Maxima Function
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 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