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