\Part[
    \eng{Calculus}
    \ger{Analysis}
    ]{cal}

    % \begin{formula}{shark}
    %     \desc{Shark-midnight formula}{\emoji{shark}-s}{}
    %     \desc[german]{Shark-Mitternachtformel}{}{}
    %     \eq{
    %         \temoji{seal}_{1,2} = \frac{-\temoji{shark}\pm \sqrt{\temoji{shark}^2-4\temoji{octopus}\temoji{tropical-fish}}}{2\temoji{octopus}}
    %     }
    % \end{formula}

    \Subsection[
        \eng{Convolution}
        \ger{Faltung / Konvolution}
        ]{conv}
        \begin{ttext}
            \eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.}
            \ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}}
        \end{ttext}
        \begin{formula}{def}
            \desc{Definition}{}{}
            \desc[german]{Definition}{}{}
            \eq{(f*g)(t) = f(t) * g(t) = \int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau}
        \end{formula}
        \begin{formula}{notation}
            \desc{Notation}{}{}
            \desc[german]{Notation}{}{}
            \eq{
                f(t) * g(t-t_0) &= (f*g)(t-t_0) \\
                f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0)
            }
        \end{formula}
        \begin{formula}{commutativity}
            \desc{Commutativity}{}{}
            \desc[german]{Kommutativität}{}{}
            \eq{f * g = g * f}
        \end{formula}

        \begin{formula}{associativity}
            \desc{Associativity}{}{}
            \desc[german]{Assoziativität]}{}{}
            \eq{(f*g)*h = f*(g*h)}
        \end{formula}

        \begin{formula}{distributivity}
            \desc{Distributivity}{}{}
            \desc[german]{Distributivität}{}{}
            \eq{f * (g + h) = f*g + f*h}
        \end{formula}

        \begin{formula}{complex_conjugate}
            \desc{Complex conjugate}{}{}
            \desc[german]{Komplexe konjugation}{}{}
            \eq{(f*g)^* = f^* * g^*}
        \end{formula}
        
    \Subsection[
        \eng{Fourier analysis}
        \ger{Fourieranalyse}
        ]{fourier}
        \Subsubsection[
            \eng{Fourier series}
            \ger{Fourierreihe}
            ]{series}
            \begin{formula}{series}
                \desc{Fourier series}{Complex representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
                \desc[german]{Fourierreihe}{Komplexe Darstellung}{}
                \eq{f(t) = \sum_{k=-\infty}^{\infty} c_k \Exp{\frac{2\pi \I kt}{T}}}
            \end{formula}
            \Eng[real]{real}
            \Ger[real]{reellwertig}
            \begin{formula}{coefficient-complex}
                \desc{Fourier coefficients}{Complex representation}{}
                \desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{}
                \eq{
                    c_k &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Exp{-\frac{2\pi \I}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
                    c_{-k} &= \overline{c_k} \quad \text{\GT{if} $f$ \GT{real}}
                }
            \end{formula}

            \begin{formula}{series_sincos}
                \desc{Fourier series}{Sine and cosine representation}{$f\in \Lebesgue^2(\R,\C)$ $T$-\GT{periodic}}
                \desc[german]{Fourierreihe}{Sinus und Kosinus Darstellung}{}
                \eq{f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} \left(a_k \Cos{\frac{2\pi}{T}kt} + b_k\Sin{\frac{2\pi}{T}kt}\right)}
            \end{formula}
            \begin{formula}{coefficient}
                \desc{Fourier coefficients}{Sine and cosine representation\\If $f$ has point symmetry: $a_{k>0}=0$, if $f$ has axial symmetry: $b_k=0$}{}
                \desc[german]{Fourierkoeffizienten}{Sinus und Kosinus Darstellung\\Wenn $f$ punktsymmetrisch: $a_{k>0}=0$, wenn $f$ achsensymmetrisch: $b_k=0$}{}
                \eq{
                    a_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Cos{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge0\\
                    b_k &= \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t)\,\Sin{-\frac{2\pi}{T}kt}\d t \quad\text{\GT{for}}\,k\ge1\\
                    a_k &= c_k + c_{-k}      \quad\text{\GT{for}}\,k\ge0\\
                    b_k &= \I(c_k - c_{-k})  \quad\text{\GT{for}}\,k\ge1
                }
            \end{formula}
        \TODO{cleanup}
        


