Section: 17 | Special Functions |
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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.

# 8 SPECIAL FUNCTIONS

## 8.1 ORTHOGONAL POLYNOMIALS

1. Legendre  Symbol:${P}_{n}\left(x\right)$   Interval: [ $-1,1$ ]

Differential Equation:$\left(1-{x}^{2}\right){y}^{\prime \prime }-2\phantom{\rule{0.2em}{0ex}}{xy}^{\prime }+n\left(n+1\right)y=0$

Explicit Expression:${P}_{n}\left(x\right)=\frac{1}{{2}^{n}}\sum _{m=0}^{\left[n/2\right]}{\left(-1\right)}^{m}\left(\begin{array}{c}n\\ m\end{array}\right)\left(\begin{array}{c}2n-2m\\ n\end{array}\right){x}^{n-2m}$

Recurrence Relation:$\left(n+1\right){P}_{n+1}\left(x\right)=\left(2n+1\right){xP}_{n}\left(x\right)-{nP}_{n-1}\left(x\right)$

Weight:  1

Standardization:${P}_{n}\left(1\right)=1$

Norm:${\int }_{-1}^{+1}\left[{P}_{n}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{2}{2n+1}$

Rodrigues' Formula:${P}_{n}\left(x\right)=\frac{\left(-1\right){}^{n}}{{2}^{n}n!}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-{x}^{2}\right){}^{n}\right\}$

Generating Function:$\begin{array}{c}{R}^{-1}=\sum _{n=0}^{\infty }{P}_{n}\left(x\right){z}^{n};-1

Inequality:$|{P}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1,\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1$ .

2. Tschebysheff, First Kind  Symbol:${T}_{n}\left(x\right)$   Interval:[−1, 1]

Differential Equation:$\left(1-{x}^{2}\right)y-x{y}^{\prime }+{n}^{2}y=0$

Explicit Expression:$\frac{n}{2}\sum _{m=0}^{\left[n/2\right]}{\left(-1\right)}^{m}\frac{\left(n-m-1\right)!}{m!\left(n-2m\right)!}{\left(2x\right)}^{n-2m}=cos\left(narccosx\right)={T}_{n}\left(x\right)$

Recurrence Relation:${T}_{n+1}\left(x\right)=2{xT}_{n}\left(x\right)-{T}_{n-1}\left(x\right)$

Weight:$\left(1-{x}^{2}\right){}^{-1/2}$

Standardization:${T}_{n}\left(1\right)=1$

Norm:${\int }_{-1}^{+1}\left(1-{x}^{2}\right){}^{-1/2}\left[{T}_{n}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\left\{\begin{array}{cc}\hfill \pi /2,\hfill & n\ne 0\\ \hfill \pi ,\hfill & n=0\end{array}$

Rodrigues' Formula:$\frac{\left(-1\right){}^{n}\left(1-{x}^{2}\right){}^{1/2}\sqrt{\pi }}{{2}^{n+1}\Gamma \left(n+\frac{1}{2}\right)}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-{x}^{2}\right){}^{n-\left(1/2\right)}\right\}={T}_{n}\left(x\right)$

Generating Function:$\frac{1-xz}{1-2xz-{z}^{2}}=\sum _{n=0}^{\infty }{T}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}{z}^{n},\phantom{\rule{0.2em}{0ex}}-1

Inequality:$|{T}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1,\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}x\le 1$ .

3. Tschebysheff, Second KindSymbol${U}_{n}\left(x\right)$   Interval: [−1, 1]

Differential Equation:$\left(1-{x}^{2}\right){y}^{″}-3x{y}^{\prime }+n\left(n+2\right)y=0$

Explicit Expression:$\begin{array}{c}{U}_{n}\left(x\right)=\sum _{m=0}^{\left[n/2\right]}\left(-1\right){}^{m}\frac{\left(m-n\right)!}{m!\left(n-2m\right)!}\left(2x\right){}^{n-2m}\hfill \\ \multicolumn{1}{c}{{U}_{n}\left(cos\theta \right)=\frac{sin\left[\left(n+1\right)\theta \right]}{sin\theta }}\\ \multicolumn{1}{c}{}\end{array}$

Recurrence Relation:${U}_{n+1}\left(x\right)=2{xU}_{n}\left(x\right)-{U}_{n-1}\left(x\right)$

Weight:$\left(1-{x}^{2}\right){}^{1/2}$

Standardization:${U}_{n}\left(1\right)=n+1$

Norm:${\int }_{-1}^{+1}\left(1-{x}^{2}\right){}^{1/2}\left[{U}_{n}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{\pi }{2}$

Rodrigues' Formula:${U}_{n}\left(x\right)=\frac{\left(-1\right){}^{n}\left(n+1\right)\sqrt{\pi }}{\left(1-{x}^{2}\right){}^{1/2}{2}^{n+1}\Gamma \left(n+\frac{3}{2}\right)}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-{x}^{2}\right){}^{n+\left(1/2\right)}\right\}$

Generating Function:$\frac{1}{1-2xz+{z}^{2}}=\sum _{n=0}^{\infty }{U}_{n}\left(x\right){z}^{n},-1

Inequality:$|{U}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}+1,\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\le \phantom{\rule{0.2em}{0ex}}1.$

4. Jacobi  Symbol:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$   Interval: [−1, 1]

Differential Equation:$\left(1-{x}^{2}\right){y}^{\prime \prime }+\left[\beta -\alpha -\left(\alpha +\beta +2\right)x\right]{y}^{\prime }+n\left(n+\alpha +\beta +1\right)y=0$

Explicit Expression:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{1}{{2}^{n}}\sum _{m=0}^{n}\left(\begin{array}{c}n+\alpha \\ m\end{array}\right)\left(\begin{array}{c}n+\beta \\ n-m\end{array}\right){\left(x-1\right)}^{n-m}{\left(x+1\right)}^{m}$

