sin(x)
------ + ln(x) cos(x)
x
/ 2 \
poly \ 3 x cos(a), [x] /
You can differentiate any function and proof well known rules like:
f(x) diff(g(x), x) + g(x) diff(f(x), x)
D(g)(h(x)) D(f)(g(h(x))) diff(h(x), x)
The differential operator D is used to represent the differentiation of functions.
cos
(D(g) + D(h)) D(f)@(g + h)
Use the function int to compute the indefinite integral of functions.
3 6
2 2560 t 4 5 16 t
1024 t - 1280 t + ------- - 320 t + 64 t - -----
3 3
ln(ln(x))
2 3
n n n
- - -- + --
6 2 3
1 psi(k + 2, 1) 1
- --- + ------------- + ----------
4 k 2 - 4 k - 4
18858053/108636528
- 2 k binomial(n, k) + k binomial(n + 1, k)
--------------------------------------------
n n n
- 2 2 + 4 k 2 - 2 n 2
1 - 2 2 - 3 4 / 6 \
- - - a x - a x + O \ x /
a
1 3 / 4 \
- + y + x + O \ x /
x
3 2
a cos(x) a sin(x) / 4 \
sin(x) + a cos(x) - --------- - --------- + O \ a /
6 2
This is an example for the generalized series expansion.
/ 5 \
ln(x) + O \ x /
The following example shows the possibility for manipulating series expansions by using arithmetic operations.
3
x / 4 \
x - -- + O \ x /
6
1 13 x / 2 \
- + ---- + O \ x /
x 6
5
3 x / 6 \
x - -- + O \ x /
2
The function limit can be used to compute limits of expressions.
cos(x)
E
The function limit is also able to handle expressions of a high complicated structure.
2
- E
1
-- 1/2 1/2 --
| 3 - 1, - 3 - 1 , 1, 1 |
-- --
-- / 2 \1/2 / 2 \1/2 --
| - b + \ - 4 a c + b / - b - \ - 4 a c + b / |
| ---------------------------, --------------------------- |
-- 2 a 2 a --
-- 1/2 1/2 --
| - 1, (1/2 I) 3 + 1/2, (- 1/2 I) 3 + 1/2 |
-- --
[{u = 4/5, v = 3/5, w = - 2/5}]
-- -- / 2 \ -- --
| | v = - 3 u + 3 , w = 2 u - 2, u = RootOf \ 4 u + u - 4 / | |
-- -- -- --
1/6 / atan(3/2) \
cos(1) exp(2) + sin(2) cosh(3) + 13 cos | --------- |
\ 3 /
Im(tan(x))
For expressions involving unknowns the function rectform is usefull.
sin(2 Re(x)) I sinh(2 Im(x))
---------------------------- + ----------------------------
cos(2 Re(x)) + cosh(2 Im(x)) cos(2 Re(x)) + cosh(2 Im(x))
sin(2 Re(x)) sinh(2 Im(x))
----------------------------, ----------------------------
cos(2 Re(x)) + cosh(2 Im(x)) cos(2 Re(x)) + cosh(2 Im(x))
Standard arithmetical operations can be performed for complex expressions in rectangular form.
cos(Im(x)) exp(Re(x)) + I sin(Im(x)) exp(Re(x))
PI cos(Im(x)) exp(Re(x)) + I PI sin(Im(x)) exp(Re(x)), 2
2 / 2
I cos(Im(x)) sin(Im(x)) exp(Re(x)) + \ cos(Im(x)) exp(
2 2 2 \
Re(x)) - sin(Im(x)) exp(Re(x)) /
(cos(Im(x)) sin(Re(x)) exp(Re(x)) cosh(Im(x)) - cos(Re(x))
sin(Im(x)) exp(Re(x)) sinh(Im(x))) + I (cos(Im(x)) cos(Re(
x)) exp(Re(x)) sinh(Im(x)) + sin(Im(x)) sin(Re(x)) exp(Re(
x)) cosh(Im(x)))