Input: f(x+h) = f(x) + h*f'(x) + h^2/2*f''(x) + O(h^3) Output: 2 h / 3\ f(x + h) = f(x) + h f'(x) + -- f''(x) + O\h / 2 Input: h = -((f'(x)/f''(x))*(1 - sqrt(1 - (2*f(x)*f''(x))/f'(x)^2))) Output: / _________________\ | / 2 f(x) f''(x)| f'(x) | / 1 - -------------| h = - ------ |1 - / 2 | f''(x) \ \/ f'(x) / Input: sqrt(42)/z=root((1+1/(1+1/(x^2+1/b)))^3,6)/(3^d/(5-e + 42/(3 + f))+sqrt((2/(1-1/(1+1/7))))+sqrt(1/(2+3)+3)^(sqrt(21/(38-w)))) Output: 6_________________ / 3 / / 1 \ / |1 + ----------| / | 1 | / | 1 + ------| / | 2 1| __ / | x + -| \/42 \/ \ b/ ---- = ------------------------------------------------------------- z ______ / 21 _________ / ------ / 2 \/ 38 - w d / --------- / _________\ 3 / 1 | / 1 | ------------- + / 1 - ----- + | / ----- + 3| 42 / 1 \\/ 2 + 3 / 5 - e + ----- / 1 + - 3 + f \/ 7 Input: int(int(int(psi^2, x = -inf .. inf), y = -inf .. inf), z = -inf .. inf) = 1 Output: oo oo oo / / / | | | 2 | | | psi dx dy dz = 1 | | | / / / -oo -oo -oo Input: A_TR = x*sqrt(x^2-1)/2 - int(sqrt(t^2-1), t = 1 .. x) Output: x ______ / / 2 | ______ x \/ x - 1 | / 2 A = ----------- - | \/ t - 1 dt TR 2 | | / 1 Input: sqrt(e) = 1+1/(1+1/(1+1/(1+1/(5+1/(1+1/(1+1/(9+1/(1+1/(1+...))))))))) Output: _ 1 \/e = 1 + --------------------------------------- 1 1 + ----------------------------------- 1 1 + ------------------------------- 1 1 + --------------------------- 1 5 + ----------------------- 1 1 + ------------------- 1 1 + --------------- 1 9 + ----------- 1 1 + ------- 1 + ... Input: e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... = 1 + sum(x^n/n!, n = 1 .. inf) Output: oo 2 3 4 ===== n x x x x \ x e = 1 + x + -- + -- + -- + ... = 1 + > -- 2! 3! 4! / n! ===== n = 1 Input: (1/4)*pi*sqrt(2) = sum((-1)^(k+1)/(4*k + 1) + (-1)^(k+1)/(4*k - 3), k = 1 .. inf) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - ... Output: oo ===== / k + 1 k + 1\ 1 __ _ \ |(-1) (-1) | 1 1 1 1 1 - || \/2 = > |--------- + ---------| = 1 + - - - - - + - + -- - ... 4 / \ 4 k + 1 4 k - 3 / 3 5 7 9 11 ===== k = 1 Input: 2/pi=sqrt(1/2)*sqrt(1/2+1/2*sqrt(1/2))*sqrt(1/2+1/2*sqrt(1/2+1/2*sqrt(1/2)))*... Output: ___________ ______________________ _ / _ / ___________ 2 /1 / 1 1 /1 / / _ -- = / - / - + - / - / 1 1 / 1 1 /1 ... __ \/ 2 \/ 2 2 \/ 2 / - + - / - + - / - || \/ 2 2 \/ 2 2 \/ 2 Input: pi = 3/4*sqrt(3) + 24*int(sqrt(x - x^2), x = 0 .. 1/4) = (3*sqrt(3))/4 + 24 * (1/12 - 1/(5*2^5) - 1/(28*2^7) - ...) Output: 1 - 4 / | ______ _ __ 3 _ | / 2 3 \/3 / 1 1 1 \ || = - \/3 + 24 | \/ x - x dx = ----- + 24 |-- - ---- - ----- - ...| 4 | 4 |12 5 7 | | \ 5 2 28 2 / / 0 Input: int(z^2, z = 1 .. root(3, 3)) * cos((3*pi)/9) = ln(root(e, 3)) Output: 3_ \/3 / __ | 2 3 || / 3_\ | z dz cos ---- = ln\\/e/ | 9 / 1 Input: x\ = (x_1 + x_2 + x_3 + ... + x_n)/n = (1/n)*sum(x_i, i = 1 .. n) Output: n x + x + x + ... + x ===== _ 1 2 3 n 1 \ x = ----------------------- = - > x n n / i ===== i = 1 Input: zeta(s) = (1 / (1 - (1/2^s))) * (1 / (1 - (1/3^s))) * (1 / (1 - (1/5^s))) * (1 / (1 - (1/7^s))) * ... = prod(1 / (1 - (1/p^s)), p_prime) Output: ===== 1 1 1 1 | | 1 zeta(s) = ------ ------ ------ ------ ... = | | ------ 1 1 1 1 | | 1 1 - -- 1 - -- 1 - -- 1 - -- | | 1 - -- s s s s p s 2 3 5 7 prime p Input: int((x^2+a)/b,x) = (1/b)*int(x^2+a,x) = (1/b)*(x^3/3 + a*x) + C Output: / | 2 / / 3 \ | x + a 1 | / 2 \ 1 |x | | ------ dx = - | \x + a/ dx = - |-- + a x| + C | b b | b \ 3 / | / / Input: sin(a)/a = cos(a/2) * cos(a/4) * cos(a/8) * cos(a/16) * ... = prod(cos(a/2^n), n = 1 .. inf) Output: oo ===== sin a a a a a | | a ----- = cos - cos - cos - cos -- ... = | | cos -- a 2 4 8 16 | | n | | 2 n = 1 Input: A_T = [sqrt(a/b), 0, 0; 0, sqrt(a/b), 0; 0, 0, sqrt(a/b)]^-1 Output: -1 / _ \ | /a | | / - 0 0 | | \/ b | | | | _ | | /a | A = | 0 / - 0 | T | \/ b | | | | _ | | /a | | 0 0 / - | \ \/ b / Input: lim(1/x^2 - (cos(x)/x)^2, x -> inf) = 1 Output: / 2\ | 1 /cos x\ | lim |-- - |-----| | = 1 | 2 \ x / | x -> oo \x /