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30.1 Introduction to Series | ||
30.2 Functions and Variables for Series | ||
30.3 Poisson series |
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Maxima contains functions taylor
and powerseries
for finding the
series of differentiable functions. It also has tools such as nusum
capable of finding the closed form of some series. Operations such as addition and multiplication work as usual on series. This section presents the global variables which control the expansion.
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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) ''s; inf i1 ==== ==== \ \ (%o6) > > g(i1 - i2) f(i2) / / ==== ==== i1 = 0 i2 = 0
Categories: Sums and products
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
Categories: Power series
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.
Categories: Power series
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
Categories: Sums and products
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
Categories: Sums and products
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
Categories: Sums and products
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 a 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)]
Categories: Power series
Default value: false
When powerdisp
is true
,
a sum is displayed with its terms in order of increasing power.
Thus a polynomial is displayed as a truncated power series,
with the constant term first and the highest power last.
By default, terms of a sum are displayed in order of decreasing power.
Categories: Display flags and variables
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
Categories: Power series
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.
Categories: Display flags and variables
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
Categories: Power series
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
Categories: Power series
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
.
Categories: Power series
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]]
Categories: Power series
Returns true
if expr is a Taylor series,
and false
otherwise.
Categories: Predicate functions · Power series
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.
Categories: Power series · Exponential and logarithm functions
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.
Categories: Power series
Simplifies coefficients of the power series expr.
taylor
calls this function.
Categories: Power series
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 precison.
Categories: Power series
Converts expr from taylor
form to canonical rational expression (CRE) form.
The effect is the same as rat (ratdisrep (expr))
, but faster.
Categories: Power series · Rational expressions
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
Categories: Power series
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)
Categories: Sums and products
Default value: false
When verbose
is true
,
powerseries
prints progress messages.
Categories: Power series
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Converts a into a Poisson encoding.
Categories: Poisson series
Converts a from Poisson encoding to general
representation. If a is not in Poisson form, outofpois
carries out the conversion,
i.e., the return value is outofpois (intopois (a))
.
This function is thus a canonical simplifier
for sums of powers of sine and cosine terms of a particular type.
Categories: Poisson series
Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.
Categories: Poisson series
Functionally identical to intopois (a^b)
.
b must be a positive integer.
Categories: Poisson series
Integrates in a similarly restricted sense (to
poisdiff
). Non-periodic terms in b are dropped if b is in the trig
arguments.
Categories: Poisson series
Default value: 5
poislim
determines the domain of the coefficients in
the arguments of the trig functions. The initial value of 5
corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it
can be set to [-2^(n-1)+1, 2^(n-1)].
Categories: Poisson series
will map the functions sinfn on the sine terms and cosfn on the cosine terms of the Poisson series given. sinfn and cosfn are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.
Categories: Poisson series
Is functionally identical to intopois (a + b)
.
Categories: Poisson series
Converts a into a Poisson series for a in general representation.
Categories: Poisson series
The symbol /P/
follows the line label of Poisson series
expressions.
Categories: Poisson series
Substitutes a for b in c. c is a Poisson series.
(1) Where B is a variable u, v, w, x, y, or z,
then a must be an
expression linear in those variables (e.g., 6*u + 4*v
).
(2) Where b is other than those variables, then a must also be free of those variables, and furthermore, free of sines or cosines.
poissubst (a, b, c, d, n)
is a special type of substitution which
operates on a and b as in type (1) above, but where d is a Poisson
series, expands cos(d)
and sin(d)
to order n so as to provide the
result of substituting a + d
for b in c. The idea is that d is an
expansion in terms of a small parameter. For example,
poissubst (u, v, cos(v), %e, 3)
yields cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6)
.
Categories: Poisson series
Is functionally identical to intopois (a*b)
.
Categories: Poisson series
is a reserved function name which (if the user has defined
it) gets applied during Poisson multiplication. It is a predicate
function of 6 arguments which are the coefficients of the u, v, ..., z
in a term. Terms for which poistrim
is true
(for the coefficients of
that term) are eliminated during multiplication.
Categories: Poisson series
Prints a Poisson series in a readable format. In common
with outofpois
, it will convert a into a Poisson encoding first, if
necessary.
Categories: Poisson series · Display functions
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