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appendixB: transform and integral tables

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\section{Transform Tables} % =============================================================================
% appB_transform_tables.tex
% Appendix B: Transform and Integral Tables
% =============================================================================
\section{Transform and Integral Tables}
\label{app:transform_tables} \label{app:transform_tables}
This appendix provides Laplace and Fourier transform tables for quick reference. This appendix provides comprehensive reference tables for Laplace transforms (\cref{ch:laplace_transforms}), Fourier series formulas (\cref{ch:fourier_series}), and common integrals used throughout the handbook. All entries are presented without derivation; see the referenced chapters for proofs and worked examples.
\subsection{Laplace Transform Table} \subsection{Laplace Transform Table}
\label{sec:appB_laplace_table} \label{sec:appB_laplace_table}
The Laplace transform is defined in \cref{eq:laplace_definition} as
\[
\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t)\,\diff t,
\]
provided the integral converges. The following table lists the most frequently encountered transforms.
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\caption{Common Laplace Transforms} \caption{Common Laplace Transforms}
\label{tab:laplace_transforms} \label{tab:laplace_transforms}
\begin{tabular}{l l} \begin{tabular}{l l l}
\toprule \toprule
\textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} \\ \textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} & \textbf{Conditions} \\
\midrule \midrule
$1$ & TBD \\ $1$ & $\dfrac{1}{s}$ & $s > 0$ \\[8pt]
$t^n$ & TBD \\ $t$ & $\dfrac{1}{s^{2}}$ & $s > 0$ \\[8pt]
$e^{at}$ & TBD \\ $t^{n}$ & $\dfrac{n!}{s^{n+1}}$ & $s > 0,\; n \in \N$ \\[8pt]
$\sin(\omega t)$ & TBD \\ $e^{at}$ & $\dfrac{1}{s-a}$ & $s > a$ \\[8pt]
$\cos(\omega t)$ & TBD \\ $t\,e^{at}$ & $\dfrac{1}{(s-a)^{2}}$ & $s > a$ \\[8pt]
$e^{at}\sin(\omega t)$ & TBD \\ $\sin(bt)$ & $\dfrac{b}{s^{2}+b^{2}}$ & $s > 0$ \\[8pt]
$e^{at}\cos(\omega t)$ & TBD \\ $\cos(bt)$ & $\dfrac{s}{s^{2}+b^{2}}$ & $s > 0$ \\[8pt]
$\sinh(at)$ & TBD \\ $\sinh(bt)$ & $\dfrac{b}{s^{2}-b^{2}}$ & $s > |b|$ \\[8pt]
$\cosh(at)$ & TBD \\ $\cosh(bt)$ & $\dfrac{s}{s^{2}-b^{2}}$ & $s > |b|$ \\[8pt]
$t^n e^{at}$ & TBD \\ $t\,\sin(bt)$ & $\dfrac{2bs}{(s^{2}+b^{2})^{2}}$ & $s > 0$ \\[8pt]
$t \sin(\omega t)$ & TBD \\ $t\,\cos(bt)$ & $\dfrac{s^{2}-b^{2}}{(s^{2}+b^{2})^{2}}$ & $s > 0$ \\[8pt]
$t \cos(\omega t)$ & TBD \\ $e^{at}\sin(bt)$ & $\dfrac{b}{(s-a)^{2}+b^{2}}$ & $s > a$ \\[8pt]
$u(t-a)$ (unit step) & TBD \\ $e^{at}\cos(bt)$ & $\dfrac{s-a}{(s-a)^{2}+b^{2}}$ & $s > a$ \\[8pt]
$\delta(t-a)$ (Dirac delta) & TBD \\ $u_{c}(t) \text{ (unit step)}$ & $\dfrac{e^{-cs}}{s}$ & $s > 0$ \\[8pt]
$\delta(t-a) \text{ (Dirac)}$ & $e^{-as}$ & $a \ge 0$ \\[8pt]
$t^{n}e^{at}$ & $\dfrac{n!}{(s-a)^{n+1}}$ & $s > a,\; n \in \N$ \\[8pt]
$\sin^{2}(bt)$ & $\dfrac{2b^{2}}{s(s^{2}+4b^{2})}$ & $s > 0$ \\[8pt]
$\cos^{2}(bt)$ & $\dfrac{s^{2}+2b^{2}}{s(s^{2}+4b^{2})}$ & $s > 0$ \\[8pt]
\bottomrule \bottomrule
\end{tabular} \end{tabular}
\end{table} \end{table}
\subsection{Fourier Transform Table} \subsection{Laplace Transform Properties}
\label{sec:appB_fourier_table} \label{sec:appB_laplace_properties}
The following algebraic and operational properties make the Laplace transform a powerful tool for solving linear differential equations.
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
\caption{Common Fourier Transforms} \caption{Laplace Transform Properties}
\label{tab:fourier_transforms} \label{tab:laplace_properties}
\begin{tabular}{l l} \begin{tabular}{l l}
\toprule \toprule
\textbf{$f(x)$} & \textbf{$\hat{f}(k)$} \\ \textbf{Property} & \textbf{Formula} \\
\midrule \midrule
$e^{iax}$ & TBD \\ Linearity & $\mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s)$ \\[6pt]
$e^{-a|x|}$ & TBD \\ First shift & $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ \\[6pt]
$\mathrm{rect}(x)$ & TBD \\ Second shift & $\mathcal{L}\{u_{c}(t)\,f(t-c)\} = e^{-cs}F(s)$ \\[6pt]
$\delta(x)$ & TBD \\ Derivative & $\mathcal{L}\{f'(t)\} = s F(s) - s f(0)$ \\[6pt]
Gaussian $e^{-ax^2}$ & TBD \\ Second derivative & $\mathcal{L}\{f''(t)\} = s^{2}F(s) - s f(0) - f'(0)$ \\[6pt]
$n$th derivative & $\mathcal{L}\{f^{(n)}(t)\} = s^{n}F(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)$ \\[6pt]
Integral & $\displaystyle \mathcal{L}\!\left\{\int_{0}^{t} f(\tau)\,\diff\tau\right\} = \frac{F(s)}{s}$ \\[8pt]
Convolution & $\mathcal{L}\{(f*g)(t)\} = F(s)\,G(s)$ \\[6pt]
$t$-multiplication & $\mathcal{L}\{t^{n}f(t)\} = (-1)^{n}\dfrac{\mathrm{d}^{n}F}{\mathrm{d}s^{n}}$ \\[8pt]
Frequency differentiation & $\mathcal{L}\{t\,f(t)\} = -\dfrac{\mathrm{d}F}{\mathrm{d}s}$ \\[8pt]
Final value theorem & $\displaystyle \lim_{t\to\infty} f(t) = \lim_{s\to 0} sF(s)$ \\[8pt]
Initial value theorem & $\displaystyle \lim_{t\to 0^{+}} f(t) = \lim_{s\to\infty} sF(s)$ \\[8pt]
\bottomrule \bottomrule
\end{tabular} \end{tabular}
\end{table} \end{table}
The convolution integral $(f*g)(t)$ appearing in the table above is defined by
\[
(f*g)(t) = \int_0^t f(t-\tau)\,g(\tau)\,\diff\tau.
\]
\subsection{Fourier Series Formulas}
\label{sec:appB_fourier_series}
Fourier series decompose a periodic function $f(x)$ of period $2L$ into sine and cosine harmonics, as developed in \cref{ch:fourier_series}. The following tables collect the most essential formulas.
\begin{table}[htbp]
\centering
\caption{Fourier Series on $[-L, L]$}
\label{tab:fourier_series}
\begin{tabular}{l l}
\toprule
\textbf{Formula} & \textbf{Expression} \\
\midrule
Full series & $\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Bigl[a_n \cos\!\Bigl(\frac{n\pi x}{L}\Bigr) + b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\Bigr]$ \\[12pt]
$a_0$ & $\displaystyle a_0 = \frac{1}{L}\int_{-L}^{L} f(x)\,\diff x$ \\[8pt]
$a_n$ & $\displaystyle a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
$b_n$ & $\displaystyle b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
Half-range cosine & $\displaystyle a_n = \frac{2}{L}\int_{0}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
Half-range sine & $\displaystyle b_n = \frac{2}{L}\int_{0}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
Complex coefficients & $\displaystyle c_n = \frac{1}{2L}\int_{-L}^{L} f(x)\,e^{-i n\pi x/L}\,\diff x$ \\[8pt]
Complex series & $\displaystyle f(x) = \sum_{n=-\infty}^{\infty} c_n\,e^{i n\pi x/L}$ \\[8pt]
Parseval's identity & $\displaystyle \frac{1}{L}\int_{-L}^{L} |f(x)|^{2}\,\diff x = \frac{a_0^{2}}{2} + \sum_{n=1}^{\infty} \bigl(a_n^{2} + b_n^{2}\bigr)$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}
\subsection{Common Integral Table}
\label{sec:appB_common_integrals}
The following integrals are used throughout the handbook, particularly in separation of variables, integrating factor methods, and Fourier coefficient calculations.
