Kevin Kreutz

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: \(\exp(x) = e^x\)

  1. \[e^0 = 1\]
  2. \[\forall x,y \in \mathbb{R}: e^{x+y} = e^x \cdot e^y\]
  3. \(\forall x \in \mathbb{R}: e^x \neq 0\) and \(e^{-x} = \frac{1}{e^x}\)
  4. \[\forall x \in \mathbb{R}: e^x > 0\]
  5. $\exp$ is strictly monotone increasing
  6. \(\exp: \mathbb{R} \rightarrow (0,\infty)\) is bijective
  7. \[(e^x)' = e^x\]

Logarithmic function

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

  1. \[e^{\ln(x)} = x \text{ for all } x > 0\]
  2. \[\ln(e^x) = x \text{ for all } x \in \mathbb{R}\]
  3. \[\ln(x \cdot y) = \ln(x) + \ln(y)\]
  4. \[\ln(x/y) = \ln(x) - \ln(y)\]
  5. \[\ln(x^a) = a \cdot \ln(x)\]
  6. \[\ln(1) = 0, \quad \ln(e) = 1\]
  7. \[(\ln(x))' = \frac{1}{x}\]

Trigonometric functions

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

  1. \(\sin^2(x) + \cos^2(x) = 1\) (Pythagorean identity)
  2. \[\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)\]
  3. \[\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)\]
  4. \[(\sin(x))' = \cos(x)\]
  5. \[(\cos(x))' = -\sin(x)\]
  6. 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:

  1. If $\lim a_n = L$ and $\lim b_n = M$, then $\lim (a_n + b_n) = L + M$
  2. $\lim (a_n \cdot b_n) = L \cdot M$
  3. If $M \neq 0$: $\lim \frac{a_n}{b_n} = \frac{L}{M}$
  4. 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:

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

  1. Divergence Test: If $\lim a_n \neq 0$, then $\sum a_n$ diverges
  2. Comparison Test: If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges
  3. 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
  4. 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:

  1. Sums, products, and quotients of continuous functions are continuous
  2. Composition of continuous functions is continuous
  3. 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 $f(c) = y$
  4. 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

  1. Sum Rule: $(f + g)’ = f’ + g’$
  2. Product Rule: $(f \cdot g)’ = f’ \cdot g + f \cdot g’$
  3. Quotient Rule: $\left(\frac{f}{g}\right)’ = \frac{f’ \cdot g - f \cdot g’}{g^2}$
  4. Chain Rule: $(f \circ g)’(x) = f’(g(x)) \cdot g’(x)$
  5. 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

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

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

Standard Antiderivatives

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