analysis

goniometric functions:

sin(a+b) = sin(a) * cos(b) + cos(a)*sin(b)
cos(a+b) = cos(a) * cos(b) - sin(a)*sin(b)

sin(a) + sin(b) = 2 * sin((a+b)/2) * cos((a-b)/2)
  sin(x+y) + sin(x-y) = 2 * sin(x) * cos(y)
cos(a) + cos(b) = 2 * cos((a+b)/2) * cos((a-b)/2)
  cos(x+y) + cos(x-y) = 2 * cos(x) * cos(y)
cos(a) - cos(b) = -2 * sin((a+b)/2) * sin((a-b)/2)
  cos(x+y) - cos(x-y) = -2 * sin(x) * sin(y)

sin^2(a) = (1 - cos(2a))/2
cos^2(a) = (1 + cos(2a))/2

1+tan^2(a) = 1/cos^2(a)

tan(a+b) = ( tan(a) + tan(b) ) / (1 - tan(a)*tan(b))

exp(x) = sum( x^i/i!, i=0..n-1 ) + O(x^n)
       = 1 + x + x^2/2! + x^3/3! + ...

sin(x) = sum( (-1)^i*x^(2i+1)/(2i+1)!, i=0..n-1 ) + O(x^(2n+1))
       = x - x^3/3! + x^5/5! - x^7/7! + ...

cos(x) = sum( (-1)^i*x^(2i)/(2i)!, i=0..n-1 ) + O(x^(2n))
       = 1 - x^2/2! + x^4/4! - x^6/6! + ...

1/(1-x) = sum( x^i, i=0..n-1 ) + O(x^n)
        = 1 + x + x^2 + x^3 + ...

ln(1+x) = -sum( (-x)^i/i, i=1..n-1 ) + O(x^n)
        = x - x^2/2 + x^3/3 - x^4/4 + ...

arctan(x) = sum( (-1)^i*x^(2*i+1)/(2*i+1), i=0..n-1 ) + O(x^(2n+1))
          = x - x^3/3 + x^5/5 - x^7/7 + ...

(1+x)^a = sum( binom(a,i)*x^i, i=0..n-1 ) + O(x^n)
        = 1 + ax + a(a-1)*x^2/2! + a(a-1)(a-2)*x^3/3! + ...

   binom(a,0) = 1
   binom(a,k) = prod( a-i, i=0..k-1 ) / i !


1/(1-x)   = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8

1/(1-i*x) = 1 + ix - x^2 -i*x^3 + x^4 + i*x^5 - x^6 -i*x^7 + x^8
x/(1+x^2) =      x       -  x^3       +   x^5       -  x^7 
1/(1+x^2) = 1      - x^2        + x^4         - x^6        + x^8

x/(1-x^2) =      x       +  x^3       +   x^5       +  x^7 
1/(1-x^2) = 1      + x^2        + x^4         + x^6        + x^8

1/(1-i*x)  = (1+i*x)/(1+x)


ln(1+x)       =  x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6  ix^7/7 - x^8/8
ln(1+ix)      = ix + x^2/2 -ix^3/3 - x^4/4 +ix^5/5 + x^6/6 +ix^7/7 + x^8/8
arctan(x)     =  x         - x^3/3         + x^5/5         + x^7/7
ln(1+x^2)/2   =    + x^2/2         - x^4/4         + x^6/6         + x^8/8


-ln(1-x)      =  x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8
-ln(1-ix)     = ix - x^2/2 -ix^3/3 + x^4/4 +ix^5/5 - x^6/6 -ix^7/7 + x^8/8
-ln(1+x^2)/2  =    - x^2/2         + x^4/4         - x^6/6         + x^8/8
arctan(x)     =  x         - x^3/3         + x^5/5         - x^7/7


-ln(1-ix) = i*arctan(x) - ln(1+x^2)/2 
(ln(1+ix) - ln(1-ix))/2 = i * arctan(x)



sinh(x) =      x          + x^3/3!          + x^5/5!          + x^7/7! + ...
cosh(x) =  1     + x^2/2!          + x^4/4!          + x^6/6! + ...

exp(x) =   1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + ...

exp(i*x) = 1 +ix - x^2/2! -ix^3/3! + x^4/4! +ix^5/5! - x^6/6! -ix^7/7! + ...

sin(x) =       x          - x^3/3!          + x^5/5!          - x^7/7! + ...
cos(x) =   1     - x^2/2!          + x^4/4!          - x^6/6! + ...

exp(i*x) = cos(x)+i*sin(x)
exp(x) = cosh(x) + sinh(x)

c      ->   0
x^n    ->  n*x^(n-1)
exp(x) -> exp(x)
ln(x)  -> 1/x
sin(x) -> cos(x)
cos(x) -> -sin(x)
tan(x) -> 1/cos^2(x)
arcsin(x) -> 1/sqrt(1-x^2)
arccos(x) -> -1/sqrt(1-x^2)
arctan(x) -> 1/(1+x^2)
ln(x+sqrt(x^2+a^2)) -> 1/sqrt(x^2+a^2)



(a*f(x)+b*g(x))' = a*f'(x) + b*g'(x)

(f(x)*g(x))' = f'(x) * g(x) + f(x) * g'(x)

(f(x)/g(x))' = ( f'(x) * g(x) - f(x) * g'(x) ) / g^2(x)

f(g(x)) ' = f'(g(x))*g'(x)