Section: 17 | Transforms |
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 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# 7 TRANSFORMS

## 7.1 FOURIER TRANSFORMS

For a piecewise continuous function $F\left(x\right)$ over a finite interval $0\le x\le \pi$ ; the finite Fourier cosine transform of $F\left(x\right)$ is ${f}_{c}\left(n\right)={\int }_{0}^{\pi }F\left(x\right)cosnx\phantom{\rule{0.2em}{0ex}}dx\mathrm{ }\left(n=0,\phantom{\rule{0.2em}{0ex}}1,\phantom{\rule{0.2em}{0ex}}2,\dots \right)$ If $x$ ranges over the interval $0\le x\le L$ , the substitution ${x}^{\prime }=\frac{\pi x}{L}$ allows the use of this definition, also. The inverse transform is written. $\stackrel{̲}{F}\left(x\right)=\frac{1}{\pi }{f}_{c}\left(0\right)-\frac{2}{\pi }\sum _{n=1}^{x}{f}_{c}\left(n\right)cosnx\mathrm{ }\left(0 where $F\left(x\right)=\frac{F\left(x+∊\right)+F\left(x-∊\right)}{2}$ . Note that $F\left(x+\right)=F\left(x-\right)=F\left(x\right)$ at points of continuity. The formula ${f}_{c}^{\left(2\right)}\left(n\right)={\int }_{0}^{\pi }{F}^{\prime \prime }\left(x\right)cosnx\phantom{\rule{0.2em}{0ex}}dx=-{n}^{2}{f}_{c}\left(n\right)-{F}^{\prime }\left(0\right)+\left(-1\right){}^{n}{F}^{\prime }\left(\pi \right)$ makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform of $F\left(x\right)$ is ${f}_{s}\left(n\right)={\int }_{0}^{\pi }F\left(x\right)sinnx\phantom{\rule{0.2em}{0ex}}dx\mathrm{ }\left(n=1,2,3,\dots \right)$ and $\stackrel{̲}{F}\left(x\right)=\frac{2}{\pi }\sum _{n=1}^{\infty }{f}_{s}\left(n\right)sinnx\mathrm{ }\left(0 Corresponding to equation (33) we have ${f}_{s}^{\left(2\right)}\left(n\right)={\int }_{0}^{\pi }{F}^{\prime \prime }\left(x\right)sinnx\phantom{\rule{0.2em}{0ex}}dx=-{n}^{2}{f}_{s}\left(n\right)-n\phantom{\rule{0.2em}{0ex}}F\left(0\right)-n\left(-1\right){}^{n}F\left(\pi \right)$ If $F\left(x\right)$ is defined for $x\le 0$ and is piecewise continuous over any finite interval, and if ${\int }_{0}^{x}F\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ is absolutely convergent, then ${f}_{c}\left(\alpha \right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{x}F\left(x\right)cos\left(\alpha x\right)\phantom{\rule{0.2em}{0ex}}dx$ is the Fourier cosine transform of $F\left(x\right)$ . Furthermore, $\stackrel{̲}{F}\left(x\right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{x}{f}_{c}\left(\alpha \right)cos\left(\alpha x\right)\phantom{\rule{0.2em}{0ex}}d\alpha .$ If $\underset{x\to \infty }{lim}{d}^{n}F/{dx}^{n}=0$ , then an important property of the Fourier cosine transform is ${f}_{c}^{\left(2r\right)}\left(\alpha \right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{x}\left(\frac{{d}^{2r}F}{{dx}^{2r}}\right)cos\left(\alpha x\right)\phantom{\rule{0.2em}{0ex}}dx=-\sqrt{\frac{2}{\pi }}\sum _{n=0}^{r-1}\left(-1\right){}^{n}{a}_{2r-2n-1}{\alpha }^{2n}+\left(-1\right){}^{r}{\alpha }^{2r}{f}_{c}\left(\alpha \right)$ where $\underset{x\to \infty }{lim}{d}^{r}F/{dx}^{r}={a}_{r,}$ which is often useful. Under the same conditions, define the Fourier sine transform of $F\left(x\right)$ as follows ${f}_{s}\left(\alpha \right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{x}F\left(x\right)sin\left(\alpha x\right)\phantom{\rule{0.2em}{0ex}}dx$ with $\stackrel{̲}{F}\left(x\right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{x}{f}_{s}\left(\alpha \right)sin\left(\alpha x\right)\phantom{\rule{0.2em}{0ex}}d\alpha$ Corresponding to (34) we have ${f}_{s}^{\left(2r\right)}\left(\alpha \right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{\infty }\left(\frac{{d}^{2r}F}{{dx}^{2r}}sin\left(\alpha x\right)\right)\phantom{\rule{0.2em}{0ex}}dx=-\sqrt{\frac{2}{\pi }}\sum _{n=1}^{r}\left(-1\right){}^{n}{\alpha }^{2n-1}{a}_{2r-2n}+\left(-1\right){}^{r-1}{\alpha }^{2r}{f}_{s}\left(\alpha \right)$ Similarly, if $F\left(x\right)$ is defined for $-\infty , and if ${\int }_{-\infty }^{\infty }F\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ is absolutely convergent, then $f\left(\alpha \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }F\left(x\right){e}^{iax}\phantom{\rule{0.2em}{0ex}}dx$ is the Fourier transform of $F\left(x\right)$ , and $\stackrel{̲}{F}\left(x\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f\left(\alpha \right){e}^{-iax}\phantom{\rule{0.2em}{0ex}}d\alpha$ Also, if   $\underset{|x|\to \infty }{lim}|\frac{{d}^{n}F}{{dx}^{n}}|=0\mathrm{ }\left(n=1,2,\dots ,r-1\right)$   then ${f}^{\left(r\right)}\left(\alpha \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{F}^{\left(r\right)}\left(x\right){e}^{i\alpha x}\phantom{\rule{0.2em}{0ex}}dx=\left(-i\alpha \right){}^{r}f\left(\alpha \right)$

