tries to retain
as many terms as are known to be correct.
When psexpand is true,
an extended rational function expression is displayed fully expanded.
The switch ratexpand has the same effect.
When psexpand is false,
a multivariate expression is displayed just as in the rational function package.
When psexpand is multi,
then terms with the same total degree in the variables are grouped together.
See also the taylor_logexpand switch for controlling expansion.
Examples:
(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
2 2
(a + 1) x (a + 2 a + 1) x
(%o1)/T/ 1 + --------- - -----------------
2 8
3 2 3
(3 a + 9 a + 9 a - 1) x
+ -------------------------- + . . .
48
(%i2) %^2;
3
x
(%o2)/T/ 1 + (a + 1) x - -- + . . .
6
(%i3) taylor (sqrt (x + 1), x, 0, 5);
2 3 4 5
x x x 5 x 7 x
(%o3)/T/ 1 + - - -- + -- - ---- + ---- + . . .
2 8 16 128 256
(%i4) %^2;
(%o4)/T/ 1 + x + . . .
(%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
inf
/===\
! ! i 2.5
! ! (x + 1)
! !
i = 1
(%o5) -----------------
2
x + 1
(%i6) ev (taylor(%, x, 0, 3), keepfloat);
2 3
(%o6)/T/ 1 + 2.5 x + 3.375 x + 6.5625 x + . . .
(%i7) taylor (1/log (x + 1), x, 0, 3);
2 3
1 1 x x 19 x
(%o7)/T/ - + - - -- + -- - ----- + . . .
x 2 12 24 720
(%i8) taylor (cos(x) - sec(x), x, 0, 5);
4
2 x
(%o8)/T/ - x - -- + . . .
6
(%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
(%o9)/T/ 0 + . . .
(%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
2 4
1 1 11 347 6767 x 15377 x
(%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
6 4 2 15120 604800 7983360
x 2 x 120 x
+ . . .
(%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
2 2 4 2 4
k x (3 k - 4 k ) x
(%o11)/T/ 1 - ----- - ----------------
2 24
6 4 2 6
(45 k - 60 k + 16 k ) x
- -------------------------- + . . .
720
(%i12) taylor ((x + 1)^n, x, 0, 4);
2 2 3 2 3
(n - n) x (n - 3 n + 2 n) x
(%o12)/T/ 1 + n x + ----------- + --------------------
2 6
4 3 2 4
(n - 6 n + 11 n - 6 n) x
+ ---------------------------- + . . .
24
(%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
3 2
y y
(%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
6 2
3 2
y y 2 1 y 3
+ (- - + -- + . . .) x + (- - + -- + . . .) x + . . .
2 12 6 12
(%i14) taylor (sin (y + x), [x, y], 0, 3);
3 2 2 3
x + 3 y x + 3 y x + y
(%o14)/T/ y + x - ------------------------- + . . .
6
(%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
1 y 1 1 1 2
(%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
y 6 2 6 3
y y
1 3
+ (- -- + . . .) x + . . .
4
y
(%i16) taylor (1/sin (y + x), [x, y], 0, 3);
3 2 2 3
1 x + y 7 x + 21 y x + 21 y x + 7 y
(%o16)/T/ ----- + ----- + ------------------------------- + . . .
x + y 6 360
Default value: 3
If there are still no nonzero terms, taylor doubles the degree of the
expansion of g(x) so long as the degree of the expansion is
less than or equal to n 2^taylordepth.
Returns information about the Taylor series expr. The return value is a list of lists. Each list comprises the name of a variable, the point of expansion, and the degree of the expansion.
taylorinfo returns false if expr is not a Taylor series.
Example:
(%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
2 2
(%o1)/T/ - (y - a) - 2 a (y - a) + (1 - a )
2 2
+ (1 - a - 2 a (y - a) - (y - a) ) x
2 2 2
+ (1 - a - 2 a (y - a) - (y - a) ) x
2 2 3
+ (1 - a - 2 a (y - a) - (y - a) ) x + . . .
(%i2) taylorinfo(%);
(%o2) [[y, a, inf], [x, 0, 3]]
Returns true if expr is a Taylor series,
and false otherwise.
Default value: true
taylor_logexpand controls expansions of logarithms in
taylor series.
When taylor_logexpand is true, all logarithms are expanded fully
so that zero-recognition problems involving logarithmic identities do not
disturb the expansion process. However, this scheme is not always
mathematically correct since it ignores branch information.
When taylor_logexpand is set to false, then the only expansion of
logarithms that occur is that necessary to obtain a formal power series.
Default value: true
taylor_order_coefficients controls the ordering of
coefficients in a Taylor series.
When taylor_order_coefficients is true,
coefficients of taylor series are ordered canonically.
Simplifies coefficients of the power series expr.
taylor calls this function.
Default value: true
When taylor_truncate_polynomials is true,
polynomials are truncated based upon the input truncation levels.
Otherwise,
polynomials input to taylor are considered to have infinite precision.
Converts expr from taylor form to canonical rational expression
(CRE) form. The effect is the same as rat (ratdisrep (expr)), but
faster.
Annotates the internal representation of the general expression expr so that it is displayed as if its sums were truncated Taylor series. expr is not otherwise modified.
Example:
(%i1) expr: x^2 + x + 1;
2
(%o1) x + x + 1
(%i2) trunc (expr);
2
(%o2) 1 + x + x + . . .
(%i3) is (expr = trunc (expr));
(%o3) true
Returns the first backward difference
f(n) - f(n - 1).
Thus unsum in a sense is the inverse of sum.
See also nusum.
Examples:
(%i1) g(p) := p*4^n/binomial(2*n,n);
n
p 4
(%o1) g(p) := ----------------
binomial(2 n, n)
(%i2) g(n^4);
4 n
n 4
(%o2) ----------------
binomial(2 n, n)
(%i3) nusum (%, n, 0, n);
4 3 2 n
2 (n + 1) (63 n + 112 n + 18 n - 22 n + 3) 4 2
(%o3) ------------------------------------------------ - ------
693 binomial(2 n, n) 3 11 7
(%i4) unsum (%, n);
4 n
n 4
(%o4) ----------------
binomial(2 n, n)
Default value: false
When verbose is true,
powerseries prints progress messages.
Next: Introduction to Fourier series, Previous: Introduction to Series, Up: Sums, Products, and Series [Contents][Index]