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 must be names of simple variables.
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]