## 7.2 TABLE OF FOURIER COSINE TRANSFORMS

$F\left(\omega \right)={f}_{c}\left(f\right)\left(\omega \right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{\infty }f\left(x\right)cos\left(\omega x\right)dx$ , $\omega >0.$

No.$f\left(x\right)$ $F\left(\omega \right)$
1. $\left\{\begin{array}{ll}1\hfill & 0a\hfill \end{array}$ $\sqrt{\frac{2}{\pi }}\frac{sina\omega }{\omega }$
2. ${x}^{p-1}\left(0 $\sqrt{\frac{2}{\pi }}\frac{\Gamma \left(p\right)}{{\omega }^{p}}cos\frac{p\pi }{2}$
3. $\left\{\begin{array}{ll}cosx\hfill & 0a\hfill \end{array}$ $\frac{1}{\sqrt{2\pi }}\left(\frac{sin\left[a\left(1-\omega \right)\right]}{1-\omega }+\frac{sin\left[a\left(1+\omega \right)\right]}{1+\omega }\right)$
4. ${e}^{-x}$ $\sqrt{\frac{2}{\pi }}\frac{1}{1+{\omega }^{2}}$
5. ${e}^{-{x}^{2}/2}$ ${e}^{-{\omega }^{2}/2}$
6. $cos\frac{{x}^{2}}{2}$ $cos\left(\frac{{\omega }^{2}}{2}-\frac{\pi }{4}\right)$
7. $sin\frac{{x}^{2}}{2}$ $cos\left(\frac{{\omega }^{2}}{2}+\frac{\pi }{4}\right)$

## 7.3 TABLE OF FINITE COSINE TRANSFORMS

${f}_{c}\left(n\right)={\int }_{0}^{\pi }F\left(x\right)\phantom{\rule{0.2em}{0ex}}cosnx\phantom{\rule{0.2em}{0ex}}dx$ , for $n=0,1,2,\dots .$

