Complex Numbers

$$ \mathbb{C} = {x + y \cdot i | x,y \in \mathbb{R}} $$ Set of complex numbers
$$ i^2 = -1 $$

Euler’s Formula

$$ e^{i\theta} = \cos(\theta) + i \cdot \sin(\theta)$$
Euler’s identity: $$ e^{i\pi} = -1 $$

Elementary Functions

Exponential function

$$ \exp : \mathbb{R} \rightarrow \mathbb{R}^+ $$
We write: $$ \exp(x) = e^x $$

$$ e^0 = 1 $$
$$ \forall x,y \in \mathbb{R}: e^{x+y} = e^x \cdot e^y $$
$$ \forall x \in \mathbb{R}: e^x \neq 0 $$ and $$ e^{-x} = \frac{1}{e^x} $$
$$ \forall x \in \mathbb{R}: e^x > 0 $$
$\exp$ is strictly monotone increasing
$$ \exp: \mathbb{R} \rightarrow (0,\infty) $$ is bijective
$$ (e^x)’ = e^x $$

Logarithmic function

$$ \ln : (0,\infty) \rightarrow \mathbb{R} $$
The natural logarithm is the inverse function of the exponential:

$$ e^{\ln(x)} = x \text{ for all } x > 0 $$
$$ \ln(e^x) = x \text{ for all } x \in \mathbb{R} $$
$$ \ln(x \cdot y) = \ln(x) + \ln(y) $$
$$ \ln(x/y) = \ln(x) - \ln(y) $$
$$ \ln(x^a) = a \cdot \ln(x) $$
$$ \ln(1) = 0, \quad \ln(e) = 1 $$
$$ (\ln(x))’ = \frac{1}{x} $$

Trigonometric functions

$$ \sin, \cos : \mathbb{R} \rightarrow [-1,1] $$

$$ \sin^2(x) + \cos^2(x) = 1 $$ (Pythagorean identity)
$$ \sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y) $$
$$ \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) $$
$$ (\sin(x))’ = \cos(x) $$
$$ (\cos(x))’ = -\sin(x) $$
Domain: $\mathbb{R}$, Period: $2\pi$

$$ \tan(x) = \frac{\sin(x)}{\cos(x)}, \quad x \neq \frac{\pi}{2} + k\pi, k \in \mathbb{Z} $$

Sequences and Limits

Definition of a Sequence

A sequence is a function $$ a: \mathbb{N} \rightarrow \mathbb{R} $$
Notation: $$ (a_n){n \in \mathbb{N}} \text{ or } (a_n){n=1}^{\infty} $$

Convergence of Sequences

A sequence $(a_n)$ converges to $L \in \mathbb{R}$ if: $$ \forall \epsilon > 0 , \exists N \in \mathbb{N} : \forall n > N : |a_n - L| < \epsilon $$ We write: $$ \lim_{n \to \infty} a_n = L $$

Properties:

If $\lim a_n = L$ and $\lim b_n = M$, then $\lim (a_n + b_n) = L + M$
$\lim (a_n \cdot b_n) = L \cdot M$
If $M \neq 0$: $\lim \frac{a_n}{b_n} = \frac{L}{M}$
Squeeze theorem: If $a_n \leq c_n \leq b_n$ and $\lim a_n = \lim b_n = L$, then $\lim c_n = L$

Monotone Sequences

Monotone Convergence Theorem:

Every monotone increasing sequence that is bounded above converges
Every monotone decreasing sequence that is bounded below converges

Special Important Limits

$$ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e $$ $$ \lim_{n \to \infty} \frac{a^n}{n!} = 0 \text{ for any } a > 0 $$ $$ \lim_{n \to \infty} \sqrt[n]{n} = 1 $$

Series

Definition

An infinite series is: $$ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \ldots $$ The $N$-th partial sum is: $$ S_N = \sum_{n=1}^{N} a_n $$

A series converges if $\lim_{N \to \infty} S_N$ exists and is finite.

