\Section[
    \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{Fourier analysis}
        \ger{Fourieranalyse}
    ]{fourier}
        \Subsubsection[
            \eng{Fourier series}
            \ger{Fourierreihe}
        ]{series}
        \begin{formula}{series} \absLabel[fourier_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} \absLabel[fourier_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}


        \Subsubsection[
            \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{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}{}{$f(x_i) = 0$}
            \desc[german]{Dirac-Delta einer Funktion}{}{}
            \eq{\delta(f(x)) = \sum_i \frac{\delta(x-x_i)}{\abs{f^\prime(x_i)}}}
        \end{formula}

        \begin{formula}{geometric_series}
            \desc{Geometric series}{}{$\abs{q}<1$}
            \desc[german]{Geometrische Reihe}{}{}
            \eq{\sum_{k=0}^{\infty}q^k = \frac{1}{1-q}}
        \end{formula}


    \Subsection[
        \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}

    \Subsection[
        \eng{Vector calculus}
        \ger{Vektor Analysis}
    ]{vec}
        \begin{formula}{laplace}
            \desc{Laplace operator}{}{}
            \desc[german]{Laplace-Operator}{}{}
            \eq{\laplace = \Grad^2 = \pdv[2]{}{x} + \pdv[2]{}{y} + \pdv[2]{}{z}}
        \end{formula}
        \Subsubsection[
            \eng{Spherical symmetry}
            \ger{Kugelsymmetrie}
        ]{sphere}
            \begin{formula}{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}{laplace}
                \desc{Laplace operator}{}{}
                \desc[german]{Laplace-Operator}{}{}
                \eq{\Grad^2 = \laplace = \frac{1}{r^2} \pdv{}{r} \left(r^2 \pdv{}{r}\right)}
            \end{formula}

            \begin{formula}{p-norm}
                \desc{$p$-norm}{}{}
                \desc[german]{$p$-Norm}{}{}
                \eq{\norm{\vecx}_p \equiv \left(\sum_{i=1}^{n} \abs{x_i}^p\right)^\frac{1}{p}}
            \end{formula}


    \Subsection[
        \eng{Integrals}
        \ger{Integralrechnung}
    ]{integral}
        \begin{formula}{partial}
            \desc{Partial integration}{}{}
            \desc[german]{Partielle integration}{}{}
            \eq{
                \int_a^b f^\prime(x)\cdot g(x) \d x= \left[f(x)\cdot g(x)\right]_a^b - \int_a^b f(x)\cdot g^\prime(x) \d x
            }
        \end{formula}

        \begin{formula}{substitution}
            \desc{Integration by substitution}{}{}
            \desc[german]{Integration durch Substitution}{}{}
            \eq{
                \int_a^b f(g(x))\,g^\prime(x) \d x = \int_{g(a)}^{g(b)} f(z) \d z
            }
        \end{formula}

        \begin{formula}{gauss}
            \desc{Gauss's theorem / Divergence theorem}{Divergence in a volume equals the flux through the surface}{$A = \partial V$}
            \desc[german]{Satz von Gauss}{Divergenz in einem Volumen ist gleich dem Fluss durch die Oberfläche}{}
            \eq{
                \iiint_V \Div{\vec{F}} \d V = \oiint_A \vec{F} \cdot \d\vec{A}
            }
        \end{formula}

        \begin{formula}{stokes}
            \desc{Stokes's theorem}{}{$S = \partial A$}
            \desc[german]{Klassischer Satz von Stokes}{}{}
            \eq{\int_A (\Rot{\vec{F}}) \cdot \d\vec{S} = \oint_{S} \vec{F} \cdot \d \vec{r}}
        \end{formula}
        \Subsubsection[
            \eng{List of common integrals}
            \ger{Liste nützlicher Integrale}
        ]{list}
            % Put links to other integrals here
            \fqEqRef{cal:log:integral}

            \begin{formula}{arcfunctions}
                \desc{Arcsine, arccosine, arctangent}{}{}
                \desc[german]{Arkussinus, Arkuskosinus, Arkustangens}{}{}
                \eq{
                    \int \frac{1}{\sqrt{1-x^2}} \d x  = \arcsin x \\
                    \int -\frac{1}{\sqrt{1-x^2}} \d x = \arccos x \\
                    \int \frac{1}{x^2+1} \d x         = \arctan x
                }
            \end{formula}
            \begin{formula}{archyperbolicfunctions}
                \desc{Arcsinh, arccosh, arctanh}{}{}
                % \desc[german]{Arkussinus, Arkuskosinus, Arkustangens}{}{}
                \eq{
                    \int \frac{1}{\sqrt{x^2+1}} \d x &= \arsinh x \\
                    \int \frac{1}{\sqrt{x^2-1}} \d x &= \arcosh x \quad\eqnote{\GT{for} $(x > 1)$}\\
                    \int \frac{1}{1-x^2} \d x        &= \artanh x \quad\eqnote{\GT{for} $(\abs{x} < 1)$}\\
                    \int \frac{1}{1-x^2} \d x        &= \arcoth x \quad\eqnote{\GT{for} $(\abs{x} > 1)$}
                }
            \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}

            \begin{formula}{gamma_function}
                \desc{Gamma function}{}{}
                \desc[german]{Gamma-Funktion}{}{}
                \eq{
                    \Gamma(n) &= (n-1)! \\
                    \Gamma(z) &= \int_0^\infty t^{z-1} \e^{-t} \d t \\
                    \Gamma(z+1) &= z\Gamma(z)
                }
            \end{formula}
            \begin{formula}{upper_incomplete_gamma_function}
                \desc{Upper incomplete gamma function}{}{}
                \desc[german]{Unvollständige Gamma-Funktion der unteren Grenze}{}{}
                \eq{\Gamma(s,x) = \int_x-^\infty t^{s-1}\e^{-t} \d t}
            \end{formula}
            \begin{formula}{lower_incomplete_gamma_function}
                \desc{Lower incomplete gamma function}{}{}
                \desc[german]{Unvollständige Gamma-Funktion der oberen Grenze}{}{}
                \eq{\gamma(s,x) = \int_0^x t^{s-1}\e^{-t} \d t}
            \end{formula}

            \begin{formula}{beta_function}
                \desc{Beta function}{Complete beta function}{}
                \desc[german]{Beta-Funktion}{}{}
                \eq{
                    \txB(z_1,z_2) &= \int_0^1 t^{z_1-1} (1-t)^{z_2-1} \d t \\
                    \txB(z_1, z_2) &= \frac{\Gamma(z_1) \Gamma(z_2)}{\Gamma(z_1+z_2)}
                }
            \end{formula}
            \begin{formula}{incomplete_beta_function}
                \desc{Incomplete beta function}{Complete beta function}{}
                \desc[german]{Unvollständige Beta-Funktion}{}{}
                \eq{\txB(x; z_1,z_2) = \int_0^x t^{z_1-1} (1-t)^{z_2-1} \d t}
            \end{formula}


            \TODO{differential equation solutions}
