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The Maxima on-line user's manual

Algebra Calculator

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Taylor Calculator

Taylor

Function: taylor (<expr>, <x>, <a>, <n>)

taylor (sqrt (x + 1), x, 0, 5);

Function: taylor (<expr>, [<x_1>, <x_2>, ...], <a>, <n>)

Function: taylor (<expr>, [<x>, <a>, <n>, asymp])

Function: taylor (<expr>, [<x_1>, <x_2>, ...], [<a_1>, <a_2>, ...], [<n_1>, <n_2>, ...])

Function: 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.

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

There are also some inexact matches for taylor. Try ?? taylor to see them.

(%o1)                                true
(%i2) 

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