Geometric Series

$$ \sum_{n=0}^{\infty} r^n = \frac{1}{1-r} \quad \text{for } |r| < 1 $$

Convergence Tests

Divergence Test: If $\lim a_n \neq 0$, then $\sum a_n$ diverges
Comparison Test: If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges
Ratio Test: Let $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$
- If $L < 1$: series converges absolutely
- If $L > 1$: series diverges
- If $L = 1$: test is inconclusive
Root Test: Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$
- If $L < 1$: series converges absolutely
- If $L > 1$: series diverges

Harmonic Series

$$ \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots \quad \text{DIVERGES} $$

Power Series

$$ \sum_{n=0}^{\infty} c_n(x - a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \ldots $$ has a radius of convergence $R$, and converges for all $x$ with $|x - a| < R$.

Limits and Continuity

Limit of a Function

$$ \lim_{x \to a} f(x) = L $$ if: $$ \forall \epsilon > 0 , \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon $$

Continuity

A function $f$ is continuous at $a$ if: $$ \lim_{x \to a} f(x) = f(a) $$

Properties of continuous functions:

Sums, products, and quotients of continuous functions are continuous
Composition of continuous functions is continuous
Intermediate Value Theorem: If $f$ is continuous on $[a,b]$ and $y$ is between $f(a)$ and $f(b)$, then $\exists c \in (a,b)$ such that $f(c) = y$
Extreme Value Theorem: If $f$ is continuous on $[a,b]$, then $f$ attains its maximum and minimum values on $[a,b]$

Differentiation

Definition of the Derivative

$$ f’(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $$

If this limit exists, we say $f$ is differentiable at $a$.

Differentiation Rules

Sum Rule: $(f + g)’ = f’ + g'$
Product Rule: $(f \cdot g)’ = f’ \cdot g + f \cdot g'$
Quotient Rule: $\left(\frac{f}{g}\right)’ = \frac{f’ \cdot g - f \cdot g’}{g^2}$
Chain Rule: $(f \circ g)’(x) = f’(g(x)) \cdot g’(x)$
Power Rule: $(x^n)’ = n \cdot x^{n-1}$

Mean Value Theorem

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists c \in (a,b)$ such that: $$ f’(c) = \frac{f(b) - f(a)}{b - a} $$

Monotonicity and Local Extrema

If $f’(x) > 0$ on an interval, then $f$ is strictly increasing on that interval
If $f’(x) < 0$ on an interval, then $f$ is strictly decreasing on that interval
If $f’(a) = 0$ and $f’’(a) < 0$, then $a$ is a local maximum
If $f’(a) = 0$ and $f’’(a) > 0$, then $a$ is a local minimum

Integration

Riemann Integral

The Riemann integral of $f$ on $[a,b]$ is: $$ \int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(\xi_i) \Delta x_i $$ where the interval is partitioned and $\Delta x_i \to 0$.

If this limit exists, $f$ is Riemann integrable on $[a,b]$.

Fundamental Theorem of Calculus

Part 1: If $f$ is continuous on $[a,b]$, then: $$ F(x) = \int_a^x f(t) , dt $$ is differentiable and $F’(x) = f(x)$.

Part 2: If $F$ is an antiderivative of $f$ on $[a,b]$, then: $$ \int_a^b f(x) , dx = F(b) - F(a) $$

Integration Rules

Linearity: $\int (af + bg) = a\int f + b\int g$
Integration by parts: $\int u , dv = uv - \int v , du$
Substitution: $\int f(g(x)) \cdot g’(x) , dx = \int f(u) , du$ where $u = g(x)$

Standard Antiderivatives

$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$ $$ \int \frac{1}{x} , dx = \ln|x| + C $$ $$ \int e^x , dx = e^x + C $$ $$ \int \sin(x) , dx = -\cos(x) + C $$ $$ \int \cos(x) , dx = \sin(x) + C $$ $$ \int \frac{1}{1+x^2} , dx = \arctan(x) + C $$