\begin{table}[htbp]
\centering
\caption{Common Indefinite Integrals}
\label{tab:common_integrals}
\begin{tabular}{l l}
\toprule
\textbf{Integrand} & \textbf{Result} \\
\midrule
$\displaystyle \int x^{n}\,\diff x$ & $\displaystyle \frac{x^{n+1}}{n+1} + C \qquad (n \neq -1)$ \\[8pt]
$\displaystyle \int \frac{1}{x}\,\diff x$ & $\displaystyle \ln|x| + C$ \\[8pt]
$\displaystyle \int e^{ax}\,\diff x$ & $\displaystyle \frac{1}{a}e^{ax} + C$ \\[8pt]
$\displaystyle \int \sin(ax)\,\diff x$ & $\displaystyle -\frac{1}{a}\cos(ax) + C$ \\[8pt]
$\displaystyle \int \cos(ax)\,\diff x$ & $\displaystyle \frac{1}{a}\sin(ax) + C$ \\[8pt]
$\displaystyle \int \sec^{2}(ax)\,\diff x$ & $\displaystyle \frac{1}{a}\tan(ax) + C$ \\[8pt]
$\displaystyle \int \csc^{2}(ax)\,\diff x$ & $\displaystyle -\frac{1}{a}\cot(ax) + C$ \\[8pt]
$\displaystyle \int \tan(ax)\,\diff x$ & $\displaystyle -\frac{1}{a}\ln|\cos(ax)| + C$ \\[8pt]
$\displaystyle \int \frac{1}{\sqrt{a^{2}-x^{2}}}\,\diff x$ & $\displaystyle \arcsin\!\Bigl(\frac{x}{a}\Bigr) + C$ \\[8pt]
$\displaystyle \int \frac{1}{a^{2}+x^{2}}\,\diff x$ & $\displaystyle \frac{1}{a}\arctan\!\Bigl(\frac{x}{a}\Bigr) + C$ \\[8pt]
$\displaystyle \int \frac{1}{x\sqrt{x^{2}-a^{2}}}\,\diff x$ & $\displaystyle \frac{1}{a}\,\mathrm{arcsec}\!\Bigl(\frac{|x|}{a}\Bigr) + C$ \\[8pt]
$\displaystyle \int \sqrt{a^{2}-x^{2}}\,\diff x$ & $\displaystyle \frac{x}{2}\sqrt{a^{2}-x^{2}} + \frac{a^{2}}{2}\arcsin\!\Bigl(\frac{x}{a}\Bigr) + C$ \\[10pt]
$\displaystyle \int e^{ax}\sin(bx)\,\diff x$ & $\displaystyle \frac{e^{ax}}{a^{2}+b^{2}}\bigl(a\sin(bx)-b\cos(bx)\bigr) + C$ \\[10pt]
$\displaystyle \int e^{ax}\cos(bx)\,\diff x$ & $\displaystyle \frac{e^{ax}}{a^{2}+b^{2}}\bigl(a\cos(bx)+b\sin(bx)\bigr) + C$ \\[10pt]
$\displaystyle \int x e^{ax}\,\diff x$ & $\displaystyle \frac{e^{ax}}{a^{2}}\bigl(ax-1\bigr) + C$ \\[8pt]
$\displaystyle \int x \sin(ax)\,\diff x$ & $\displaystyle \frac{\sin(ax)}{a^{2}} - \frac{x\cos(ax)}{a} + C$ \\[8pt]
$\displaystyle \int x \cos(ax)\,\diff x$ & $\displaystyle \frac{\cos(ax)}{a^{2}} + \frac{x\sin(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \ln(x)\,\diff x$ & $\displaystyle x\ln(x) - x + C$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}
\begin{keyresult}
\textbf{Integration by parts reminder.} For products of functions that appear in Fourier coefficient computations:
\[
\int u\,\mathrm{d}v = uv - \int v\,\mathrm{d}u.
\]
This formula underlies the $x e^{ax}$, $x \sin(ax)$, and $x \cos(ax)$ entries in \cref{tab:common_integrals} and is essential for evaluating Fourier coefficients of piecewise-linear functions in \cref{ch:fourier_series}.
\end{keyresult}