No.${f}_{c}\left(n\right)$ $F\left(x\right)$
1. $\left(-1\right){}^{n}{f}_{c}\left(n\right)$ $F\left(\pi -x\right)$
2. $\left\{\begin{array}{ll}\pi \hfill & n=0\hfill \\ 0\hfill & n=1,2,\dots \hfill \end{array}$ 1
3. $\left\{\begin{array}{ll}0\hfill & n=0\hfill \\ \frac{2}{n}sin\frac{n\pi }{2}\hfill & n=1,2,\dots \hfill \end{array}$ $\left\{\begin{array}{ll}1\hfill & \text{for}\phantom{\rule{0.2em}{0ex}}\text{0}
4. $\left\{\begin{array}{ll}\frac{{\pi }^{2}}{2}\hfill & n=0\hfill \\ {\left(-1\right)}^{n}-1/{n}^{2}\hfill & n=1,2\dots \hfill \end{array}$ $x$
5. $\left\{\begin{array}{ll}\frac{{\pi }^{2}}{6}\hfill & n=0\hfill \\ {\left(-1\right)}^{n}-1/{n}^{2}\hfill & n=1,2\dots \hfill \end{array}$ $\frac{{x}^{2}}{2\pi }$
6. $\frac{\left(-1\right){}^{n}{e}^{c}\pi -1}{{n}^{2}+{c}^{2}}$ $\frac{1}{c}{e}^{cx}$
7. $\frac{k}{{n}^{2}-{k}^{2}}\left[\left(-1\right){}^{n}cos\pi k-1\right]$ with $k\ne 0,1,2,\dots$ $sinkx$
8. $\left\{\begin{array}{ll}0\hfill & m=1,2,\dots \hfill \\ \frac{{\left(-1\right)}^{n+m}-1}{{n}^{2}-{m}^{2}}\hfill & m\ne 1,2\dots \hfill \end{array}$ $\frac{1}{m}sinmx$
9. $\frac{1}{{n}^{2}-{k}^{2}}$ with $k\ne 0,1,2,\dots$ $-\frac{cosk\left(\pi -x\right)}{ksink\pi }$
10. $\left\{\begin{array}{ll}\pi /2\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}n=m\hfill \\ 0\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}n\ne m\hfill \end{array}$ $cosmx\mathrm{ }\left(m=1,2,\dots \right)$

## 7.4 TABLE OF FOURIER SINE TRANSFORMS

$F\left(\omega \right)={F}_{s}\left(f\right)\left(\omega \right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{\infty }f\left(x\right)sin\left(\omega x\right)\phantom{\rule{0.2em}{0ex}}dx,\mathrm{ }\omega >0.$

No.$f\left(x\right)$ $F\left(\omega \right)$
1. $\left\{\begin{array}{ll}1\hfill & 0a\hfill \end{array}$ $\sqrt{\frac{2}{\pi }}\frac{1-cos\omega a}{\omega }$
2. ${x}^{p-1}\left(0 $\sqrt{\frac{2}{\pi }}\frac{\Gamma \left(p\right)}{{\omega }^{p}}sin\frac{p\pi }{2}$
3. $\left\{\begin{array}{ll}sinx\hfill & 0a\hfill \end{array}$ $\frac{1}{\sqrt{2\pi }}\left(\frac{sin\left[a\left(1-\omega \right)\right]}{1-\omega }-\frac{sin\left[a\left(1+\omega \right)\right]}{1+\omega }\right)$
4. ${e}^{-x}$ $\sqrt{\frac{2}{\pi }}\frac{\omega }{1+{\omega }^{2}}$
5. ${xe}^{-{x}^{2}/2}$ $\omega {e}^{-{\omega }^{2}/2}$

## 7.5 TABLE OF FINITE SINE TRANSFORMS

${f}_{s}\left(n\right)={\int }_{0}^{\pi }F\left(x\right)\phantom{\rule{0.2em}{0ex}}sinnx\phantom{\rule{0.2em}{0ex}}dx$ , for $n=1,2,\dots$ .

No.${f}_{s}\left(n\right)$ $F\left(x\right)$
1. $\left(-1\right){}^{n+1}{f}_{s}\left(n\right)$ $F\left(\pi -x\right)$
2. $1/n$ $\pi -x/\pi$
3. $\left(-1\right){}^{n+1}/n$ $x/\pi$
4. $1-\left(-1\right){}^{n}/n$ $1$
5. $\frac{2}{{n}^{2}}sin\frac{n\pi }{2}$ $\left\{\begin{array}{ll}x\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}0
6. $\left(-1\right){}^{n+1}/{n}^{3}$ $x\left({\pi }^{2}-{x}^{2}\right)/6\pi$
7. $1-\left(-1\right){}^{n}/{n}^{3}$ $x\left(\pi -x\right)/2$
8. $\frac{{\pi }^{2}\left(-1\right){}^{n-1}}{n}-\frac{2\left[1-\left(-1\right){}^{n}\right]}{{n}^{3}}$ ${x}^{2}$
9. $\frac{n}{{n}^{2}+{c}^{2}}\left[1-\left(-1\right){}^{n}{e}^{c\pi }\right]$ ${e}^{cx}$
10. $\frac{n}{{n}^{2}+{c}^{2}}$ $\frac{sinhc\left(\pi -x\right)}{sinhc\pi }$
11. $\frac{n}{{n}^{2}-{k}^{2}}$ with $k\ne 0,1,2,\dots$ $\frac{sink\left(\pi -x\right)}{sink\pi }$
12. $\left\{\begin{array}{ll}\pi /2\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}n=m\hfill \\ 0\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}n\ne m,m=1,2,\dots \hfill \end{array}$ $sin\phantom{\rule{0.2em}{0ex}}mx$
13. $\frac{n}{{n}^{2}-{k}^{2}}\left[1-\left(-1\right){}^{n}cosk\pi \right]$ with $k\ne 1,2,\dots$ (0 if $n=k$ ) $coskx$
14. $\frac{{b}^{n}}{n}$ with $|b|\le 1$ $\frac{2}{\pi }arctan\frac{bsinx}{1-bcosx}$
15. $\frac{1-\left(-1\right){}^{n}}{n}{b}^{n}$ with $|b|\le 1$ $\frac{2}{\pi }arctan\frac{2bsinx}{1-{b}^{2}}$

