Mathematic definitions and Formulas

António Rafael C. Paiva

This document is also available in pdf.
May 16, 2009

Contents

  1. Derivatives
  2. Trigonometrics
  3. Continuity
  4. Probabilities
  5. Counting formulas (for probabilities)
  6. Newton expansion
  7. Limits
  8. Arithmetic progressions
  9. Geometric progressions
  10. Statistics for experimental data analysis

1 Derivatives

Definition:

f′(x0) =  lim  f(x)--f-(x0-),  x ∈ D
        x→x0    x- x0

where f′(x0) is the slope of the line tangent to f  at x0   .

A function is said to have derivable at a point x0   if it has finite derivate at that point, which happens if the side derivates are equal. In that case

 ′ -      ′ +     ′
f(x0 ) = f (x0 ) = f (x0).

Properties of the derivatives:

Derivates of some trigonometric functions:

2 Trigonometrics

The fundamental equation of trigonometrics:

  2        2
sin (x) + cos(x) = 1

Sine and cosine as linear combinations of complex exponentials:

         eix - e-ix             eix + e-ix
sin(x) = ---2i----  ,  cos(x ) =----2----

Useful formulas:

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

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

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

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

                    (     )    (      )
sin(a)- sin(b) = 2 sin a---b  cos  a+--b
                       2           2

                    (a + b)    ( a-  b)
sin(a)+ sin(b) = 2 sin --2--  cos  --2--

                     (     )     (     )
cos(a)+ cos(b) = 2 cos a+-b- cos  a---b
                        2           2

                      (     )    (      )
                       a---b       a-+-b
cos(a) - cos(b) = - 2 sin   2    cos    2

sin(2x) = 2sin (x )cos(x )

cos(2x) = 2 cos2(x) - 1

3 Continuity

A function f(x)  is said to be continous at some point a  iff :

∃ lim  f(x) = f (a)
  x→a

4 Probabilities

Some properties:

p(∅) = 0

p(A) = 1 - p(A )

p(A ∪ B) = p(A) + p(B) - p(A ∩B )

If two events are incompatibles or dijoint:

p(A ∩ B ) = 0

If two events are independent:

p(A  ∩B ) = p(A )p(B)

Conditional probability:

          p(A ∩ B )
p(B ∕A) = ---------
            p(A )

Binomial probability law:

P =  Cnkpk(1 - p)n-k

5 Counting formulas (for probabilities)

Factorial:

n! = n(n- 1)(n - 2)⋅⋅⋅1

Ordered combinations without repetitions:

n         n!
  Ap = (n---p)!

Ordered combinations with repetitions:

nA ′p = np

Unordered combinations without repetitions:

          n!
nCp = ---------
      p!(n- p )!

Some properties of unordered combinations without repetitions:

6 Newton expansion

General formula:

    n   n   n   n   n- 1   n    n-1 2       n       n-1   n   n
(a+ b)  =  C0a  +  C1a    b+  C2a    b + ⋅⋅⋅+  Cn -1ab   +   Cnb

Some useful formulas from the particular case, n = 2  :

(a + b)2 = a2 + 2ab+ b2

(a - b)2 = a2 - 2ab+ b2

(a + b)(a- b) = a2 - b2

7 Limits

Definition:

∃ lim f (x ) = b  ⇔    lim  f (x ) = lim  f(x) = b
  x→a                x→a -       x→a+

Some trigonometric limits:

The Neper number related limits:

and the following theorems

8 Arithmetic progressions

Definition:

u =  u + (n - 1)× r
 n    1

Sum of n elements of the progression:

Sn =  u1 +-un-× n
         2

9 Geometric progressions

Definition:

un =  u1 × rn-1

Sum of n elements of the progression:

          1 - rn
Sn =  u1 ×------
           1 - r

Sum of all the elements:

S =  lim  Sn = --u1-, if |r| < 1
    n→ ∞      1 - r

10 Statistics for experimental data analysis

Variance of some data around the mean:

       ∑       ∑
σ2 = n---x2i---(--xi)2
         (n - 1)n2

Principle of the error propagation:

     ∘  (---)2-------(---)2-----------
Δz =     ∂z-   Δx2 +   ∂z-  Δy2 +  ⋅⋅⋅
         ∂x            ∂y

Linear regression (y = mx  + b  ):

      ∑         ∑     ∑
m =  n---x∑iyi -2-(-∑xi)(-yi)-
        n   xi - (  xi)2

     ∑     ∑        ∑     ∑
    (--x2i)(---yi)---(--xi)(--xiyi)
b =       n ∑  x2- (∑  xi)2
                i

       ┌│ --∑-------∑---------∑------∑----∑---2-
       │ n   y2i - (  yi)2 - (n-xi∑yi-(2--x∑i)(-2yi))
Δm  =  ∘ -------------(--∑---2-n-∑xi-(-)xi)-----
               (n - 2) n   x i - ( xi)2

      ∘ ------
        ∑  x2
Δb  =   ----iΔm
          n

             ∑        ∑     ∑
r = ∘------n---xiyi --(--xi)(---yi)--------
      (n∑  x2- (∑  x )2) (n∑  y2- (∑  y)2)
            i       i         i       i