        \Subsubsection[
            \eng{Fourier transformation}
            \ger{Fouriertransformation}
            ]{trafo}
            \begin{formula}{transform}
                \desc{Fourier transform}{}{$\hat{f}:\R^n \mapsto \C$, $\forall f\in L^1(\R^n)$}
                \desc[german]{Fouriertransformierte}{}{}
                \eq{\hat{f}(k) \coloneq \frac{1}{\sqrt{2\pi}^n} \int_{\R^n} \e^{-\I kx}f(x)\d x}
            \end{formula}

            \Eng[linear_in]{linear in}
            \Ger[linear_in]{linear in}
            \GT{for} $f\in L^1(\R^n)$:
            \begin{enumerate}[i)]
                \item $f \mapsto \hat{f}$ \GT{linear_in} $f$
                \item $g(x) = f(x-h) \qRarrow \hat{g}(k) = \e^{-\I kn}\hat{f}(k)$
                \item $g(x) = \e^{ih\cdot x}f(x) \qRarrow \hat{g}(k) = \hat{f}(k-h)$
                \item $g(\lambda) = f\left(\frac{x}{\lambda}\right) \qRarrow \hat{g}(k)\lambda^n \hat{f}(\lambda k)$
            \end{enumerate}


    \Subsection[
        \eng{Misc}
        \ger{Verschiedenes}
        ]{misc}
        
        \begin{formula}{stirling-approx}
            \desc{Stirling approximation}{}{}
            \desc[german]{Stirlingformel}{}{}
            \eq{\ln (N!) \approx N \ln(N) - N + \Order(\ln(N))}
        \end{formula}

        \begin{formula}{error-function}
            \desc{Error function}{$\erf: \C \to \C$ and complementary error function $\erfc$}{}
            \desc[german]{Fehlerfunktion}{$\erf: \C \to \C$ und komplementäre Fehlerfunktion $\erfc$}{}
            \eq{
                \erf(x)  &= \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \d t \\
                \erfc(x) &= 1 - \erf(x)\\
                         &= \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \d t
            }
        \end{formula}

        \begin{formula}{delta_of_function}
            \desc{Dirac-Delta of a function}{}{$g(x_0) = 0$}
            \desc[german]{Dirac-Delta einer Funktion}{}{}
            \eq{\delta(f(x)) = \frac{\delta(x-x_0)}{\abs{g'(x_0)}}}
        \end{formula}


\Section[
    \eng{Logarithm}
    \ger{Logarithmus}
]{log}
    \begin{formula}{identities}
        \desc{Logarithm identities}{}{}
        \desc[german]{Logarithmus Identitäten}{Logarithmus Rechenregeln}{}
        \eq{
            \log(xy) &= \log(x) + \log(y) \\
            \log \left(\frac{x}{y}\right) &= \log(x) - \log(y) \\
            \log \left(x^d\right) &= d\log(x) \\
            \log \left(\sqrt[y]{x}\right) &= \frac{\log(x)}{y} \\
            x^{\log(y)} &= y^{\log(x)}
        }
    \end{formula}
    \begin{formula}{intergral}
        \desc{Integral of natural logarithm}{}{}
        \desc[german]{Integral des natürluchen Logarithmus}{}{}
        \eq{
            \int \ln(x) \d x &= x \left(\ln(x) -1\right) \\
            \int \ln(ax + b) \d x &= \frac{ax+b}{a} \left(\ln(ax + b) -1\right)
        }
    \end{formula}

\Section[
    \eng{List of common integrals}
    \ger{Liste nützlicher Integrale}
    ]{integrals}

    % Put links to other integrals here
    \fqEqRef{cal:log:integral}

    \begin{formula}{spherical-coordinates}
        \desc{Spherical coordinates}{}{}
        \desc[german]{Kugelkoordinaten}{}{}
        \eq{
            x &= r \sin\phi,\cos\theta \\
            y &= r \cos\phi,\cos\theta \\
            z &= r \sin\theta
        }
    \end{formula}
    \begin{formula}{spheical-coordinates-int}
        \desc{Integration in spherical coordinates}{}{}
        \desc[german]{Integration in Kugelkoordinaten}{}{}
        \eq{\iiint\d x \d y \d z= \int_0^{\infty} \!\! \int_0^{2\pi} \!\! \int_0^\pi \d r \d\phi\d\theta \, r^2\sin\theta}
    \end{formula}

    \begin{formula}{riemann_zeta}
        \desc{Riemann Zeta Function}{}{}
        \desc[german]{Riemannsche Zeta-Funktion}{}{}
        \eq{\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{(1-2^{(1-s)})\Gamma(s)} \int_0^\infty \d\eta \frac{\eta^{(s-1)}}{\e^\eta + 1}}
    \end{formula}




