\appendix

\section{Solution Method Summary}
\label{app:solution_summary}

This appendix provides comprehensive quick-reference tables for every solution method covered in the handbook. Use these tables to identify the correct method for a given equation type and to recall key formulas.

\subsection{First-Order ODE Methods}
\label{sec:appA_first_order}

\begin{table}[htbp]
\centering
\caption{First-Order Solution Methods (see \cref{ch:first_order})}
\label{tab:appA_first_order_methods}
\begin{tabular}{p{2.5cm} p{4.5cm} p{4.5cm} p{3.5cm}}
\toprule
\textbf{Method} & \textbf{Equation Form} & \textbf{Solution / Procedure} & \textbf{When to Use} \\
\midrule
Separable &
$\displaystyle \frac{\diff y}{\diff x} = g(x)\,h(y)$ &
$\displaystyle \int \frac{\diff y}{h(y)} = \int g(x)\,\diff x + C$ &
RHS factors as $g(x)\cdot h(y)$ \\[8pt]
Linear &
$\displaystyle y' + p(x)y = g(x)$ &
$y = \dfrac{\displaystyle \int \mu(x)g(x)\,\diff x + C}{\mu(x)}$, \qquad $\mu = \exp\!\bigl(\int p(x)\,\diff x\bigr)$ &
Can be written in standard linear form \\[8pt]
Exact &
$M(x,y)\,\diff x + N(x,y)\,\diff y = 0$, \qquad $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ &
Find $\psi(x,y)$ such that $\dfrac{\partial \psi}{\partial x} = M$, $\dfrac{\partial \psi}{\partial y} = N$; solution is $\psi(x,y) = C$ &
Exactness test $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ passes \\[8pt]
Bernoulli &
$\displaystyle y' + p(x)y = g(x)\,y^n$ &
Substitute $v = y^{1-n}$; equation becomes linear in $v$ &
RHS is a power of $y$ times a function of $x$ \\[8pt]
Homogeneous &
$\displaystyle \frac{\diff y}{\diff x} = F\!\left(\frac{y}{x}\right)$ &
Substitute $y = vx$; equation becomes separable in $v$ &
RHS is a function of the ratio $y/x$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Second-Order Linear Homogeneous (Constant Coefficients)}
\label{sec:appA_second_order_hom}

Consider the equation $a\,y'' + b\,y' + c\,y = 0$ with $a,b,c \in \R$, $a \neq 0$.
\begin{table}[htbp]
\centering
\caption{Characteristic Equation Cases (see \cref{ch:second_order_homogeneous})}
\label{tab:appA_homogeneous_cases}
\begin{tabular}{p{3cm} p{3cm} p{6cm} p{3cm}}
\toprule
\textbf{Discriminant} & \textbf{Roots of } $ar^2+br+c=0$ & \textbf{General Solution} & \textbf{Phase / Behavior} \\
\midrule
$b^2 - 4ac > 0$ \small (distinct real) &
$r_1 = \dfrac{-b+\sqrt{b^2-4ac}}{2a}$, \ $r_2 = \dfrac{-b-\sqrt{b^2-4ac}}{2a}$ &
$y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$ &
Exponential growth/decay; overdamped \\[12pt]
$b^2 - 4ac = 0$ \small (repeated) &
$r = -\dfrac{b}{2a}$ &
$y = c_1 e^{rx} + c_2 x\,e^{rx}$ &
Critically damped \\[12pt]
$b^2 - 4ac < 0$ \small (complex conjugate) &
$r = \alpha \pm i\beta$, \qquad $\alpha = -\dfrac{b}{2a}$, \ $\beta = \dfrac{\sqrt{4ac-b^2}}{2a}$ &
$y = e^{\alpha x}\bigl[c_1\cos(\beta x) + c_2\sin(\beta x)\bigr]$ &
Damped oscillation; underdamped \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{keyresult}
  \textbf{Superposition principle.} For $L[y]=0$ linear homogeneous, any linear combination of solutions is again a solution. A \emph{fundamental set} $\{y_1,y_2\}$ has $W(y_1,y_2)\neq 0$, and the general solution is $y = c_1 y_1 + c_2 y_2$.
\end{keyresult}

\subsection{Second-Order Linear Nonhomogeneous}
\label{sec:appA_second_order_nonhom}

Consider $a\,y'' + b\,y' + c\,y = g(x)$. The general solution is $y = y_h + y_p$, where $y_h$ is the homogeneous solution and $y_p$ is a particular solution.

\begin{table}[htbp]
\centering
\caption{Nonhomogeneous Solution Methods (see \cref{ch:second_order_nonhomogeneous})}
\label{tab:appA_nonhomogeneous_methods}
\begin{tabular}{p{3cm} p{6cm} p{4cm}}
\toprule
\textbf{Method} & \textbf{Formula / Procedure} & \textbf{When to Use} \\
\midrule
Undetermined Coefficients &
See \cref{tab:appA_undetermined_guess} for guess table. Plug $y_p$ into ODE, solve for unknown coefficients. &
$g(x)$ is a polynomial, exponential, sine/cosine, or finite sums/products thereof \\[12pt]
Variation of Parameters &
$\textstyle y_p(x) = -y_1(x)\!\int \frac{y_2(x)\,g(x)}{a\,W(x)}\,\diff x + y_2(x)\!\int \frac{y_1(x)\,g(x)}{a\,W(x)}\,\diff x$

where $\textstyle W = y_1 y_2' - y_2 y_1'$. &
Any $g(x)$ for which the integrals can be evaluated; requires the fundamental set $\{y_1,y_2\}$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{table}[htbp]
\centering
\caption{Guess Table for Undetermined Coefficients (see \cref{ch:second_order_nonhomogeneous})}
\label{tab:appA_undetermined_guess}
\begin{tabular}{p{4.5cm} p{5.5cm}}
\toprule
\textbf{Form of } $g(x)$ & \textbf{Guess for } $y_p(x)$ \\
\midrule
$P_n(x)$ (polynomial of degree $n$) &
$A_n x^n + A_{n-1}x^{n-1} + \cdots + A_1 x + A_0$ \\[8pt]
$A\,e^{kx}$ &
$C\,e^{kx}$ \\[6pt]
$A\sin(\omega x)$ or $A\cos(\omega x)$ &
$C_1\cos(\omega x) + C_2\sin(\omega x)$ \\[6pt]
$A\,e^{kx}\sin(\omega x)$ or $A\,e^{kx}\cos(\omega x)$ &
$e^{kx}\bigl[C_1\cos(\omega x) + C_2\sin(\omega x)\bigr]$ \\[8pt]
Sums of the above &
Sum of the corresponding guesses \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{keyresult}
  \textbf{Modification rule.} If any term of the guess $y_p$ is already a solution of the homogeneous equation (i.e.\ duplicates a term in $y_h$), multiply the \emph{entire} corresponding group by $x$. If $x$-multiplication still duplicates, multiply by $x^2$, and so on.
\end{keyresult}