## 7.6 TABLE OF FOURIER TRANSFORMS

$F\left(\omega \right)=F\left(f\right)\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f\left(x\right){e}^{i\omega x}\phantom{\rule{0.2em}{0ex}}dx$

No.$f\left(x\right)$ $F\left(\omega \right)$
1. $\delta \left(x\right)$ $1/\sqrt{2\pi }$
2. $\delta \left(x-\tau \right)$ ${e}^{i\omega \tau }/\sqrt{2\pi }$
3. ${\delta }^{\left(n\right)}\left(x\right)$ $\left(-i\omega \right){}^{n}/\sqrt{2\pi }$
4. $H\left(x\right)=\left\{\begin{array}{ll}1\hfill & x>0\hfill \\ 0\hfill & x<0\hfill \end{array}$ $-\frac{1}{i\omega \sqrt{2\pi }}+\sqrt{\frac{\pi }{2}}\delta \left(\omega \right)$
5. $sgn\left(x\right)=\left\{\begin{array}{ll}1\hfill & x>0\hfill \\ -1\hfill & x<0\hfill \end{array}$ $-\sqrt{\frac{2}{\pi }}\frac{1}{i\omega }$
6. $\left\{\begin{array}{ll}1\hfill & |x|\phantom{\rule{0.2em}{0ex}}a\hfill \end{array}$ $\sqrt{\frac{2}{\pi }}\frac{sina\omega }{\omega }$
7. $\left\{\begin{array}{ll}e\Omega t\hfill & |x|\phantom{\rule{0.2em}{0ex}}a\hfill \end{array}$ $\sqrt{\frac{2}{\pi }}\frac{sina\left(\Omega +\omega \right)}{\Omega +\omega }$
8. ${e}^{-a|x|}a>0$ $-\sqrt{\frac{2}{\pi }}\frac{a}{{a}^{2}+{\omega }^{2}}$
9. $\frac{sinax}{x}$ $\left\{\begin{array}{ll}\sqrt{\frac{\pi }{2}}\hfill & |\omega |\phantom{\rule{0.2em}{0ex}}a\hfill \end{array}$
10. $\left\{\begin{array}{ll}{e}^{iax}\hfill & pq\hfill \end{array}$ $\frac{i}{\sqrt{2\pi }}{e}^{ip\left(\omega +a\right)}-{e}^{iq\left(\omega +a\right)}/\omega +a$
11. $\left\{\begin{array}{ll}{e}^{-cx+iax}\hfill & x>0\hfill \\ 0\hfill & x<0\hfill \end{array}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\left(c>0\right)$ $\frac{i}{\sqrt{2\pi }\left(\omega +a+ic\right)}$
12. ${e}^{-{px}^{2}}\text{Re}p>0$ $\frac{1}{\sqrt{2p}}{e}^{-{\omega }^{2}/4p}$
13. $cos{px}^{2}$ $\frac{1}{\sqrt{2p}}cos\left(\frac{{\omega }^{2}}{4p}-\frac{\pi }{4}\right)$
14. $sin{px}^{2}$ $\frac{1}{\sqrt{2p}}cos\left(\frac{{\omega }^{2}}{4p}+\frac{\pi }{4}\right)$
15. $|x|{}^{-p}\left(0 $\sqrt{\frac{2}{\pi }}\frac{\Gamma \left(1-p\right)sin\frac{p\pi }{2}}{|\omega |{}^{1-p}}$
16. ${e}^{-a|x|}/\sqrt{|x|}$ $\frac{\sqrt{\sqrt{{a}^{2}+{\omega }^{2}}+a}}{\sqrt{{\omega }^{2}+{a}^{2}}}$
17. $\frac{coshax}{cosh\pi x}\left(-\pi $\sqrt{\frac{2}{\pi }}\frac{cos\frac{a}{2}cosh\frac{\omega }{2}}{cosa+cosh\omega }$
18. $\frac{sinhax}{sinh\pi x}\left(-\pi $\frac{1}{\sqrt{2\pi }}\frac{sina}{cosa+cosh\omega }$
19. $\left\{\begin{array}{ll}\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}\hfill & |x|a\hfill \end{array}$ $\sqrt{\frac{\pi }{2}}{J}_{0}\left(a\omega \right)$
20. $\frac{sin\left[b\sqrt{{a}^{2}+{x}^{2}}\right]}{\sqrt{{a}^{2}+{x}^{2}}}$ $\left\{\begin{array}{ll}0\hfill & |\omega |>b\hfill \\ \sqrt{\frac{\pi }{2}}{J}_{0}\left(a\sqrt{{b}^{2}-{\omega }^{2}}\right)\hfill & |\omega |