Recurrence Relation:$\begin{array}{cc}\hfill & 2\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+\alpha +\beta +1\right)\phantom{\rule{0.2em}{0ex}}\left(2n+\alpha +\beta \right){P}_{n+1}^{\left(\alpha ,\beta \right)}\left(x\right)\hfill \\ \multicolumn{1}{c}{}& \mathrm{ }=\left(2n+\alpha +\beta +1\right)\left[\left({\alpha }^{2}-{\beta }^{2}\right)+\left(2n+\alpha +\beta +2\right)\hfill \\ \multicolumn{1}{c}{}& \mathrm{ }×\left(2n+\alpha +\beta \right)x\right]{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)\hfill \\ \multicolumn{1}{c}{}& \mathrm{ }-2\left(n+\alpha \right)\phantom{\rule{0.2em}{0ex}}\left(n+\beta \right)\phantom{\rule{0.2em}{0ex}}\left(2n+\alpha +\beta +2\right){P}_{n-1}^{\left(\alpha ,\beta \right)}\left(x\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$

Weight:$\left(1-x\right){}^{\alpha }\left(1+x\right){}^{\beta };\alpha ,\beta >1$

Standardization:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{n+\alpha }{n}$

Norm:${\int }_{-1}^{+1}\left(1-x\right){}^{\alpha }\left(1+x\right){}^{\beta }\left[{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{{2}^{\alpha +\beta +1}\Gamma \left(n+\alpha +1\right)\Gamma \left(n+\beta +1\right)}{\left(2n+\alpha +\beta +1\right)n!\Gamma \left(n+\alpha +\beta +1\right)}$

Rodrigues' Formula:${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{\left(-1\right){}^{n}}{{2}^{n}n!\left(1-x\right){}^{\alpha }\left(1+x\right){}^{\beta }}\frac{{d}^{n}}{{dx}^{n}}\left\{\left(1-x\right){}^{n+\alpha }\left(1+x\right){}^{n+\beta }\right\}$

Generating Function:$\begin{array}{c}{R}^{-1}\left(1-z+R\right){}^{-\alpha }\left(1+z+R\right){}^{-\beta }=\sum _{n=0}^{\infty }{2}^{-\alpha -\beta }{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right){z}^{n},\\ R=\sqrt{1-2xz+{z}^{2}},\mathrm{ }|z|<1\end{array}$

Inequality:$\underset{-1\le x\le 1}{max}|{P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)|=\left\{\begin{array}{l}\left(\begin{array}{l}n+q\hfill \\ n\hfill \end{array}\right)~{n}^{q}\phantom{\rule{0.2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}q=max\left(\alpha ,\beta \right)\ge -\frac{1}{2}\hfill \\ |{P}_{n}^{\left(\alpha ,\beta \right)}\left({x}^{\prime }\right)|~{n}^{-1/2}\phantom{\rule{0.2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}q<-\frac{1}{2}\hfill \\ {x}^{\prime }\phantom{\rule{0.2em}{0ex}}\text{is one of the two maximum points nearest}\hfill \\ \frac{\beta -\alpha }{\alpha +\beta +1}\hfill \end{array}$

5. Generalized Laguerre  Symbol:${L}_{n}^{\left(\alpha \right)}\left(x\right)$   Interval:$\left[0,\infty \right]$

Differential Equation:${xy}^{\prime \prime }+\left(\alpha +1-x\right){y}^{\prime }+ny=0$

Explicit Expression:${L}_{n}^{\left(\alpha \right)}\left(x\right)=\sum _{m=0}^{n}{\left(-1\right)}^{m}\left(\begin{array}{c}n+\alpha \\ n-m\end{array}\right)\frac{1}{m!}{x}^{m}$

Recurrence Relation:$\left(n+1\right){L}_{n}^{\left(\alpha \right)}+1\left(x\right)=\left[\left(2n+\alpha +1\right)-x\right]{L}_{n}^{\left(\alpha \right)}\left(x\right)-\left(n+\alpha \right){L}_{n}^{\left(\alpha \right)}-1\left(x\right)$

Weight:${x}^{\alpha }{e}^{-x},\alpha >-1$

Standardization:${L}_{n}^{\left(\alpha \right)}\left(x\right)=\frac{\left(-1\right){}^{n}}{n!}{x}^{n}+\dots$

Norm:${\int }_{0}^{\infty }{x}^{\alpha }{e}^{-x}\left[{L}_{n}^{\left(\alpha \right)}\left(x\right)\right]{}^{2}\phantom{\rule{0.2em}{0ex}}dx=\frac{\Gamma \left(n+\alpha +1\right)}{n!}$

Rodrigues' Formula:${L}_{n}^{\left(\alpha \right)}\left(x\right)=\frac{1}{n!{x}^{\alpha }{e}^{-x}}\frac{{d}^{n}}{{dx}^{n}}\left\{{x}^{n+\alpha }{e}^{-x}\right\}$

Generating Function:$\left(1-z\right){}^{-\alpha -1}exp\left(\frac{xz}{z-1}\right)=\sum _{n=0}^{\infty }{L}_{n}^{\left(\alpha \right)}\left(x\right){z}^{n}$

Inequality:$\begin{array}{c}{L}_{n}^{\left(\alpha \right)}\left(x\right)\le \frac{\Gamma \left(n+\alpha +1\right)}{n!\Gamma \left(\alpha +1\right)}{e}^{x/2};\mathrm{ }\begin{array}{c}x\ge 0\\ \alpha >0\end{array}\\ |{L}_{n}^{\left(a\right)}\left(x\right)|\le \left[2-\frac{\Gamma \left(\alpha +n+1\right)}{n!\Gamma \left(\alpha +1\right)}\right]{e}^{x/2};\mathrm{ }\begin{array}{c}\hfill x\ge 0\hfill \\ \hfill -1<\alpha <0\hfill \end{array}\end{array}$

6. Hermite  Symbol:${H}_{n}\left(x\right)$   Interval:$\left[-\infty ,\infty \right]$

Differential Equation:${y}^{\prime \prime }-2{xy}^{\prime }+2ny=0$

Explicit Expression:${H}_{n}\left(x\right)=\sum _{m=0}^{\left[n/2\right]}\frac{\left(-1\right){}^{m}n!\left(2x\right){}^{n-2m}}{m!\left(n-2m\right)!}$

Recurrence Relation:${H}_{n+1}\left(x\right)=2{xH}_{n}\left(x\right)-2{nH}_{n-1}\left(x\right)$

Weight:${e}^{-{x}^{2}}$

Standardization:${H}_{n}\left(1\right)={2}^{n}{x}^{n}+\dots$

Norm:${\int }_{-\infty }^{\infty }{e}^{-{x}^{2}}{\left[{H}_{n}\left(x\right)\right]}^{2}dx={2}^{n}n!\sqrt{\pi }$

Rodrigues' Formula:${H}_{n}\left(x\right)=\left(-1\right){}^{n}{e}^{{x}^{2}}\frac{{d}^{n}}{{dx}^{n}}\left({e}^{-{x}^{2}}\right)$

Generating Function:${e}^{-{x}^{2}+2zx}=\sum _{n=0}^{\infty }{H}_{n}\left(x\right)\frac{{z}^{n}}{n!}$

Inequality:$|{H}_{n}\left(x\right)|{e}^{{x}^{2}/2}k{2}^{n/2}\sqrt{n!}\phantom{\rule{0.2em}{0ex}}k\approx 1.086435$

## 8.2 TABLES OF ORTHOGONAL POLYNOMIALS

In the following, $\left\{{H}_{n},{L}_{n},{P}_{n},{T}_{n},{U}_{n}\right\}$ represent the ${n}^{\text{th}}$ order Hermite, Laguerre, Legendre, Tschebysheff (first kind), and Tschebysheff (second kind) polynomials.

 ${H}_{0}=1$ ${x}^{10}=\left(30240{H}_{0}+75600{H}_{2}+25200{H}_{4}+2520{H}_{6}+90{H}_{8}+{H}_{10}\right)/1024$ ${H}_{1}=2x$ ${x}^{9}=\left(15120{H}_{1}+10080{H}_{3}+1512{H}_{5}+72{H}_{7}+{H}_{9}\right)/512$ ${H}_{2}=4{x}^{2}-2$ ${x}^{8}=\left(1680{H}_{0}+3360{H}_{2}+840{H}_{4}+56{H}_{6}+{H}_{8}\right)/256$ ${H}_{3}=8{x}^{3}-12x$ ${x}^{7}=\left(840{H}_{1}+420{H}_{3}+42{H}_{5}+{H}_{7}\right)/128$ ${H}_{4}=16{x}^{4}-48{x}^{2}+12$ ${x}^{6}=\left(120{H}_{0}+180{H}_{2}+30{H}_{4}+{H}_{6}\right)/64$ ${H}_{5}=32{x}^{5}-160{x}^{3}+120x$ ${x}^{5}=\left(60{H}_{1}+20{H}_{3}+{H}_{5}\right)/32$ ${H}_{6}=64{x}^{6}-480{x}^{4}+720{x}^{2}-120$ ${x}^{4}=\left(12{H}_{0}+12{H}_{2}+{H}_{4}\right)/16$ ${H}_{7}=128{x}^{7}-1344{x}^{5}+3360{x}^{3}-1680x$ ${x}^{3}=\left(6{H}_{1}+{H}_{3}\right)/8$ ${H}_{8}=256{x}^{8}-3584{x}^{6}+13440{x}^{4}-13440{x}^{2}+1680$ ${x}^{2}=\left(2{H}_{0}+{H}_{2}\right)/4$ ${H}_{9}=512{x}^{9}-9216{x}^{7}+48384{x}^{5}-80640{x}^{3}+30240x$ $x=\left({H}_{1}\right)/2$ ${H}_{10}=1024{x}^{10}-23040{x}^{8}+161280{x}^{6}-403200{x}^{4}+302400{x}^{2}-30240$ $1={H}_{0}$ ${L}_{0}=1$ ${x}^{6}=720{L}_{0}-4320{L}_{1}+10800{L}_{2}-14400{L}_{3}+10800{L}_{4}-4320{L}_{5}+720{L}_{6}$ ${L}_{1}=-x+1$ ${x}^{5}=120{L}_{0}-600{L}_{1}+1200{L}_{2}-1200{L}_{3}+600{L}_{4}-120{L}_{5}$ ${L}_{2}=\left({x}^{2}-4x+2\right)/2$ ${x}^{4}=24{L}_{0}-96{L}_{1}+144{L}_{2}-96{L}_{3}+24{L}_{4}$ ${L}_{3}=\left(-{x}^{3}+9{x}^{2}-18x+6\right)/6$ ${x}^{3}=6{L}_{0}-18{L}_{1}+18{L}_{2}-6{L}_{3}$ ${L}_{4}=\left({x}^{4}-16{x}^{3}+72{x}^{2}-96x+24\right)/24$ ${x}^{2}=2{L}_{0}-4{L}_{1}+2{L}_{2}$ ${L}_{5}=\left(-{x}^{5}+25{x}^{4}-200{x}^{3}+600{x}^{2}-600x+120\right)/120$ $x={L}_{0}-{L}_{1}$ ${L}_{6}=\left({x}^{6}-36{x}^{5}+450{x}^{4}-2400{x}^{3}+5400{x}^{2}-4320x+720\right)/720$ $1={L}_{0}$ ${P}_{0}=1$ ${x}^{10}=\left(4199{P}_{0}+16150{P}_{2}+15504{P}_{4}+7904{P}_{6}+2176{P}_{8}+256{P}_{10}\right)/46189$ ${P}_{1}=x$ ${x}^{9}=\left(3315{P}_{1}+4760{P}_{3}+2992{P}_{5}+960{P}_{7}+128{P}_{9}\right)/12155$ ${P}_{2}=\left(3{x}^{2}-1\right)/2$ ${x}^{8}=\left(715{P}_{0}+2600{P}_{2}+2160{P}_{4}+832{P}_{6}+128{P}_{8}\right)/6435$ ${P}_{3}=\left(5{x}^{3}-3x\right)/2$ ${x}^{7}=\left(143{P}_{1}+182{P}_{3}+88{P}_{5}+16{P}_{7}\right)/429$ ${P}_{4}=\left(35{x}^{4}-30{x}^{2}+3\right)/8$ ${x}^{6}=\left(33{P}_{0}+110{P}_{2}+72{P}_{4}+16{P}_{6}\right)/231$ ${P}_{5}=\left(63{x}^{5}-70{x}^{3}+15x\right)/8$ ${x}^{5}=\left(27{P}_{1}+28{P}_{3}+8{P}_{5}\right)/63$ ${P}_{6}=\left(231{x}^{6}-315{x}^{4}+105{x}^{2}-5\right)/16$ ${x}^{4}=\left(7{P}_{0}+20{P}_{2}+8{P}_{4}\right)/35$ ${P}_{7}=\left(429{x}^{7}-693{x}^{5}+315{x}^{3}-35x\right)/16$ ${x}^{3}=\left(3{P}_{1}+2{P}_{3}\right)/5$ ${P}_{8}=\left(6435{x}^{8}-12012{x}^{6}+6930{x}^{4}-1260{x}^{2}+35\right)/128$ ${x}^{2}=\left({P}_{0}+2{P}_{2}\right)/3$ ${P}_{9}=\left(12155{x}^{9}-25740{x}^{7}+18018{x}^{5}-4620{x}^{3}+315x\right)/128$ $x={P}_{1}$ ${P}_{10}=\left(46189{x}^{10}-109395{x}^{8}+90090{x}^{6}-30030{x}^{4}+3465{x}^{2}-63\right)/256$ $1={P}_{0}$ ${T}_{0}=1$ ${x}^{10}=\left(126{T}_{0}+210{T}_{2}+120{T}_{4}+45{T}_{6}+10{T}_{8}+{T}_{10}\right)/512$ ${T}_{1}=x$ ${x}^{9}=\left(126{T}_{1}+84{T}_{3}+36{T}_{5}+9{T}_{7}+{T}_{9}\right)/256$ ${T}_{2}=2{x}^{2}-1$ ${x}^{8}=\left(35{T}_{0}+56{T}_{2}+28{T}_{4}+8{T}_{6}+{T}_{8}\right)/128$ ${T}_{3}=4{x}^{3}-3x$ ${x}^{7}=\left(35{T}_{1}+21{T}_{3}+7{T}_{5}+{T}_{7}\right)/64$ ${T}_{4}=8{x}^{4}-8{x}^{2}+1$ ${x}^{6}=\left(10{T}_{0}+15{T}_{2}+6{T}_{4}+{T}_{6}\right)/32$ ${T}_{5}=16{x}^{5}-20{x}^{3}+5x$ ${x}^{5}=\left(10{T}_{1}+5{T}_{3}+{T}_{5}\right)/16$ ${T}_{6}=32{x}^{6}-48{x}^{4}+18{x}^{2}-1$ ${x}^{4}=\left(3{T}_{0}+4{T}_{2}+{T}_{4}\right)/8$ ${T}_{7}=64{x}^{7}-112{x}^{5}+56{x}^{3}-7x$ ${x}^{3}=\left(3{T}_{1}+{T}_{3}\right)/4$ ${T}_{8}=128{x}^{8}-256{x}^{6}+160{x}^{4}-32{x}^{2}+1$ ${x}^{2}=\left({T}_{0}+{T}_{2}\right)/2$ ${T}_{9}=256{x}^{9}-576{x}^{7}+432{x}^{5}-120{x}^{3}+9x$ $x={T}_{1}$ ${T}_{10}=512{x}^{10}-1280{x}^{8}+1120{x}^{6}-400{x}^{4}+50{x}^{2}-1$ $1={T}_{0}$ ${U}_{0}=1$ ${x}^{10}=\left(42{U}_{0}+90{U}_{2}+75{U}_{4}+35{U}_{6}+9{U}_{8}+{U}_{10}\right)/1024$ ${U}_{1}=2x$ ${x}^{9}=\left(42{U}_{1}+48{U}_{3}+27{U}_{5}+8{U}_{7}+{U}_{9}\right)/512$ ${U}_{2}=4{x}^{2}-1$ ${x}^{8}=\left(14{U}_{0}+28{U}_{2}+20{U}_{4}+7{U}_{6}+{U}_{8}\right)/256$ ${U}_{3}=8{x}^{3}-4x$ ${x}^{7}=\left(14{U}_{1}+14{U}_{3}+6{U}_{5}+{U}_{7}\right)/128$ ${U}_{4}=16{x}^{4}-12{x}^{2}+1$ ${x}^{6}=\left(5{U}_{0}+9{U}_{2}+5{U}_{4}+{U}_{6}\right)/64$ ${U}_{5}=32{x}^{5}-32{x}^{3}+6x$ ${x}^{5}=\left(5{U}_{1}+4{U}_{3}+{U}_{5}\right)/32$ ${U}_{6}=64{x}^{6}-80{x}^{4}+24{x}^{2}-1$ ${x}^{4}=\left(2{U}_{0}+3{U}_{2}+{U}_{4}\right)/16$ ${U}_{7}=128{x}^{7}-192{x}^{5}+80{x}^{3}-8x$ ${x}^{3}=\left(2{U}_{1}+{U}_{3}\right)/8$ ${U}_{8}=256{x}^{8}-448{x}^{6}+240{x}^{4}-40{x}^{2}+1$ ${x}^{2}=\left({U}_{0}+{U}_{2}\right)/4$ ${U}_{9}=512{x}^{9}-1024{x}^{7}+672{x}^{5}-160{x}^{3}+10x$ $x=\left({U}_{1}\right)/2$ ${U}_{10}=1024{x}^{10}-2304{x}^{8}+1792{x}^{6}-560{x}^{4}+60{x}^{2}-1$ $1={U}_{0}$

## 8.3 BESSEL FUNCTIONS

1. Bessel's differential equation for a real variable $x$ is ${x}^{2}\frac{{d}^{2}y}{{dx}^{2}}+x\frac{dy}{dx}+\left({x}^{2}-{n}^{2}\right)y=0$
2. When $n$ is not an integer, two independent solutions of the equation are ${J}_{n}\left(x\right)$ and ${J}_{-n}\left(x\right)$ where Jn(x)=∑k=0∞(−1)kk!Γ(n+k+1)(x2)n+2k
3. If $n$ is an integer then ${J}_{n}\left(x\right)=\left(-1\right){}^{n}{J}_{n}\left(x\right)$ , where ${J}_{n}\left(x\right)=\frac{{x}^{n}}{{2}^{n}n!}\left\{1-\frac{{x}^{2}}{{2}^{2}·1!\left(n+1\right)}+\frac{{x}^{4}}{{2}^{4}·2!\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+2\right)}+\frac{{x}^{6}}{{2}^{6}·3!\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+2\right)\phantom{\rule{0.2em}{0ex}}\left(n+3\right)}+\dots \right\}$
4. For $n=0$ and $n=1$ , this formula becomes $\begin{array}{cc}{J}_{0}\left(x\right)\hfill & =1-\frac{{x}^{2}}{{2}^{2}\left(1!\right){}^{2}}+\frac{{x}^{4}}{{2}^{4}\left(2!\right){}^{2}}-\frac{{x}^{6}}{{2}^{6}\left(3!\right){}^{2}}+\frac{{x}^{8}}{{2}^{8}\left(4!\right){}^{2}}-\dots \hfill \\ \multicolumn{1}{c}{{J}_{1}\left(x\right)}& =\frac{x}{2}-\frac{{x}^{3}}{{2}^{3}·1!2!}+\frac{{x}^{5}}{{2}^{5}·2!3!}-\frac{{x}^{7}}{{2}^{7}·3!4!}+\frac{{x}^{9}}{{2}^{9}·4!5!}-\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$
5. Table of zeros for ${J}_{0}\left(x\right)$ and ${J}_{1}\left(x\right)$ . Define $\left\{{\alpha }_{n},{\beta }_{n}\right\}$ by ${J}_{0}\left({\alpha }_{n}\right)=0$ and ${J}_{1}\left({\beta }_{n}\right)=0$ .
Roots ${\alpha }_{n}$ ${J}_{1}\left({\alpha }_{n}\right)$ Roots ${\beta }_{n}$ ${J}_{0}\left({\beta }_{n}\right)$
2.4048 0.5191 0.0000 1.0000
5.5201 $-0.3403$ 3.8317 $-0.4028$
8.6537 0.2715 7.0156 0.3001
11.7915 $-0.2325$ 10.1735 $-0.2497$
14.9309 0.2065 13.3237 0.2184
18.0711 $-0.1877$ 16.4706 $-0.1965$
21.2116 0.1733 19.6159 0.1801
6. Recurrence formulas $\begin{array}{cc}{J}_{n-1}\left(x\right)+{J}_{n+1}\left(x\right)=\frac{2n}{x}{J}_{n}\left(x\right)\hfill & {nJ}_{n}\left(x\right)+{xJ}_{n}^{\prime }\left(x\right)={xJ}_{n-1}\left(x\right)\hfill \\ \multicolumn{1}{c}{{J}_{n-1}\left(x\right)-{J}_{n+1}\left(x\right)=2{J}_{n}^{\prime }\left(x\right)}& {nJ}_{n}\left(x\right)-{xJ}_{n}^{\prime }\left(x\right)={xJ}_{n+1}\left(x\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$
7. If ${J}_{n}$ is written for ${J}_{n}\left(x\right)$ and ${J}_{n}^{\left(k\right)}$ is written for $\frac{{d}^{k}}{{dx}^{k}}\left\{{J}_{n}\left(x\right)\right\}$ , then the following derivative relationships are important $\begin{array}{c}{J}_{0}^{\left(r\right)}=-{J}_{1}^{\left(r-1\right)}\hfill \\ \multicolumn{1}{c}{{J}_{0}^{\left(2\right)}=-{J}_{0}+\frac{1}{x}{J}_{1}=\frac{1}{2}\left({J}_{2}-{J}_{0}\right)}\\ \multicolumn{1}{c}{{J}_{0}^{\left(3\right)}=\frac{1}{x}{J}_{0}+\left(1-\frac{2}{{x}^{2}}\right){J}_{1}=\frac{1}{4}\left(-{J}_{3}+3{J}_{1}\right)}\\ \multicolumn{1}{c}{{J}_{0}^{\left(4\right)}=\left(1-\frac{3}{{x}^{2}}\right){J}_{0}-\left(\frac{2}{x}-\frac{6}{{x}^{3}}\right){J}_{1}=\frac{1}{8}\left({J}_{4}-4{J}_{2}+3{J}_{0}\right),\text{etc}.}\\ \multicolumn{1}{c}{}\end{array}$
8. Half-order Bessel functions $\begin{array}{c}{J}_{\frac{1}{2}}\left(x\right)=\sqrt{\frac{2}{\pi x}}sinx\hfill \\ \multicolumn{1}{c}{{J}_{-\frac{1}{2}}\left(x\right)=\sqrt{\frac{2}{\pi x}}cosx}\\ \multicolumn{1}{c}{{J}_{n+\frac{3}{2}}\left(x\right)=-{x}^{n+\frac{1}{2}}\frac{d}{dx}\left\{{x}^{-\left(n+\frac{1}{2}\right)}{J}_{n+\frac{1}{2}}\left(x\right)\right\}}\\ \multicolumn{1}{c}{{J}_{n-\frac{1}{2}}\left(x\right)={x}^{-\left(n+\frac{1}{2}\right)}\frac{d}{dx}\left\{{x}^{n+\frac{1}{2}}{J}_{n+\frac{1}{2}}\left(x\right)\right\}}\\ \multicolumn{1}{c}{}\end{array}$
$n$ ${\left(\frac{\pi x}{2}\right)}^{\frac{1}{2}}{J}_{n+\frac{1}{2}}\left(x\right)$ ${\left(\frac{\pi x}{2}\right)}^{\frac{1}{2}}{J}_{-\left(n+\frac{1}{2}\right)}\left(x\right)$
0 $sinx$ $cosx$
1 $\frac{sinx}{x}-cosx$ $-\frac{cosx}{x}-sinx$
2 $\left(\frac{3}{{x}^{2}}-1\right)sinx-\frac{3}{x}cosx$ $\left(\frac{3}{{x}^{2}}-1\right)cosx+\frac{3}{x}sinx$
3 $\left(\frac{15}{{x}^{3}}-\frac{6}{x}\right)sinx-\left(\frac{15}{{x}^{2}}-1\right)cosx$ $-\left(\frac{15}{{x}^{3}}-\frac{6}{x}\right)cosx-\left(\frac{15}{{x}^{2}}-1\right)sinx$
9. Additional solutions to Bessel's equation are ${Y}_{n}\left(x\right)$ (also called Weber's function, and sometimes denoted by ${N}_{n}\left(x\right)$ ) and ${H}_{n}^{\left(1\right)}\left(x\right)$ and ${H}_{n}^{\left(2\right)}\left(x\right)$ (also called Hankel functions) These solutions are defined as follows Yn(x)={Jn(x)cos(nπ)−J−n(x)sin(nπ)nnot an integerHn(1)(x)=Jn(x)+iYn(x)limv→nJv(x)cos(vπ)−J−v(x)sin(vπ)nan integerHn(2)(x)=Jn(x)−iYn(x) The additional properties of these functions may all be derived from the above relations and the known properties of ${J}_{n}\left(x\right)$ .
10. Complete solutions to Bessel's equation may be written as ${c}_{1}{J}_{n}\left(x\right)+{c}_{2}{J}_{-n}\left(x\right)$ when $n$ is not an integer or, for any value of $n$ , ${c}_{1}{J}_{n}\left(x\right)+{c}_{2}{Y}_{n}\left(x\right)$ or ${c}_{1}{H}_{n}^{\left(1\right)}x+{c}_{2}{H}_{n}^{\left(2\right)}\left(x\right)$ .
11. The modified (or hyperbolic) Bessel's differential equation is ${x}^{2}\frac{{d}^{2}y}{{dx}^{2}}+x\frac{dy}{dx}-\left({x}^{2}+{n}^{2}\right)y=0$
12. When $n$ is not an integer, two independent solutions of the equation are ${I}_{n}\left(x\right)$ and ${I}_{-n}\left(x\right)$ , where ${I}_{n}\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!\Gamma \left(n+k+1\right)}{\left(\frac{x}{2}\right)}^{n+2k}$
13. If $n$ is an integer, ${I}_{n}\left(x\right)={I}_{-n}\left(x\right)=\frac{{x}^{n}}{{2}^{n}n!}\left(1+\frac{{x}^{2}}{{2}^{2}·1!\left(n+1\right)}+\frac{{x}^{4}}{{2}^{4}·2!\left(n+1\right)\left(n+2\right)}+\frac{{x}^{6}}{{2}^{6}·3!\left(n+1\right)\phantom{\rule{0.2em}{0ex}}\left(n+2\right)\phantom{\rule{0.2em}{0ex}}\left(n+3\right)}+\dots \right)$
14. For $n=0$ and $n=1$ , this formula becomes $\begin{array}{cc}{I}_{0}\left(x\right)\hfill & =1+\frac{{x}^{2}}{{2}^{2}\left(1!\right){}^{2}}+\frac{{x}^{4}}{{2}^{4}\left(2!\right){}^{2}}+\frac{{x}^{6}}{{2}^{6}\left(3!\right){}^{2}}+\frac{{x}^{8}}{{2}^{8}\left(4!\right){}^{2}}+\dots \hfill \\ \multicolumn{1}{c}{{I}_{1}\left(x\right)}& =\frac{x}{2}+\frac{{x}^{3}}{{2}^{3}·1!2!}+\frac{{x}^{5}}{{2}^{5}·2!3!}+\frac{{x}^{7}}{{2}^{7}·3!4!}+\frac{{x}^{9}}{{2}^{9}·4!5!}+\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$
15. Another solution to the modified Bessel's equation is Kn(x)={12πI−n(x)−In(x)sin(nπ)nnot an integerlimv→n12πI−v(x)−Iv(x)sin(vπ)nan integer This function is linearly independent of ${I}_{n}\left(x\right)$ for all values of $n$ . Thus the complete solution to the modified Bessel's equation may be written as ${c}_{1}{I}_{n}\left(x\right)+{c}_{2}{I}_{-n}\left(x\right)\mathrm{ }\text{when}n\phantom{\rule{0.2em}{0ex}}\text{is not an integer}$ or ${c}_{1}{I}_{n}\left(x\right)+{c}_{2}{K}_{n}\left(x\right)\mathrm{ }\text{for any value of}n$
16. The following relations hold among the various Bessel functions: $\begin{array}{c}{I}_{n}\left(z\right)={i}^{-m}{J}_{m}\left(iz\right)\hfill \\ \multicolumn{1}{c}{{Y}_{n}\left(iz\right)=\left(i\right){}^{n+1}{I}_{n}\left(z\right)-\frac{2}{\pi }{i}^{-n}{K}_{n}\left(z\right)}\end{array}$ Most of the properties of the modified Bessel function may be deduced from the known properties of ${J}_{n}\left(x\right)$ by use of these relations and those previously given.
17. Recurrence formulas $\begin{array}{cc}{I}_{n-1}\left(x\right)-{I}_{n+1}\left(x\right)=\frac{2n}{x}{I}_{n}\left(x\right)\hfill & {I}_{n-1}\left(x\right)+{I}_{n+1}\left(x\right)=2{I}_{n}^{\prime }\left(x\right)\hfill \\ \multicolumn{1}{c}{{I}_{n-1}\left(x\right)-\frac{n}{x}{I}_{n}\left(x\right)={I}_{n}^{\prime }\left(x\right)}& {I}_{n}^{\prime }\left(x\right)={I}_{n+1}\left(x\right)+\frac{n}{x}{I}_{n}\left(z\right)\hfill \\ \multicolumn{1}{c}{}\end{array}$

## 8.4 FACTORIAL FUNCTION

For non-negative integers $n$ , the factorial of $n$ , denoted $n!$ , is the product of all positive integers less than or equal to $n$ ; $n!=n·\left(n-1\right)·\left(n-2\right)\dots 2·1$ . If $n$ is a negative integer ( $n=-1,-2,\dots$ ) then $n!=±\infty$ .

Approximations to $n!$ for large $n$ include Stirling's formula $n!\approx \sqrt{2\pi e}{\left(\frac{n}{e}\right)}^{n+\frac{1}{2}},$ and Burnsides's formula $n!\approx \sqrt{2\pi }{\left(\frac{n+\frac{1}{2}}{e}\right)}^{n+\frac{1}{2}}.$

$n$ $n!$ $log{}_{10}n!$ $n$ $n!$ $log{}_{10}n!$
$0$ $1$ $0.00000$ $1$ $1$ $0.00000$
$2$ $2$ $0.30103$ $3$ $6$ $0.77815$
$4$ $24$ $1.38021$ $5$ $120$ $2.07918$
$6$ $720$ $2.85733$ $7$ $5040$ $3.70243$
$8$ $40320$ $4.60552$ $9$ $3.6288×{10}^{5}$ $5.55976$
$10$ $3.6288×{10}^{6}$ $6.55976$ $11$ $3.9917×{10}^{7}$ $7.60116$
$12$ $4.7900×{10}^{8}$ $8.68034$ $13$ $6.2270×{10}^{9}$ $9.79428$
$14$ $8.7178×{10}^{10}$ $10.94041$ $15$ $1.3077×{10}^{12}$ $12.11650$
$16$ $2.0923×{10}^{13}$ $13.32062$ $17$ $3.5569×{10}^{14}$ $14.55107$
$18$ $6.4024×{10}^{15}$ $15.80634$ $19$ $1.2165×{10}^{17}$ $17.08509$
$20$ $2.4329×{10}^{18}$ $18.38612$ $25$ $1.5511×{10}^{25}$ $25.19065$
$30$ $2.6525×{10}^{32}$ $32.42366$ $40$ $8.1592×{10}^{47}$ $47.91165$
$50$ $3.0414×{10}^{64}$ $64.48307$ $60$ $8.3210×{10}^{81}$ $81.92017$
$70$ $1.1979×{10}^{100}$ $100.07841$ $80$ $7.1569×{10}^{118}$ $118.85473$
$90$ $1.4857×{10}^{138}$ $138.17194$ $100$ $9.3326×{10}^{157}$ $157.97000$
$110$ $1.5882×{10}^{178}$ $178.20092$ $120$ $6.6895×{10}^{198}$ $198.82539$
$130$ $6.4669×{10}^{219}$ $219.81069$ $150$ $5.7134×{10}^{262}$ $262.75689$
$500$ $1.2201×{10}^{1134}$ $1134.0864$ $1000$ $4.0239×{10}^{2567}$ $2567.6046$

## 8.5 Gamma Function

Definition:$\Gamma \left(n\right)=\underset{0}{\overset{\infty }{\int }}{t}^{n-1}{e}^{-t}\phantom{\rule{0.2em}{0ex}}dt\mathrm{ }n>0$

Recursion Formula:

Special Values:$\begin{array}{cc}\Gamma \left(1/2\right)\hfill & =\sqrt{\pi }\hfill \\ \multicolumn{1}{c}{\Gamma \left(m+\frac{1}{2}\right)}& =\frac{1·3·5\dots \left(2m-1\right)}{{2}^{m}}\sqrt{\pi }\mathrm{ }m=1,2,3,\dots \hfill \\ \multicolumn{1}{c}{\Gamma \left(-m+\frac{1}{2}\right)}& =\frac{\left(-1\right){}^{m}{2}^{m}\sqrt{\pi }}{1·3·5\dots \left(2m-1\right)}\mathrm{ }m=1,2,3,\dots \hfill \\ \multicolumn{1}{c}{}\end{array}$

Special Formulas:$\begin{array}{cc}\Gamma \left(x+1\right)\hfill & =\underset{k\to \infty }{lim}\frac{1·2·3\dots k}{\left(x+1\right)\phantom{\rule{0.2em}{0ex}}\left(x+2\right)\dots \left(x+k\right)}{k}^{x}\hfill \\ \multicolumn{1}{c}{\frac{1}{\Gamma \left(x\right)}}& ={xe}^{\gamma x}\underset{m=1}{\overset{\infty }{\Pi }}\left\{\left(1+\frac{x}{m}\right){e}^{-x/m}\right\}\mathrm{ }\left(\gamma \phantom{\rule{0.2em}{0ex}}\text{is Euler"s constant)}\hfill \\ \multicolumn{1}{c}{}\end{array}$

Properties:$\begin{array}{cc}{\Gamma }^{\prime }\left(1\right)\hfill & ={\int }_{0}^{\infty }{e}^{\gamma x}\text{ln}x\phantom{\rule{0.2em}{0ex}}dx=-\gamma \hfill \\ \multicolumn{1}{c}{\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}}& =-\gamma +\left(\frac{1}{1}-\frac{1}{x}\right)+\left(\frac{1}{2}-\frac{1}{x+1}\right)+\dots +\left(\frac{1}{n}-\frac{1}{x+n-1}\right)+\dots \hfill \\ \multicolumn{1}{c}{\Gamma \left(x+1\right)}& =\sqrt{2\pi x}\phantom{\rule{0.2em}{0ex}}{x}^{x}{e}^{-x}\left\{1+\frac{1}{12x}+\frac{1}{288{x}^{2}}-\frac{139}{51,840{x}^{3}}+\dots \right\}\mathrm{ }\left(\mathit{Stirling"s asymptotic series}\right)\hfill \end{array}$

$n$ $\Gamma \left(n\right)$ $n$ $\Gamma \left(n\right)$ $n$ $\Gamma \left(n\right)$ $n$ $\Gamma \left(n\right)$
1.00 1.00000 1.25 .90640 1.50 .88623 1.75 .91906
1.01 .99433 1.26 .90440 1.51 .88659 1.76 .92137
1.02 .98884 1.27 .90250 1.52 .88704 1.77 .92376
1.03 .98355 1.28 .90072 1.53 .88757 1.78 .92623
1.04 .97844 1.29 .89904 1.54 .88818 1.79 .92877
1.05 .97350 1.30 .89747 1.55 .88887 1.80 .93138
1.06 .96874 1.31 .89600 1.56 .88964 1.81 .93408
1.07 .96415 1.32 .89464 1.57 .89049 1.82 .93685
1.08 .95973 1.33 .89338 1.58 .89142 1.83 .93969
1.09 .95546 1.34 .89222 1.59 .89243 1.84 .94261
1.10 .95135 1.35 .89115 1.60 .89352 1.85 .94561
1.11 .94740 1.36 .89018 1.61 .89468 1.86 .94869
1.12 .94359 1.37 .88931 1.62 .89592 1.87 .95184
1.13 .93993 1.38 .88854 1.63 .89724 1.88 .95507
1.14 .93642 1.39 .88785 1.64 .89864 1.89 .95838
1.15 .93304 1.40 .88726 1.65 .90012 1.90 .96177
1.16 .92980 1.41 .88676 1.66 .90167 1.91 .96523
1.17 .92670 1.42 .88636 1.67 .90330 1.92 .96877
1.18 .92373 1.43 .88604 1.68 .90500 1.93 .97240
1.19 .92089 1.44 .88581 1.69 .90678 1.94 .97610
1.20 .91817 1.45 .88566 1.70 .90864 1.95 .97988
1.21 .91558 1.46 .88560 1.71 .91057 1.96 .98374
1.22 .91311 1.47 .88563 1.72 .91258 1.97 .98768
1.23 .91075 1.48 .88575 1.73 .91466 1.98 .99171
1.24 .90852 1.49 .88595 1.74 .91683 1.99 .99581
2.00 1.00000

## 8.6 BETA FUNCTION

Definition:$B\left(m,n\right)={\int }_{0}^{1}{t}^{m-1}\left(1-t\right){}^{n-1}dt\mathrm{ }m>0,\phantom{\rule{0.2em}{0ex}}n>0$

Relationship with Gamma function:$B\left(m,n\right)=\frac{\Gamma \left(m\right)\Gamma \left(n\right)}{\Gamma \left(m+n\right)}$

Properties:$\begin{array}{c}B\left(m,n\right)=B\left(n,m\right)\\ B\left(m,n\right)=2{\int }_{0}^{\pi /2}sin{}^{2m-1}\theta cos{}^{2n-1}\theta \phantom{\rule{0.2em}{0ex}}d\theta \\ B\left(m,n\right)={\int }_{0}^{\infty }\frac{{t}^{m-1}}{\left(1+t\right){}^{m+n}}dt\\ B\left(m,n\right)={r}^{n}\left(r+1\right){}^{m}{\int }_{0}^{1}\frac{{t}^{m-1}\left(1-t\right){}^{n-1}}{\left(r+t\right){}^{m+n}}\phantom{\rule{0.2em}{0ex}}dt\end{array}$

## 8.7 ERROR FUNCTION

Definition:$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_{0}^{x}{e}^{-{t}^{2}}dt$

Series:$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}\left(x-\frac{{x}^{3}}{3}+\frac{1}{2!}\frac{{x}^{5}}{5}-\frac{1}{3!}\frac{{x}^{7}}{7}+\dots \right)$

Property:$\text{erf}\left(x\right)=-\text{erf}\left(-x\right)$

Relationship with Normal Probability Function$f\left(t\right)$ :  ${\int }_{0}^{x}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt=\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)$ To evaluate $\text{erf}\left(2.3\right)$ , one proceeds as follows: For $\frac{x}{\sqrt{2}}=2.3$ , one finds $x=\left(2.3\right)\phantom{\rule{0.2em}{0ex}}\left(\sqrt{2}\right)=3.25$ . In the normal probability function table, one finds the entry 0.4994 opposite the value 3.25. Thus $\text{erf}\left(2.3\right)=2\left(0.4994\right)=0.9988$ . $\text{erfc}\left(z\right)=1-\text{erf}\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_{z}^{\infty }{e}^{-{t}^{2}}dt$ is known as the complementary error function.

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