\subsection{Laplace Transform Summary}
\label{sec:appA_laplace}

\begin{table}[htbp]
\centering
\caption{Common Laplace Transforms (see \cref{ch:laplace_transforms})}
\label{tab:appA_laplace_transforms}
\begin{tabular}{l l}
\toprule
\textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} \\
\midrule
$1$ & $\dfrac{1}{s}$ \\[8pt]
$t^n$ \quad ($n \in \N$) & $\dfrac{n!}{s^{n+1}}$ \\[8pt]
$e^{at}$ & $\dfrac{1}{s-a}$ \\[8pt]
$\sin(\omega t)$ & $\dfrac{\omega}{s^2+\omega^2}$ \\[8pt]
$\cos(\omega t)$ & $\dfrac{s}{s^2+\omega^2}$ \\[8pt]
$\sinh(at)$ & $\dfrac{a}{s^2-a^2}$ \\[8pt]
$\cosh(at)$ & $\dfrac{s}{s^2-a^2}$ \\[8pt]
$e^{at}\sin(\omega t)$ & $\dfrac{\omega}{(s-a)^2+\omega^2}$ \\[8pt]
$e^{at}\cos(\omega t)$ & $\dfrac{s-a}{(s-a)^2+\omega^2}$ \\[8pt]
$t\,\sin(\omega t)$ & $\dfrac{2\omega s}{(s^2+\omega^2)^2}$ \\[8pt]
$t\,\cos(\omega t)$ & $\dfrac{s^2-\omega^2}{(s^2+\omega^2)^2}$ \\[8pt]
$u(t-a)$ \small (unit step) & $\dfrac{e^{-as}}{s}$ \\[8pt]
$\delta(t-a)$ \small (Dirac delta) & $e^{-as}$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{table}[htbp]
\centering
\caption{Laplace Transform Properties (see \cref{ch:laplace_transforms})}
\label{tab:appA_laplace_properties}
\begin{tabular}{p{3.2cm} p{7.5cm}}
\toprule
\textbf{Property} & \textbf{Formula} \\
\midrule
Linearity &
$\mathcal{L}\{a\,f(t) + b\,g(t)\} = a\,F(s) + b\,G(s)$ \\[10pt]
First Shift (exponential) &
$\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ \\[10pt]
Second Shift (time delay) &
$\mathcal{L}\{u(t-a)\,f(t-a)\} = e^{-as}F(s)$ \\[10pt]
Derivatives &
$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$ \\[4pt]
& $\mathcal{L}\{f''(t)\} = s^2 F(s) - s\,f(0) - f'(0)$ \\[4pt]
& $\mathcal{L}\{f^{(n)}(t)\} = s^n F(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)$ \\[10pt]
Multiplication by $t^n$ &
$\mathcal{L}\{t^n f(t)\} = (-1)^n \dfrac{\mathrm{d}^n}{\mathrm{d}s^n}F(s)$ \\[10pt]
Convolution &
$\mathcal{L}\{f * g\} = F(s)\,G(s)$, \quad $(f*g)(t) = \displaystyle\int_0^t f(\tau)\,g(t-\tau)\,\diff\tau$ \\[10pt]
Integral &
$\displaystyle \mathcal{L}\!\left\{\int_0^t f(\tau)\,\diff\tau\right\} = \frac{F(s)}{s}$ \\[10pt]
\bottomrule
\end{tabular}
\end{table}

\begin{keyresult}
  \textbf{Solving IVPs with Laplace.} (1) Take the Laplace transform of the ODE (using derivative formulas to incorporate initial conditions). (2) Solve the resulting algebraic equation for $Y(s)$. (3) Find $y(t) = \mathcal{L}^{-1}\{Y(s)\}$ via partial fractions and the transform table.
\end{keyresult}

\subsection{Systems of Linear ODEs}
\label{sec:appA_systems}

For $\mathbf{x}' = A\,\mathbf{x}$ with $A$ a constant $n \times n$ matrix (see \cref{ch:systems}):

\begin{table}[htbp]
\centering
\caption{Eigenvalue Cases for 2$\times$2 Systems (see \cref{ch:systems})}
\label{tab:appA_systems_cases}
\begin{tabular}{p{2.8cm} p{3.5cm} p{6.5cm}}
\toprule
\textbf{Eigenvalue Case} & \textbf{Eigenvalues / Vectors} & \textbf{General Solution} \\
\midrule
Real distinct &
$\lambda_1 \neq \lambda_2 \in \R$; eigenvectors $\mathbf{v}_1, \mathbf{v}_2$ &
$\mathbf{x}(t) = c_1 e^{\lambda_1 t}\,\mathbf{v}_1 + c_2 e^{\lambda_2 t}\,\mathbf{v}_2$ \\[12pt]
Complex conjugate &
$\lambda = \alpha \pm i\beta$; eigenvector $\mathbf{a} \pm i\mathbf{b}$ &
$\mathbf{x}(t) = c_1\bigl[e^{\alpha t}\cos(\beta t)\,\mathbf{a} - e^{\alpha t}\sin(\beta t)\,\mathbf{b}\bigr]$
$+ c_2\bigl[e^{\alpha t}\sin(\beta t)\,\mathbf{a} + e^{\alpha t}\cos(\beta t)\,\mathbf{b}\bigr]$ \\[12pt]
Repeated (diagonalizable) &
$\lambda_1=\lambda_2=\lambda$; two independent eigenvectors $\mathbf{v}_1,\mathbf{v}_2$ &
$\mathbf{x}(t) = c_1 e^{\lambda t}\,\mathbf{v}_1 + c_2 e^{\lambda t}\,\mathbf{v}_2$ \\[10pt]
Repeated (defective) &
$\lambda$ (single eigenvector $\mathbf{v}_1$; generalized eigenvector $\mathbf{v}_2$) &
$\mathbf{x}(t) = c_1 e^{\lambda t}\,\mathbf{v}_1 + c_2 e^{\lambda t}\bigl(t\,\mathbf{v}_1 + \mathbf{v}_2\bigr)$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{keyresult}
  \textbf{Matrix exponential.} The solution to $\mathbf{x}' = A\mathbf{x},\; \mathbf{x}(0)=\mathbf{x}_0$ is $\mathbf{x}(t) = e^{At}\,\mathbf{x}_0$, where $e^{At} = \displaystyle\sum_{k=0}^{\infty} \frac{(At)^k}{k!}$. When $A$ is diagonalizable, $e^{At} = P\,e^{\Lambda t}\,P^{-1}$ with $\Lambda$ diagonal.
\end{keyresult}

