Maxima – 167 – Somme, prodotti e serie – 4

petsciibots-5

Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.504.

taylor (expr, x, a, n)
taylor (expr, [x_1, x_2, ...], a, n)
taylor (expr, [x, a, n, 'asymp])
taylor (expr, [x_1, x_2, ...], [a_1, a_2, ...], [n_1, n_2, ...])
taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_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.

(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                           2             2       3      2             3
             (a + 1) x   (a  + 2 a + 1) x    (3 a  + 9 a  + 9 a - 1) x
(%o1)/T/ 1 + --------- - ----------------- + -------------------------- + . . .
                 2               8                       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        6       4       2   6
              k  x    (3 k  - 4 k ) x    (45 k  - 60 k  + 16 k ) x
(%o11)/T/ 1 - ----- - ---------------- - -------------------------- + . . .
                2            24                     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                        3
              y                       y                   y    y            2
(%o13)/T/ ((- --) + y + . . .) + (1 - -- + . . .) x + ((- -) + -- + . . .) x
              6                       2                   2    12
                                                          2
                                                    1    y            3
                                              + ((- -) + -- + . . .) x  + . . .
                                                    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);
           y   1                1     1               1            2
(%o15)/T/ (- + - + . . .) + ((- --) + - + . . .) x + (-- + . . .) x
           6   y                 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

taylordepth
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.

taylorinfo (expr)
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.

(%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                2                         2
(%o1)/T/ ((1 - a ) - 2 a (y - a) - (y - a) )
              2                        2
 + ((- (y - a) ) - 2 a (y - a) + (1 - a )) x
              2                        2    2
 + ((- (y - a) ) - 2 a (y - a) + (1 - a )) x
              2                        2    3
 + ((- (y - a) ) - 2 a (y - a) + (1 - a )) x  + . . .
(%i2) taylorinfo(%);
(%o2)                      [[x, 0, 3], [y, a, inf]]

taylorp (expr)
Returns true if expr is a Taylor series, and false otherwise.

taylor_logexpand
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.

taylor_order_coefficients
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.

taylor_simplifier (expr)
Simplifies coefficients of the power series expr. taylor calls this function.

taylor_truncate_polynomials
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.

taytorat (expr)
Converts expr from taylor form to canonical rational expression (CRE) form. The effect is the same as rat (ratdisrep (expr)), but faster.

trunc (expr)
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.

(%i3) expr: x^2 + x + 1;
                                   2
(%o3)                             x  + x + 1
(%i4) trunc (expr);
                                       2
(%o4)                         1 + x + x  + . . .
(%i5) is (expr = trunc (expr));
(%o5)                                true

unsum (f, n)
Returns the first backward difference f(n) - f(n - 1). Thus unsum in a sense is the inverse of sum. See also nusum.

(%i6) g(p) := p*4^n/binomial(2*n,n);
                                            n
                                         p 4
(%o6)                      g(p) := ----------------
                                   binomial(2 n, n)
(%i7) g(n^4);
                                     4  n
                                    n  4
(%o7)                          ----------------
                               binomial(2 n, n)
(%i8) nusum (%, n, 0, n);
                           4        3       2              n
            2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4     2
(%o8)       ------------------------------------------------ - ---
                          693 binomial(2 n, n)                 231
(%i9) unsum (%, n);
                                     4  n
                                    n  4
(%o9)                          ----------------
                               binomial(2 n, n)

verbose
Default value: false.

When verbose is true, powerseries prints progress messages.

Posta un commento o usa questo indirizzo per il trackback.

Trackback

Rispondi

Inserisci i tuoi dati qui sotto o clicca su un'icona per effettuare l'accesso:

Logo WordPress.com

Stai commentando usando il tuo account WordPress.com. Chiudi sessione /  Modifica )

Google photo

Stai commentando usando il tuo account Google. Chiudi sessione /  Modifica )

Foto Twitter

Stai commentando usando il tuo account Twitter. Chiudi sessione /  Modifica )

Foto di Facebook

Stai commentando usando il tuo account Facebook. Chiudi sessione /  Modifica )

Connessione a %s...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d blogger hanno fatto clic su Mi Piace per questo: