Mathematic definitions and Formulas

António Rafael C. Paiva

Date: December 13, 2004

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Contents

• Arithmetic progression
• Geometric progression
• Counting formulas (for probabilities)
• Probabilities
• Newton expansion
• Limits
• Continuity
• Derivates
• Trigonometrics
• Experimental statistics

Arithmetic progression

Definition:

$$u_n = u_1 + (n-1) \times r$$

Sum of $n$ elements of the progression:

$$S_n = \frac{u_1+u_n}{2} \times n$$

Geometric progression

Definition:

$$u_n = u_1 \times r^{n-1}$$

Sum of $n$ elements of the progression:

$$S_n = u_1 \times \frac{1-r^n}{1-r}$$

Sum of all the elements:

$$S = \lim_{n\rightarrow\infty} S_n = \frac{u_1}{1-r},\$$   $$\mbox{if $\vert r\vert<1$}$$$$$$

Counting formulas (for probabilities)

Factorial:

$$n!= n(n-1)(n-2)\cdots1$$

Ordered combinations without repetitions:

$${}^nA_p = \frac{n!}{(n-p)!}$$

Ordered combinations with repetitions:

$${}^n{A'}_p = n^p$$

Unordered combinations without repetitions:

$${}^nC_p = \frac{n!}{p!(n-p)!}$$

Some properties of unordered combinations without repetitions:

• ${}^nC_0 = 1$
• ${}^nC_n = 1$
• ${}^nC_1 = n$
• ${}^nC_p = {}^nC_{n-p}$
• ${}^nC_p + {}^nC_{p+1} = {}^{n+1}C_{p+1}$

Probabilities

Some properties:

$$p(\emptyset) = 0$$

$$p(\overline{A}) = 1 - p(A)$$

$$p(A \cup B) = p(A) + p(B) - p(A \cap B)$$

If two events are incompatibles or dijoint:

$$p(A \cap B) = 0$$

If two events are independent:

$$p(A \cap B) = p(A)p(B)$$

Conditional probability:

$$p(B/A) = \frac{p(A \cap B)}{p(A)}$$

Binomial probability law:

$$P = C_k^n p^k (1-p)^{n-k}$$

Newton expansion

General formula:

$$(a+b)^n = {}^nC_0 a^n + {}^nC_1 a^{n-1} b + {}^nC_2 a^{n-1} b^2 + % \cdots + {}^nC_{n-1} a b^{n-1} + {}^nC_n b^n$$

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

$$(a+b)^2 = a^2 + 2ab + b^2$$

$$(a-b)^2 = a^2 - 2ab + b^2$$

$$(a+b)(a-b) = a^2 - b^2$$

Limits

Definition:

$$\exists \lim_{x \rightarrow a} f(x) = b \quad \Leftrightarrow \quad \lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = b$$

Some trigonometric limits:

• $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$
• $\lim_{x \rightarrow 0} \frac{x}{\sin x} = 1$
• $\lim_{x \rightarrow 0} \frac{1 - \cos x}{x} = 0$
• $\lim_{x \rightarrow 0} \frac{\tan x}{x} = 1$
• $\lim_{x \rightarrow 0} \frac{x}{\tan x} = 1$

The Neper number related limits:

• $\lim \left( 1 + \frac{1}{n} \right)^n = e$
• $\lim \left( 1 + \frac{k}{n} \right)^n = e^k$
• $\lim \left( 1 - \frac{k}{n} \right)^n = e^{-k}$
• $\lim \left( 1 + \frac{x}{a_n} \right)^{a_n} = e^x,$   if $a_n \rightarrow \pm \infty$ and $x \in \mathbb{R}$$$

and the following theorems

• Theorem 1:
  If $\lim a_n = 1$ and $\lim b_n = +\infty$ then, $\lim a_n^{b_n} = e^{\lim b_n (a_n - 1)}$.
• Theorem 2:
  If $g(n) > 0$ for all $n \in \mathbb{N}$ then, $\lim g(n)^{f(n)} = \lim g(n)^{\lim f(n)}$.

Continuity

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

$$\exists \lim_{x \rightarrow a} f(x) = f(a)$$

Derivates

Definition:

$$f'(x_0) = \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0},\quad x \in D$$

where $f'(x_0)$ is the slope of a line tangent to function $f$, at $x_0$.

A function is said to have derivable at a point $x_0$ if it has finite derivate at that point, which happens if the side derivates are equal. In that case

$$f'(x_0^-) = f'(x_0^+) = f'(x_0).$$

Properties of the derivatives:

• $(f(x) \pm g(x))' = f'(x) \pm g'(x)$
• $(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$
• $\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$
• $(f\circ g)(x) = f(g(x)) = f'(g(x)) g'(x)$

Derivates of some trigonometric functions:

• $(\sin u)' = u' \cos u$
• $(\cos u)' = -u' \sin u$
• $(\tan u)' = \frac{u'}{\cos^2 u}$

Trigonometrics

The fundamental equation of trigonometrics:

$$\sin^2(x) + \cos^2(x) = 1$$

Sine and cosine as linear combinations of complex exponentials:

$$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \quad,% \quad \cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$

Useful formulas:

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

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

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

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

$$\sin(a) - \sin(b) = 2\sin\left(\frac{a-b}{2}\right) \cos\left(\frac{a+b}{2}\right)$$

$$\sin(a) + \sin(b) = 2\sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)$$

$$\cos(a) + \cos(b) = 2\cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)$$

$$\cos(a) - \cos(b) = -2\sin\left(\frac{a-b}{2}\right) \cos\left(\frac{a+b}{2}\right)$$

$$\sin(2x) = 2\sin(x)\cos(x)$$

$$\cos(2x) = 2\cos^2(x) - 1$$

Experimental statistics

Variance of some data around the mean:

$$\sigma^2 = \frac{n\sum x_i^2 - (\sum x_i)^2}{(n-1)n^2}$$

Principle of the error propagation:

$$\Delta z = \sqrt{ \left(\frac{\partial z}{\partial x}\right)^2\!\Delta x^2 + \left(\frac{\partial z}{\partial y}\right)^2\!\Delta y^2 + \cdots}$$

Linear regression ($y=mx+b$):

$$m = \frac{n\sum x_i y_i - (\sum x_i)(\sum y_i)}% {n\sum x_i^2 - (\sum x_i)^2}$$

$$b = \frac{(\sum x_i^2)(\sum y_i) - (\sum x_i)(\sum x_i y_i)}% {n\sum x_i^2 - (\sum x_i)^2}$$

$$\Delta m = \sqrt{\frac{n\sum y_i^2 - (\sum y_i)^2 - \frac{\left(... ...n\sum x_i^2 - (\sum x_i)^2}}% {(n-2)\left(n\sum x_i^2 - (\sum x_i)^2\right)}}$$

$$\Delta b = \sqrt{\frac{\sum x_i^2}{n}}\:\Delta m$$

$$r = \frac{n\sum x_i y_i - (\sum x_i)(\sum y_i)} {\sqrt{\left(n\sum x_i^2 - (\sum x_i)^2\right)% \left(n\sum y_i^2 - (\sum y_i)^2\right)}}$$

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Mathematic definitions and Formulas

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