## 7.7 TABLE OF FUNCTIONAL RELATIONS FOR FOURIER TRANSFORMS

$F\left(\omega \right)=F\left(f\right)\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f\left(x\right){e}^{i\omega x}\phantom{\rule{0.2em}{0ex}}dx$

No.$f\left(x\right)$ $F\left(\omega \right)$
1. $ag\left(x\right)+bh\left(x\right)$ $aG\left(\omega \right)+bH\left(\omega \right)$
2. $f\left(ax\right)a\ne 0,\text{Im}a=0$ $\frac{1}{|a|}F\left(\frac{\omega }{a}\right)$
3. $f\left(-x\right)$ $F\left(-\omega \right)$
4. $\stackrel{̲}{f\left(x\right)}$ $\stackrel{̲}{F\left(-\omega \right)}$
5. $f\left(x-\tau \right)\text{Im}\tau =0$ ${e}^{i\omega \tau }F\left(\omega \right)$
6. ${e}^{i\Omega x}f\left(x\right)\text{Im}\Omega =0$ $F\left(\omega +\Omega \right)$
7. $F\left(x\right)$ $f\left(-\omega \right)$
8. $\frac{{d}^{n}}{{dx}^{n}}f\left(x\right)$ $\left(-i\omega \right){}^{n}F\left(\omega \right)$
9. $\left(ix\right){}^{n}f\left(x\right)$ $\frac{{d}^{n}}{d{\omega }^{n}}F\left(\omega \right)$

## 7.8 TABLE OF MULTIDIMENSIONAL FOURIER TRANSFORMS

$F\left(u\right)={\left(2\pi \right)}^{-n/2}\int \underset{{ℝ}^{n}}{\cdots }\int f\left(x\right){e}^{i\left(x\cdot u\right)}dx$

No.$f\left(x\right)$ $F\left(u\right)$
Two dimensions: let $x=\left(x,y\right)$ and $u=\left(u,v\right).$
1. $f\left(ax,by\right)$ $\frac{1}{|ab|}F\left(\frac{u}{a},\frac{v}{b}\right)$
2. $f\left(x-a,y-b\right)$ ${e}^{i\left(au+bv\right)}F\left(u,v\right)$
3. ${e}^{i\left(ax+by\right)}f\left(x,y\right)$ $F\left(u+a,v+b\right)$
4. $F\left(x,y\right)$ $\left(2\pi \right){}^{2}F\left(-u,-v\right)$
5. $\delta \left(x-a\right)\delta \left(y-b\right)$ $\frac{1}{2\pi }{e}^{-i\left(au+bv\right)}$
6. ${e}^{-{x}^{2}/4a-{y}^{2}/4b}a,b>0$ $2\sqrt{ab}\phantom{\rule{0.2em}{0ex}}{e}^{-{au}^{2}-{bv}^{2}}$
7. $\left\{\begin{array}{ll}1\hfill & |x| $\frac{2sinausinbv}{\pi uv}$
8. $\left\{\begin{array}{ll}1\hfill & |x| $\frac{2sinau}{\pi uv}\delta \left(v\right)$
9. $\left\{\begin{array}{ll}1\hfill & {x}^{2}+{y}^{2}<{a}^{2}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\left(\text{circle}\right)$ $\frac{{aJ}_{1}\left(a\sqrt{{u}^{2}+{v}^{2}}\right)}{\sqrt{{u}^{2}+{v}^{2}}}$
Three dimensions: let $x=\left(x,y,z\right)$ and $u=\left(u,v,w\right).$
10. $\delta \left(x-a\right)\delta \left(y-b\right)\delta \left(z-c\right)$ $\frac{1}{\left(2\pi \right){}^{3/2}}{e}^{-i\left(au+bv+cw\right)}$
11. ${e}^{-{x}^{2}/4a-{y}^{2}/4b-{z}^{2}/4c}a,b,c>0$ ${2}^{3/2}\sqrt{abc}\phantom{\rule{0.2em}{0ex}}{e}^{-{au}^{2}-{bv}^{2}-{cw}^{2}}$
12. $\left\{\begin{array}{ll}1\hfill & |x| ${\left(\frac{2}{\pi }\right)}^{3/2}\frac{sinausinbvsincw}{uvw}$
13. $\left\{\begin{array}{ll}1\hfill & {x}^{2}+{y}^{2}+{z}^{2}<{a}^{2}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\left(\text{ball}\right)$ $\frac{sina\rho -a\rho cosa\rho }{\sqrt{2\pi }{\rho }^{3}}\mathrm{ }{\rho }^{2}={u}^{2}+{v}^{2}+{w}^{2}$

## 7.9 TABLE OF LAPLACE TRANSFORMS

$F\left(s\right)=L\left(f\right)\left(s\right)={\int }_{0}^{\infty }f\left(t\right){e}^{-st}\phantom{\rule{0.2em}{0ex}}dt.$

No.$f\left(t\right)$ $F\left(s\right)$
1. $\delta \left(t\right),\text{delta function}$ 1
2. $H\left(t\right),\text{unit step function or Heaviside function}$ $1/s$
3. $t$ $1/{s}^{2}$
4. $\frac{{t}^{n-1}}{\left(n-1\right)!}$ $1/{s}^{n}\left(n=1,2,\dots \right)$
5. $1/\sqrt{\pi t}$ $1/\sqrt{s}$
6. $2\sqrt{t/\pi }$ ${s}^{-3/2}$
7. $\frac{{2}^{n}{t}^{n-1/2}}{\sqrt{\pi }\left(2n-1\right)!!}$ ${s}^{-\left(n+1/2\right)}\left(n=1,2,\dots \right)$
8. ${t}^{k-1}$ $\frac{\Gamma \left(k\right)}{{s}^{k}}\left(k>0\right)$
9. ${e}^{at}$ $\frac{1}{s-a}$
10. $t{e}^{at}$ $\frac{1}{\left(s-a\right){}^{2}}$
11. $\frac{1}{\left(n-1\right)!}{t}^{n-1}{e}^{at}$ $\frac{1}{\left(s-a\right){}^{n}}\left(n=1,2,\dots \right)$
12. ${t}^{k-1}{e}^{at}$ $\frac{\Gamma \left(k\right)}{\left(s-a\right){}^{k}}\left(k>0\right)$
13. $\frac{1}{a-b}\left({e}^{at}-{e}^{bt}\right)$ $\frac{1}{\left(s-a\right)\left(s-b\right)}\left(a\ne b\right)$
14. $\frac{1}{a-b}\left({ae}^{at}-{be}^{bt}\right)$ $\frac{s}{\left(s-a\right)\left(s-b\right)}\left(a\ne b\right)$
15. $-\frac{\left(b-c\right){e}^{at}+\left(c-a\right){e}^{bt}+\left(a-b\right){e}^{ct}}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}$ $\frac{1}{\left(s-a\right)\left(s-b\right)\left(s-c\right)}\left(a,b,c\phantom{\rule{0.2em}{0ex}}\text{distinct}\right)$
16. $\frac{1}{a}sinat$ $\frac{1}{{s}^{2}+{a}^{2}}$
17. $cosat$ $\frac{s}{{s}^{2}+{a}^{2}}$
18. $\frac{1}{a}sinhat$ $\frac{1}{{s}^{2}-{a}^{2}}$
19. $coshat$ $\frac{s}{{s}^{2}-{a}^{2}}$
20. $\frac{1}{{a}^{2}}\left(1-cosat\right)$ $\frac{1}{s\left({s}^{2}+{a}^{2}\right)}$
21. $\frac{1}{{a}^{3}}\left(at-sinat\right)$ $\frac{1}{{s}^{2}\left({s}^{2}+{a}^{2}\right)}$
22. $\frac{1}{2{a}^{3}}\left(sinat-atcosat\right)$ $\frac{1}{\left({s}^{2}+{a}^{2}\right){}^{2}}$
23. $\frac{t}{2a}sinat$ $\frac{s}{\left({s}^{2}+{a}^{2}\right){}^{2}}$
24. $\frac{1}{2a}\left(sinat+atcosat\right)$ $\frac{{s}^{2}}{\left({s}^{2}+{a}^{2}\right){}^{2}}$
25. $tcosat$ $\frac{{s}^{2}-{a}^{2}}{\left({s}^{2}+{a}^{2}\right){}^{2}}$
26. $\frac{cosat-cosbt}{{b}^{2}-{a}^{2}}$ $\frac{s}{\left({s}^{2}+{a}^{2}\right)\left({s}^{2}+{b}^{2}\right)}\left({a}^{2}\ne {b}^{2}\right)$
27. $\frac{1}{b}{e}^{at}sinbt$ $\frac{1}{\left(s-a\right){}^{2}+{b}^{2}}$
28. ${e}^{at}cosbt$ $\frac{s-a}{\left(s-a\right){}^{2}+{b}^{2}}$
29. $sinatcoshat-cosatsinhat$ $\frac{4{a}^{3}}{{s}^{4}+4{a}^{4}}$
30. $\frac{1}{2{a}^{2}}sinatsinhat$ $\frac{s}{{s}^{4}+4{a}^{4}}$
31. $\frac{1}{2{a}^{3}}\left(sinhat-sinat\right)$ $\frac{1}{{s}^{4}-{a}^{4}}$
32. $\frac{1}{2{a}^{2}}\left(coshat-cosat\right)$ $\frac{s}{{s}^{4}-{a}^{4}}$
33. $\left(1+{a}^{2}{t}^{2}\right)sinat-cosat$ $\frac{8{a}^{3}{s}^{2}}{\left({s}^{2}+{a}^{2}\right){}^{3}}$
34. $\frac{{e}^{t}}{n!}\frac{{d}^{n}}{{dt}^{n}}\left({t}^{n}{e}^{-t}\right)$ $\frac{1}{s}{\left(\frac{s-1}{s}\right)}^{n}$
35. $\frac{1}{\sqrt{\pi t}}{e}^{at}\left(1+2at\right)$ $\frac{s}{\left(s-a\right){}^{3/2}}$
36. $\frac{1}{2\sqrt{\pi {t}^{3}}}\left({e}^{bt}-{e}^{at}\right)$ $\sqrt{s-a}-\sqrt{s-b}$
37. $\frac{1}{\sqrt{\pi t}}-{ae}^{{a}^{2}t}\text{erfc}\left(a\sqrt{t}\right)$ $\frac{1}{\sqrt{s}+a}$
38. $\frac{1}{\sqrt{\pi t}}+{ae}^{{a}^{2}t}\text{erf}\left(a\sqrt{t}\right)$ $\frac{\sqrt{s}}{s-{a}^{2}}$
39. $\frac{1}{a}{e}^{{a}^{2}t}\text{erf}\left(a\sqrt{t}\right)$ $\frac{1}{\sqrt{s}\left(s-{a}^{2}\right)}$
40. $\left\{\begin{array}{ll}0\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}0k\hfill \end{array}$ $\frac{{e}^{-ks}}{s}$
41. $\left\{\begin{array}{ll}0\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}0k\hfill \end{array}$ $\frac{{e}^{-ks}}{{s}^{2}}$
42. $\frac{{e}^{-ks}}{{s}^{p}}$   (p > 0)
43. $\left\{\begin{array}{ll}1\hfill & \text{when}\phantom{\rule{0.2em}{0ex}}0k\hfill \end{array}$ $\frac{1-{e}^{-ks}}{s}$
44. $|sinat|$ $\frac{a}{{s}^{2}+{a}^{2}}coth\frac{\pi s}{2a}$
45. ${J}_{0}\left(2\sqrt{at}\right)$ $\frac{1}{s}{e}^{-a/s}$
46. $\frac{1}{\sqrt{\pi t}}cos2\sqrt{at}$ $\frac{1}{\sqrt{s}}{e}^{-a/s}$
47. $\frac{1}{\sqrt{\pi t}}cosh2\sqrt{at}$ $\frac{1}{\sqrt{s}}{e}^{a/s}$
48. $\frac{a}{2\sqrt{\pi t}}{e}^{-{a}^{2}/4t}$ ${e}^{-a\sqrt{s}}\left(a>0\right)$
49. $\frac{1}{\sqrt{\pi t}}{e}^{-{a}^{2}/4t}$ $\frac{1}{\sqrt{s}}{e}^{-a\sqrt{s}}\left(a\ge 0\right)$
50. ${\Gamma }^{\prime }\left(1\right)-log t$ $\frac{1}{s}log s$
51. ${t}^{k-1}\left[\frac{{\Gamma }^{\prime }\left(k\right)}{|\Gamma \left(k\right){|}^{2}}-\frac{logt}{\Gamma \left(k\right)}\right]$ $\frac{1}{{s}^{k}}log s\left(k>0\right)$
52. ${e}^{at}\left[log a-\text{Ei}\left(-at\right)\right]$ $\frac{log s}{s-a}\left(a>0\right)$
53. $\frac{1}{t}\left({e}^{bt}-{e}^{at}\right)$ $log\frac{s-a}{s-b}$
54. $\frac{2}{t}\left(1-cosat\right)$ $log\frac{{s}^{2}+{a}^{2}}{{s}^{2}}$
55. $\frac{2}{t}\left(1-coshat\right)$ $log\frac{{s}^{2}-{a}^{2}}{{s}^{2}}$
56. $\frac{1}{t}sinat$ $arctan\frac{a}{s}$
57. $\frac{1}{a\sqrt{\pi }}{e}^{-{t}^{2}/4{a}^{2}}$ ${e}^{{a}^{2}{s}^{2}}\text{erfc}\left(as\right)\left(a>0\right)$
58. $\text{erf}\left(\frac{t}{2a}\right)$ $\frac{1}{s}{e}^{{a}^{2}{s}^{2}}\text{erfc}\left(as\right)\left(a>0\right)$
59. $\frac{\sqrt{a}}{\pi \sqrt{t}\left(t+a\right)}$ ${e}^{as}\text{erfc}\left(\sqrt{as}\right)\left(a>0\right)$
60. $\frac{1}{\pi t}sin\left(2a\sqrt{t}\right)$ $\text{erf}\left(\frac{a}{\sqrt{s}}\right)$

## 7.10 TABLE OF FUNCTIONAL RELATIONS FOR LAPLACE TRANSFORMS

$F\left(s\right)=L\left(f\right)\left(s\right)={\int }_{0}^{\infty }f\left(t\right){e}^{-st}\phantom{\rule{0.2em}{0ex}}dt.$

No.$f\left(t\right)$ $F\left(s\right)$
1. $af\left(t\right)+bg\left(t\right)$ $aF\left(s\right)+bG\left(s\right)$
2. ${f}^{\prime }\left(t\right)$ $sF\left(s\right)-F\left(0+\right)$
3. ${f}^{″}\left(t\right)$ ${s}^{2}F\left(s\right)-sF\left(0+\right)-{F}^{\prime }\left(0+\right)$
4. ${f}^{\left(n\right)}\left(t\right)$ ${s}^{n}F\left(s\right)-\sum _{k=0}^{n-1}{s}^{n-1-k}{F}^{\left(k\right)}\left(0+\right)$
5. ${\int }_{0}^{t}f\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau$ $\frac{1}{s}F\left(s\right)$
6. ${\int }_{0}^{t}{\int }_{0}^{\tau }f\left(u\right)\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}d\tau$ $\frac{1}{{s}^{2}}F\left(s\right)$
7. ${\int }_{0}^{t}{f}_{1}\left(t-\tau \right){f}_{2}\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau ={f}_{1}*{f}_{2}$ ${F}_{1}\left(s\right){F}_{2}\left(s\right)$
8. $tf\left(t\right)$ $-{F}^{\prime }\left(s\right)$
9. ${t}^{n}f\left(t\right)$ $\left(-1\right){}^{n}{F}^{\left(n\right)}\left(s\right)$
10. $\frac{1}{t}f\left(t\right)$ ${\int }_{s}^{\infty }F\left(z\right)\phantom{\rule{0.2em}{0ex}}dz$
11. ${e}^{at}f\left(t\right)$ $F\left(s-a\right)$
12. $f\left(t-b\right)\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.2em}{0ex}}f\left(t\right)=0\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}t<0$ ${e}^{-bs}F\left(s\right)$
13. $\frac{1}{c}f\left(\frac{t}{c}\right)$ $F\left(cs\right)$
14. $\frac{1}{c}{e}^{bt/c}f\left(\frac{t}{c}\right)$ $F\left(cs-b\right)$
15. $f\left(t+a\right)=f\left(t\right)$ ${\int }_{0}^{a}{e}^{-st}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt/1-{e}^{-as}$
16. $f\left(t+a\right)=-f\left(t\right)$ ${\int }_{0}^{a}{e}^{-st}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt/1+{e}^{-as}$
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