Table of (bidirectional) Laplace Transform Pairs and Properties

(used in ECE301 and ECE438)

Laplace Transform Pairs and Properties
Definition
(bidirectional) Laplace Transform $F(s)=\int_{-\infty}^\infty f(t) e^{-st}dt, \ s\in {\mathbb C} \$
Properties of the Laplace Transform
Function Laplace Transform ROC
$f(t) \$ $F(s) \$ ROC $R$
$af_1(t)+bf_2(t) \$ $aF_1(s)+bF_2(s) \$ at least $R_1 \cap R_2$
$af(at) \$ $F\left( \frac{s}{a} \right)$
$e^{at}f(t) \$ $F(s-a) \$
$u(t-a) = \begin{cases} f(t-a) & t>a \\ 0 & t<a \end{cases}$ $e^{-as}F(s) \$
$f'(t) \$ $sF(s)-f(0) \$
$f''(t) \$ $s^2F(s)-sf(0)-f'(0) \$
$f^{(n)}(t) \$ $s^{n}F(s)-\sum_{k=1}^ns^{n-k}f^{(k)}(0) \$
$-tf(t) \$ $F'(s) \$
$t^2f(t) \$ $F''(s) \$
$(-1)^{(ntn)}f(t) \$ $F^{(n)}(s) \$
$\int_{0}^{t} f(u) du \$ $\frac{F(s)}s \$
$\int_{0}^{t}...\int_{0}^{t}f(u)du^n = \int_{0}^{t}\frac{{(t-u)}^{n-1}}{(n-1)!} f(u)du \$ $\frac{F(s)}{s^n} \$
$\int_{0}^{t}f(u)g(t-u)du \$ $F(s)G(s) \$
$\frac{f(t)}t \$ $\int_{s}^{\infty}F(u)du \$
$f(t)=f(t+T) \$ $\frac1{1-e^{-sT}}\int_{0}^{T}e^{-su}f(u)du \$
$\frac{1}{\sqrt{{\pi}t}}\int_{0}^{\infty}e^{-\frac{u^2}4t}f(u)du$ $\frac{F(\sqrt{s})}s \$
$\int_{0}^{\infty}J_0(2\sqrt{ut})f(u)du \$ $\frac1sF\left(\frac1s\right) \$
$t^{\frac{n}2}\int_{0}^{\infty}u^{-\frac{n}2}J_n(2\sqrt{ut})f(u)du \$ $\frac1{s^{n+1}}F\left(\frac1s\right) \$
$\int_{0}^{t}J_0(2\sqrt{u(t-u)})f(u)du \$ $\frac{F(s+\frac1s)}{s^2+1} \$
$f(t^2) \$ $\frac1{2\sqrt\pi}\int_{0}^{\infty}u^{-\frac32}e^{-\frac{s^2}{4u}}F(u)du \$
$\int_{0}^{\infty}\frac{t^uf(u)}{\Gamma(u+1)}du \$ $\frac{F(\ln s)}{s\ln s} \$
$\sum_{k=1}^N \frac{P(\alpha_k)}{Q'(\alpha_k)}e^{\alpha_kt} \$ $\frac{P(s)}{Q(s)} \$
Laplace Transform Pairs
Signal Laplace Transform ROC
unit impulse/Dirac delta $\,\!\delta(t)$ 1 $\text{All}\, s \in {\mathbb C}$
unit step function $\,\! u(t)$ $\frac{1}{s}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
$\,\! -u(-t)$ $\frac{1}{s}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
$\frac{t^{n-1}}{(n-1)!}u(t)$ $\frac{1}{s^{n}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
$-\frac{t^{n-1}}{(n-1)!}u(-t)$ $\frac{1}{s^{n}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
$\,\!e^{-\alpha t}u(t)$ $\frac{1}{s+\alpha}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
$\,\! -e^{-\alpha t}u(-t)$ $\frac{1}{s+\alpha}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
$\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)$ $\frac{1}{(s+\alpha )^{n}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
$-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)$ $\frac{1}{(s+\alpha )^{n}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
$\,\!\delta (t - T)$ $\,\! e^{-sT}$ $\text{All}\,\, s\in {\mathbb C}$
$\,\cos( \omega_0 t)u(t)$ $\frac{s}{s^2+\omega_0^{2}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
$\, \sin( \omega_0 t)u(t)$ $\frac{\omega_0}{s^2+\omega_0^{2}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
$\,e^{-\alpha t}\cos( \omega_0 t) u(t)$ $\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
$\, e^{-\alpha t}\sin( \omega_0 t)u(t)$ $\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
$u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}$ $\,\!s^{n}$ $All\,\, s$
$u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times}$ $\frac{1}{s^{n}}$ $\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$ $1 \$ $\frac{1}{s} \$
$t \$ $\frac1{s^2} \$
$\frac{t^{n-1}}{(n-1)!}, \ 0!=1 \$ $\frac1{s^n}, \ n=1,2,3,... \$
$\frac{t^{n-1}}{\Gamma(n)} \$ $\frac1{s^n}, \ n>0 \$
$e^{at}\$ $\frac1{s-a}\$
$\frac{t^{n-1}e^{at}}{(n-1)!}, \ 0!=1\$ $\frac1{(s-a)^n}, \ n=1,2,3,...\$
$\frac{t^{n-1}e^{at}}{\Gamma(n)}\$ $\frac1{(s-a)^n}, \ n>0\$
$\frac{\sin {at}}{a} \$ $\frac1{s^2+a^2}\$
$\cos {at} \$ $\frac{s}{s^2+a^2} \$
$\frac{e^{bt}\sin{at}}{a} \$ $\frac1{(s-b)^2+a^2}\$
$e^{bt}\cos{at}\$ $\frac{s-b}{(s-b)^2+a^2}\$
$\left(\frac{{sh}\ {at}}{a}\right)\$ $\frac{1}{s^2-a^2} \$
${ch}\ {at}\$ $\frac{s}{s^2-a^2}\$
$\frac{e^{bt}{sh}\ {at}}a\$ $\frac1{(s-b)^2-a^2}\$
$e^{bt} {ch}\ {at}\$ $\frac{s-b}{(s-b)^2-a^2} \$
$\frac{e^{bt}-e^{at}}{b-a}\$ $\frac1{(s-a)(s-b)},\ a \ne b\$
$\frac{be^{bt}-ae^{at}}{b-a}\$ $\frac{s}{(s-a)(s-b)},\ a \ne b \$
$\frac{\sin {at}-at\cos{at}}{2a^3}\$ $\frac1{(s^2+a^2)^2}\$
$\frac{t\sin {at}}{2a}\$ $\frac{s}{(s^2+a^2)^2}\$
$\frac{\sin {at}+at\cos {at}}{2a}\$ $\frac{s^2}{(s^2+a^2)^2}\$
$\cos {at}-\frac12at\sin {at}\$ $\frac{s^3}{(s^2+a^2)^2}\$
$t\cos {at}\$ $\frac{s^2-a^2}{(s^2+a^2)^2}\$
$\frac{at\ {ch}\ {at}-{sh}\ {at}}{2a^3}\$ $\frac{1}{(s^2-a^2)^2}\$
$\frac{t\ {sh}\ {at}}{2a}\$ $\frac{s}{(s^2-a^2)^2}\$
$\frac{{sh}\ {at}+at\ {ch}\ {at}}{2a}\$ $\frac{s^2}{(s^2-a^2)^2}\$
${ch}\ {at}+\frac12at\ {sh}\ {at} \$ $\frac{s^3}{(s^2-a^2)^2}\$
$t\ {ch}\ {at}\$ $\frac{s^2+a^2}{(s^2-a^2)^2}\$
$\frac{(3-a^2t^2)\sin {at}-3at\cos {at}}{8a^5}\$ $\frac{1}{(s^2+a^2)^3}\$
$\frac{t\sin {at}-at^2\cos {at}}{8a^3}\$ $\frac{s}{(s^2+a^2)^3}\$
$\frac{(1+a^2t^2)\sin {at}-at\cos {at}}{8a^3}\$ $\frac{s^2}{(s^2+a^2)^3}\$
$\frac{3t\sin {at}+at^2\cos {at}}{8a}\$ $\frac{s^3}{(s^2+a^2)^3}\$
$\frac{(3-a^2t^2)\sin {at}+5at\cos {at}}{8a}\$ $\frac{s^4}{(s^2+a^2)^3}\$
$\frac{(8-a^2t^2)\cos {at}-7at\sin {at}}{8}\$ $\frac{s^5}{(s^2+a^2)^3}\$
$\frac{t^2\sin {at}}{2a}\$ $\frac{3s^2-a^2}{(s^2+a^2)^3}\$
$\frac12t^2\cos {at}\$ $\frac{s^3-3a^2s}{(s^2+a^2)^3}\$
$\frac16t^3\cos {at}\$ $\frac{s^4-6a^2s^2+a^4}{(s^2+a^2)^4}\$
$\frac{t^3\sin {at}}{24a}\$ $\frac{s^3-a^2s}{(s^2+a^2)^4}\$
$\frac{3+a^2t^2\ {sh}\ {at}-3at\ {ch}\ {at}}{8a^5}\$ $\frac{1}{(s^2-a^2)^3}\$
$\frac{at^2\ {ch}\ {at}-t\ {sh}\ {at}}{8a^3}\$ $\frac{s}{(s^2-a^2)^3}\$
$\frac{at\ {ch}\ {at}+(a^2t^2-1)\ {sh}\ {at}}{8a^3}\$ $\frac{s^2}{(s^2-a^2)^3}\$
$\frac{3t\ {sh}\ {at}+at^2\ {ch}\ {at}}{8a}\$ $\frac{s^3}{(s^2-a^2)^3}\$
$\frac{(3+a^2t^2)\ {sh}\ {at}+5at\ {ch}\ {at}}{8a}\$ $\frac{s^4}{(s^2-a^2)^3}\$
$\frac{(8+a^2t^2)\ {ch}\ {at}+7at\ {sh}\ {at}}{8}\$ $\frac{s^5}{(s^2-a^2)^3}\$
$\frac{t^2\ {sh}\ {at}}{2a}\$ $\frac{3s^2+a^2}{(s^2-a^2)^3}\$
$\frac12t^2\ {ch}\ {at}\$ $\frac{s^3+3a^2s}{(s^2-a^2)^3}\$
$\frac16t^3\ {ch}\ {at}\$ $\frac{s^4+6a^2s^2+a^4}{(s^2-a^2)^4}\$
$\frac{t^3\ {sh}\ {at}}{24a}\$ $\frac{s^3+a^2s}{(s^2-a^2)^4}\$
$\frac{e^{at/2}}{3a^2} \left \{ \sqrt{3} \sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{-3at/2} \right \}\$ $\frac{1}{s^3+a^3}\$
$\frac{e^{at/2}}{3a^2} \left \{ \cos {\frac{\sqrt{3}at}{2}}+ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-e^{-3at/2} \right \}\$ $\frac{s}{s^3+a^3}\$
$\frac13 \left \{ e^{-at}+ 2e^{at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\$ $\frac{s^2}{s^3+a^3}\$
$\frac{e^{-at/2}}{3a^2} \left \{e^{3at/2}- \cos {\frac{\sqrt{3}at}{2}}- \sqrt{3}\sin {\frac{\sqrt{3}at}{2}} \right \}\$ $\frac{1}{s^3-a^3}\$
$\frac{e^{-at/2}}{3a} \left \{ \sqrt{3}\sin {\frac{\sqrt{3}at}{2}}-\cos {\frac{\sqrt{3}at}{2}}+e^{3at/2} \right \}\$ $\frac{s}{s^3-a^3}\$
$\frac13 \left \{ e^{at}+ 2e^{-at/2} \cos {\frac{\sqrt{3}at}{2}} \right \}\$ $\frac{s^2}{s^3-a^3}\$
$\frac1{4a^3} \left (\sin {at}\ {ch}\ {at}-\cos {at}\ {sh}\ {at} \right )\$ $\frac{1}{s^4+4a^4}\$
$\frac{\sin {at}\ {sh}\ {at}}{2a^2}\$ $\frac{s}{s^4+4a^4}\$
$\frac1{2a} \left (\sin {at}\ {ch}\ {at}+\cos {at}\ {sh}\ {at} \right )\$ $\frac{s^2}{s^4+4a^4}\$
$\cos {at}\ {ch}\ {at}\$ $\frac{s^3}{s^4+4a^4}\$
$\frac1{2a^3} \left (\ {sh}\ {at}-\sin {at} \right )\$ $\frac{1}{s^4-a^4}\$
$\frac1{2a^2} \left (\ {ch}\ {at}-\cos {at} \right )\$ $\frac{s}{s^4-a^4}\$
$\frac1{2a} \left (\ {sh}\ {at}+\sin {at} \right )\$ $\frac{s^2}{s^4-a^4}\$
$\frac12 \left (\ {ch}\ {at}+\cos {at} \right )\$ $\frac{s^3}{s^4-a^4}\$
$\frac{e^{-bt}-e^{-at}}{2(b-a)\sqrt{\pi t^3}}\$ $\frac1{\sqrt{s+a}+\sqrt{s+b}}\$
$\frac{erf\ \sqrt{at}}{\sqrt{a}}\$ $\frac1{s\sqrt{s+a}}\$
$\frac{e^{at}\ {erf}\ \sqrt{at}}{\sqrt{a}}\$ $\frac1{\sqrt{s}(s-a)}\$
$e^{at} \left \{\frac1{\sqrt{\pi t}}-be^{b^{2}t}\ {erfc}\ (b\sqrt{t}) \right \}\$ $\frac1{\sqrt{s-a}+b}\$
$J_0(at)\$ $\frac1{\sqrt{s^2+a^2}}\$
$I_0(at)\$ $\frac1{\sqrt{s^2-a^2}}\$
$a^nJ_n(at)\$ $\frac{{\left (\sqrt{s^2+a^2}-s \right )}^n}{\sqrt{s^2+a^2}},\quad n>-1 \$
$a^nI_n(at)\$ $\frac{{\left (s- \sqrt{s^2-a^2} \right )}^n}{\sqrt{s^2-a^2}},\quad n>-1 \$
$J_0(a\sqrt{t(t+2b)})\$ $\frac{e^{b \left (s- \sqrt{s^2+a^2} \right )}}{\sqrt{s^2+a^2}} \$
$\begin{cases} J_0(a\sqrt{t^2-b^2}) & t>b \\ 0 &t<b \end{cases} \$ $\frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}} \$
$tJ_0(at)\$ $\frac1{(s^2+a^2)^{3/2}}\$
$J_0(at)-atJ_1(at)\$ $\frac{s^2}{(s^2+a^2)^{3/2}}\$
$\frac{tI_1(at)}{a}\$ $\frac1{(s^2-a^2)^{3/2}}\$
$I_0(at)+atI_1(at)\$ $\frac{s}{(s^2+a^2)^{3/2}}\$
$f(t)=n,\ n \leqq t\ <n+1,\ n=0,1,2,... \$ $\frac1{s(e^s-1)}\ =\ \frac{e^{-s}}{s(1-e^{-s})}\$
$f(t)= \sum_{k=1}^{[t]} r^k\$ $\frac1{s(e^s-r)}\ =\ \frac{e^{-s}}{s(1-re^{-s})}\$
$f(t)= r^n,\ n\leqq t<n+1,\ n=0,1,2,...\$ $\frac{s^s-1}{s(e^s-r)}\ =\ \frac{1-e^{-s}}{s(1-re^{-s})}\$
$\frac{\cos {2\sqrt{at}}}{\sqrt{ \pi t}}\$ $\frac{s^{-a/s}}{\sqrt{s}}\$
$\frac{\sin {2\sqrt{at}}}{\sqrt{ \pi a}}\$ $\frac{e^{-a/s}}{s^{3/2}}\$
$\left ( \frac{t}{a} \right )^{n/2}J_n(2\sqrt{at})\$ $\frac{e^{-a/s}}{s^n+1} \quad n>-1 \$
$\frac{e^{-a^2/4t}}{\sqrt{ \pi t}}\$ $\frac{e^{-a\sqrt{s}}}{\sqrt{s}}\$
$\frac{a}{2\sqrt{ \pi t^3}}e^{-a^2/4t}\$ $e^{-a\sqrt{s}}\$
$erf(a/2\sqrt{t})\$ $\frac{1-e^{-a\sqrt{s}}}{s}\$
$erfc(a/2\sqrt{t})\$ $\frac{e^{-a\sqrt{s}}}{s}\$
$e^{b(bt+a)}erfc \left ( b\sqrt{t}+\frac{a}{2\sqrt{t}} \right )\$ $\frac{e^{-a\sqrt{s}}}{\sqrt{s}(\sqrt{s}+b)}\$
$\frac1{\sqrt{\pi t}a^{2n+1}}\int_{0}^{\infty}u^ne^{-u^2/4a^2t}J_{2n}(2\sqrt{u})du \$ $\frac{e^{-a\sqrt{s}}}{s^{n+1}} \quad n>-1\$
$\frac{e^{-bt}-e^{-at}}{t}\$ $\ln \left ( \frac{s+a}{s+b} \right )\$
$Ci(at)\$ $\frac{\ln [(s^2+a^2)/a^2]}{2s}\$
$Ei(at)\$ $\frac{\ln [(s+a)/a]}{s}\$
$\ln t\$ $\begin{array}{lcl} -\frac{(\gamma+\ln s)}{s} \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \$