\subsection{Series Solutions}
\label{sec:appA_series}

\begin{table}[htbp]
\centering
\caption{Series Solution Methods (see \cref{ch:series_solutions})}
\label{tab:appA_series_methods}
\begin{tabular}{p{2.5cm} p{3.8cm} p{5.5cm} p{3.5cm}}
\toprule
\textbf{Method} & \textbf{Equation Type} & \textbf{Solution Form} & \textbf{Key Details} \\
\midrule
Power series &
$p(x)$ and $q(x)$ analytic at ordinary point $x_0$ in \ $y''+p(x)y'+q(x)y=0$ &
$y = \sum_{n=0}^{\infty} a_n (x-x_0)^n$ &
Substitute into ODE, equate coefficients to find recurrence for $a_n$ \\[12pt]
Euler--Cauchy &
$x^2 y'' + \alpha x y' + \beta y = 0$, \ $x > 0$ &
Try $y = x^r$; indicial equation $r(r-1)+\alpha r+\beta = 0$ &
Distinct roots $r_1\neq r_2$: $y=c_1 x^{r_1}+c_2 x^{r_2}$. Repeated: $y=x^r(c_1+c_2\ln x)$ \\[12pt]
Frobenius &
Regular singular point at $x_0$ in $y''+p(x)y'+q(x)y=0$ &
$y = x^r\sum_{n=0}^{\infty} a_n x^n$, \quad $a_0\neq 0$ &
Indicial equation from lowest power. Cases depend on $r_1-r_2$: non-integer, zero, or positive integer \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Nonlinear Systems}
\label{sec:appA_nonlinear}

\begin{table}[htbp]
\centering
\caption{Linearized Classification via Jacobian (see \cref{ch:nonlinear_systems})}
\label{tab:appA_nonlinear_classification}
\begin{tabular}{p{3.2cm} p{3cm} p{3cm} p{3.8cm}}
\toprule
\textbf{Type} & \textbf{Trace } $T=\mathrm{tr}(J)$ & \textbf{Determinant } $D=\det(J)$ & \textbf{Eigenvalue Pattern} \\
\midrule
Stable node & $T<0$ & $D>0$, $T^2-4D>0$ & $\lambda_1,\lambda_2 < 0$, real \\[6pt]
Unstable node & $T>0$ & $D>0$, $T^2-4D>0$ & $\lambda_1,\lambda_2 > 0$, real \\[6pt]
Saddle point & $-$ & $D<0$ & $\lambda_1 < 0 < \lambda_2$, real \\[6pt]
Stable spiral & $T<0$ & $D>0$, $T^2-4D<0$ & $\alpha \pm i\beta$, $\alpha < 0$ \\[6pt]
Unstable spiral & $T>0$ & $D>0$, $T^2-4D<0$ & $\alpha \pm i\beta$, $\alpha > 0$ \\[6pt]
Center & $T=0$ & $D>0$ & $\pm i\beta$ (pure imaginary) \\[6pt]
\bottomrule
\end{tabular}
\end{table}

\begin{keyresult}
  \textbf{Hartman--Grobman theorem.} Near a hyperbolic equilibrium point ($\Re(\lambda_i)\neq 0$), the nonlinear system is topologically conjugate to its linearization. The phase portrait of the linearized system accurately describes the local behavior.
\end{keyresult}

\subsection{PDE Solution Methods}
\label{sec:appA_pde}

\begin{table}[htbp]
\centering
\caption{PDE Methods and Solution Forms (see \cref{ch:heat_equation}, \cref{ch:wave_and_laplace})}
\label{tab:appA_pde_methods}
\begin{tabular}{p{3.2cm} p{2.5cm} p{2.5cm} p{6cm}}
\toprule
\textbf{Equation} & \textbf{Domain} & \textbf{Method} & \textbf{Solution Form} \\
\midrule
Heat equation \ $\displaystyle u_t = \alpha\,u_{xx}$ & $0<x<L$ &
Separation of variables + Fourier series &
$\displaystyle u(x,t) = \sum_{n=1}^{\infty} b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)
  \exp\!\Bigl(-\alpha\,\frac{n^2\pi^2}{L^2}\,t\Bigr)$ \\[12pt]
Wave equation \ $\displaystyle u_{tt} = c^2\,u_{xx}$ & $x \in \R$ (infinite) &
d'Alembert's formula &
$\displaystyle u(x,t) = \tfrac{1}{2}\bigl[f(x-ct)+f(x+ct)\bigr]
  + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,\diff s$ \\[12pt]
Wave equation \ $\displaystyle u_{tt} = c^2\,u_{xx}$ & $0<x<L$ (finite, Dirichlet) &
Separation + Fourier sine series &
$\displaystyle u(x,t) = \sum_{n=1}^{\infty} \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)
  \bigl[A_n\cos(\tfrac{n\pi c t}{L}) + B_n\sin(\tfrac{n\pi c t}{L})\bigr]$ \\[12pt]
Laplace \ $\displaystyle u_{xx}+u_{yy}=0$ & Rectangle $0<x<a$, $0<y<b$ &
Separation of variables &
$\displaystyle u(x,y) = \sum_{n=1}^{\infty} A_n \sin\!\Bigl(\frac{n\pi x}{a}\Bigr)
  \sinh\!\Bigl(\frac{n\pi y}{a}\Bigr)$ \\[10pt]
Laplace \ $\displaystyle u_{rr}+\tfrac{1}{r}u_r+\tfrac{1}{r^2}u_{\theta\theta}=0$ & Disk $r<a$ &
Polar separation &
$\displaystyle u(r,\theta) = A_0 + \sum_{n=1}^{\infty} r^n\bigl[A_n\cos(n\theta)+B_n\sin(n\theta)\bigr]$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{keyresult}
  \textbf{Separation of variables (general procedure).} For a PDE $F(x,t,u,u_x,u_t,\dots)=0$, assume $u(x,t)=X(x)T(t)$. Substitute and separate variables to obtain two ODEs, one for $X$ and one for $T$, linked by a separation constant. Solve the spatial ODE as a Sturm--Liouville eigenvalue problem (\cref{ch:boundary_value_problems}), then expand the initial/boundary data in the resulting eigenfunction basis.
\end{keyresult}

\subsection{Mechanical and Electrical Applications}
\label{sec:appA_applications}

\begin{table}[htbp]
\centering
\caption{Damping Cases for Spring-Mass Systems (see \cref{ch:mechanical_applications})}
\label{tab:appA_damping}
\begin{tabular}{p{3.2cm} p{2.8cm} p{4.5cm} p{4cm}}
\toprule
\textbf{Case} & \textbf{Condition} & \textbf{Solution $y(t)$} & \textbf{Behavior} \\
\midrule
Undamped & $c=0$ &
$y(t) = C_1\cos(\omega_0 t) + C_2\sin(\omega_0 t)$, \ $\omega_0 = \sqrt{k/m}$ &
Persistent oscillation \\[10pt]
Underdamped & $0 < c < 2\sqrt{km}$ &
$y(t) = e^{-ct/(2m)}\bigl[C_1\cos(\omega_d t) + C_2\sin(\omega_d t)\bigr]$, \ $\omega_d = \sqrt{\omega_0^2-(c/2m)^2}$ &
Decaying oscillation \\[10pt]
Critically damped & $c = 2\sqrt{km}$ &
$y(t) = C_1 e^{-\omega_0 t} + C_2 t\,e^{-\omega_0 t}$ &
Fastest decay without oscillation \\[10pt]
Overdamped & $c > 2\sqrt{km}$ &
$y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}$, \ $r_{1,2} = \dfrac{-c \pm \sqrt{c^2-4km}}{2m}$ &
Slow non-oscillatory decay \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{table}[htbp]
\centering
\caption{Mechanical--Electrical Analogy (see \cref{ch:mechanical_applications})}
\label{tab:appA_mech_elec}
\begin{tabular}{l l l l}
\toprule
\textbf{Mechanical} & \textbf{Electrical (charge)} & \textbf{Mechanical} & \textbf{Electrical (flux)} \\
\midrule
Mass $m$ & Inductance $L$ & Mass $m$ & Capacitance $C$ \\
Damping $c$ & Resistance $R$ & Damping $c$ & Conductance $G$ \\
Spring constant $k$ & Inverse capacitance $1/C$ & Spring constant $k$ & Inverse inductance $1/L$ \\
Force $F(t)$ & Voltage $E(t)$ & Force $F(t)$ & Current $I(t)$ \\
Displacement $y(t)$ & Charge $q(t)$ & Velocity $v(t)$ & Flux $\phi(t)$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Qualitative Analysis and Numerical Methods}
\label{sec:appA_qualitative}

\begin{table}[htbp]
\centering
\caption{Equilibrium Classification for $y' = f(y)$ (see \cref{ch:qualitative})}
\label{tab:appA_equilibrium}
\begin{tabular}{p{3.5cm} p{4cm} p{4.5cm}}
\toprule
\textbf{Condition at } $y^*$ (where $f(y^*)=0$) & \textbf{Stability} & \textbf{Phase Line Behavior} \\
\midrule
$f'(y^*) < 0$ & Asymptotically stable (sink) & Solutions near $y^*$ converge to $y^*$ \\[6pt]
$f'(y^*) > 0$ & Unstable (source) & Solutions near $y^*$ diverge away from $y^*$ \\[6pt]
$f'(y^*) = 0$ & Inconclusive (semi-stable or higher-order) & Requires higher-order analysis; may be a node or saddle \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\begin{table}[htbp]
\centering
\caption{Euler's Method for Numerical Approximation (see \cref{ch:qualitative})}
\label{tab:appA_euler}
\begin{tabular}{l l}
\toprule
\textbf{Algorithm} & \\
\midrule
Given $y' = f(t,y)$, $y(t_0) = y_0$, step size $h$: \\[6pt]
\quad $t_{n+1} = t_n + h$ \\[4pt]
\quad $y_{n+1} = y_n + h\,f(t_n, y_n)$ \\[10pt]
Local truncation error: $\mathcal{O}(h^2)$ \\
Global error: $\mathcal{O}(h)$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Fourier Series Summary}
\label{sec:appA_fourier}

\begin{table}[htbp]
\centering
\caption{Fourier Series Coefficients (see \cref{ch:fourier_series})}
\label{tab:appA_fourier}
\begin{tabular}{l l}
\toprule
\textbf{Series} & \textbf{Coefficients} \\
\midrule
Full range $[-L,L]$: \ $f(x) \sim \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^{\infty} \Bigl[a_n\cos(\tfrac{n\pi x}{L}) + b_n\sin(\tfrac{n\pi x}{L})\Bigr]$ &
$\displaystyle a_0 = \frac{1}{L}\int_{-L}^{L} f(x)\,\diff x$ \\[8pt]
& $\displaystyle a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
& $\displaystyle b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[10pt]
Even $f$ (cosine series): & $b_n = 0$, \ $\displaystyle a_n = \frac{2}{L}\int_{0}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[10pt]
Odd $f$ (sine series): & $a_n = 0$, \ $\displaystyle b_n = \frac{2}{L}\int_{0}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}