[Analysis] Add some notes

semester2/analysis-i/cheat-sheet-jh/parts/fields.tex

@@ -102,6 +102,8 @@ $\overline{a + bi} := a-bi$ (\tr{complex conjugate}{Komplexe Konjugation});
 Transformation polar $\rightarrow$ normal: $r \cdot \cos(\phi) + r \cdot \sin(\phi)i$.
 Transformation normal $\rightarrow$ polar: $|z| \cdot e^{i \cdot \arcsin(\frac{b}{|z|})}$;
 
+\textbf{\tr{Square root of negative number}{Quadratwurzel einer negativen Zahl}}: $\sqrt{-c} = ci$
+
 
 \begin{theorem}[]{\tr{Fundamental Theorem of Algebra}{Fundamentalsatz der Algebra}}
     \trLet $n \geq 1, n \in \N$ \tr{and let}{und sei}

semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/01_order-one.tex

@@ -1,12 +1,12 @@
 \newsectionNoPB
 \subsection{Linear differential equations of first order}
 \rmvspace
-\shortproposition Solution of $y' + ay = 0$ is of form $f(x) = z e^{-A(x)}$ with $A$ anti-derivative of $a$
+\shortproposition Solution of $y' + ay = 0$ is of form $f(x) = C e^{-A(x)}$ with $A$ anti-derivative of $a$
 
 \rmvspace
 \shade{gray}{Imhomogeneous equation} 
 \rmvspace
 \begin{enumerate}[noitemsep]
     \item Plug all values into $y_p = \int b(x) e^{A(x)}$ ($A(x)$ in the exponent instead of $-A(x)$ as in the homogeneous solution)
-    \item Solve and the final $y(x) = y_h + y_p$. For initial value problem, determine coefficient $z$
+    \item Solve and the final $y(x) = y_h + y_p$. For initial value problem, determine coefficient $C$
 \end{enumerate}

semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex

@@ -5,11 +5,14 @@ The coefficients $a_i$ are constant functions of form $a_i(x) = k$ with $k$ cons
 \shade{gray}{Homogeneous Equation}\rmvspace
 \begin{enumerate}[noitemsep]
     \item Find \bi{characteristic polynomial} (of form $\lambda^k + a_{k - 1} \lambda^{k - 1} + \ldots + a_1 \lambda + a_0$ for order $k$ lin. ODE with coefficients $a_i \in \R$).
-    \item Find the roots of polynomial. The solution space is given by $\{ z_j \cdot x^{v_j - 1} e^{\gamma_i x} \divides v_j \in \N, \gamma_i \in \R \}$ where $v_j$ is the multiplicity of the root $\gamma_i$.
+    \item Find the roots of polynomial. The solution space is given by $\{ C_j \cdot x^{v_j - 1} e^{\gamma_i x} \divides v_j \in \N, \gamma_i \in \R \}$
-          For $\gamma_i = \alpha + \beta i \in \C$, we have $z_1 \cdot e^{\alpha x}\cos(\beta x)$, $z_2 \cdot e^{\alpha x}\sin(\beta x)$, representing the two complex conjugated solutions.
+          where $v_j$ is the multiplicity of the root $\gamma_i$ and $C_j$ is a constant.
+          For $\gamma_i = \alpha + \beta i \in \C$, we have $C_1 \cdot e^{\alpha x}\cos(\beta x)$, $C_2 \cdot e^{\alpha x}\sin(\beta x)$,
+          representing the two complex conjugated solutions.
 \end{enumerate}
-
 \rmvspace
+
+The homogeneous equation will then be all the elements of the set summed up.\\
 \shade{gray}{Inhomogeneous Equation}\rmvspace
 \begin{enumerate}[noitemsep]
     \item \bi{(Case 1)} $b(x) = c x^d e^{\alpha x}$, with special cases $x^d$ and $e^{\alpha x}$: