The Maxima on-line user's manual

Algebra Calculator

Taylor

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

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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