$\frac{2(\cos {at}-\cos {bt})}{t}\$ $\ln \left ( \frac{s^2+a^2}{s^2+b^2} \right )\$
$\ln^2 t\$ $\begin{array}{lcl} \frac{{\pi}^2}{6s}+\frac{ \left (\gamma+\ln s \right )^2}{s} \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \$
$\begin{array}{lcl} - \left (\ln t+\gamma \right ) \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \$ $\frac{\ln s}{s}\$
$\begin{array}{lcl} \left ( \ln t+\gamma \right )^2-\frac16{\pi}^2 \\ \gamma = \text{Eular constant}=0.5772156... \end{array} \$ $\frac{\ln^2 s}{s}\$
$t^n\ln t\$ $\frac{\Gamma'(n+1)-\Gamma(n+1)\ln s}{s^{n+1}} \quad n>-1\$
$\frac{\sin {at}}{t}\$ ${Arc}\ {tg}\ (a/s)\$
$Si(at)\$ $\frac{{Arc}\ {tg}\ (a/s)}{s}\$
$\frac{e^{-2\sqrt{at}}}{\sqrt{\pi t}} \$ $\frac{e^{a/s}}{\sqrt{s}}\ erfc(\sqrt{a/s})\$
$\frac{2a}{\sqrt{\pi }}e^{-a^2t^2}\$ $e^{s^2/4a^2}\ erfc(s/2a)\$
$erf(at)\$ $\frac{e^{s^2/4a^2}\ erfc(s/2a)}{s}\$
$\frac1{\sqrt{\pi (t+a)}}\$ $\frac{e^{as}erfc\sqrt{as}}{\sqrt{s}}\$
$\frac1{t+a}\$ $e^{as}Ei(as)\$
$\frac1{t^2+a^2}\$ $\frac1a \left [ \cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as) \right ]\$
$\frac{t}{t^2+a^2}\$ $\sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)\$
${Arc}\ {tg}(t/a)\$ $\frac{\cos {as} \left \{ \frac{\pi }{2}-Si(as) \right \}-\sin {as}\ Ci(as)}{s}\$
$\frac12\ln \left (\frac{t^2+a^2}{a^2} \right )\$ $\frac{\sin {as} \left \{ \frac{\pi }{2}-Si(as) \right \}+\cos {as}\ Ci(as)}{s}\$
$\frac1t \ln \left ( \frac{t^2+a^2}{a^2} \right )\$ $\left [ \frac{\pi}{2}-Si(as) \right ]^2 + Ci^2(as)\$
$\mathcal{N}(t)\ =\ fonction nulle\$ $0\$
$\delta(t)\ =\ fonction delta\$ $1\$
$\delta(t-a)\$ $e^{-as}\$
$\mu(t-a)\$ $\frac{e^{-as}}{s}\$
$\frac xa+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin {\frac{n \pi x}{a}} \cos {\frac{n\pi t}{a}}\$ $\frac{{sh}\ sx}{s\ {sh}\ sa}\$
$\frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \sin {\frac{(2n-1) \pi x}{2a}} \sin {\frac{(2n-1)\pi t}{2a}}\$ $\frac{{sh}\ sx}{s\ {ch}\ sa}\$
$|fracta+\frac2{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \cos {\frac{n \pi x}{a}} \sin {\frac{n\pi t}{a}}\$ $\frac{{ch}\ sx}{s\ {sh}\ as}\$
$1+\frac4{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1} \cos {\frac{n \pi x}{a}} \cos {\frac{(2n-1)\pi t}{2a}}\$ $\frac{{ch}\ sx}{s\ {ch}\ as}\$
$\frac {xt}a+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \sin {\frac{n \pi x}{a}} \sin {\frac{n\pi t}{a}}\$ $\frac{{sh}\ sx}{s^2\ {sh}\ sa}\$
$x+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \sin {\frac{(2n-1) \pi x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\$ $\frac{{sh}\ sx}{s^2\ {ch}\ sa}\$
$\frac{t^2}{2a}+\frac{2a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos {\frac{n \pi x}{a}} \left ( 1-\cos {\frac{n \pi t}{a}} \right )\$ $\frac{{ch}\ sx}{s^2\ {sh}\ sa}\$
$t+\frac{8a}{{\pi}^2 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^2} \cos {\frac{(2n-1) \pi x}{2a}} \sin {\frac{(2n-1) \pi t}{2a}}\$ $\frac{{ch}\ sx}{s^3\ {sh}\ sa}\$
$\frac12(t^2+x^2-a^2)-\frac{16a^2}{{\pi}^3 } \sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3} \cos {\frac{(2n-1) \pi x}{2a}} \cos {\frac{(2n-1) \pi t}{2a}}\$ $\frac{{ch}\ sx}{s^3\ {ch}\ sa}\$
$\frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\$ $\frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\$
$\frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\$ $\frac{{ch}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\$
$\frac{2 \pi}{a^2} \sum_{n=1}^{\infty} (-1)^nne^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{n \pi x}{a}}\$ $\frac{{sh}\ x\sqrt{s}}{{sh}\ a\sqrt{s}}\$
$\frac{\pi}{a^2} \sum_{n=1}^{\infty} (-1)^{n-1}(2n-1)e^{-(2n-1)^2{\pi}^2t/4a^2}\cos {\frac{(2n-1) \pi x}{2a}}\$ $\frac{{ch}\ x\sqrt{s}}{{ch}\ a\sqrt{s}}\$
$\frac{2}{a} \sum_{n=1}^{\infty} (-1)^{n-1}e^{-(2n-1)^2{\pi}^2t/4a^2}\sin {\frac{(2n-1) \pi x}{2a}}\$ $\frac{{sh}\ x\sqrt{s}}{\sqrt{s}{ch}\ a\sqrt{s}}\$
$\frac1a+\frac2a\sum_{n=1}^{\infty} (-1)^ne^{-n^2{\pi}^2t/a^2}\cos {\frac{n \pi x}{a}}\$ $\frac{{ch}\ x\sqrt{s}}{\sqrt{s}{sh}\ a\sqrt{s}}\$
$\frac{x}{a}+\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n}e^{-n^2{\pi}^2t/a^2} \sin {\frac{n \pi x}{a}}\$ $\frac{{sh}\ x\sqrt{s}}{s{sh}\ a\sqrt{s}}\$
$1+\frac4{\pi}\sum_{n=1}^{\infty} \frac{(-1)^n}{2n-1}e^{-(2n-1)^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi x}{2a}}\$ $\frac{{ch}\ x\sqrt{s}}{s{ch}\ a\sqrt{s}}\$
$\frac{xt}{a}+\frac{2a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}(1-e^{-n^2{\pi}^2t/a^2})\sin {\frac{n \pi x}{a}}\$ $\frac{{sh}\ x\sqrt{s}}{s^2{sh}\ a\sqrt{s}}\$
$\frac12(x^2+a^2)+t-\frac{16a^2}{{\pi}^3}\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3}e^{-{(2n-1)}^2{\pi}^2t/a^2}\cos {\frac{(2n-1) \pi x}{2a}}\$ $\frac{{ch}\ x\sqrt{s}}{s^2{ch}\ a\sqrt{s}}\$

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Correspondence Chess Grandmaster and Purdue Alumni

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