<2-1> (2.2.1) \int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi} <2-2> (2.2.2) \int_{-\infty}^{\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}} <2-3> x=\frac{x^\prime}{\sqrt{a}} <2-4> dx=\frac{1 }{\sqrt{a}} dx^\prime <2-5> x^\prime\left(=\sqrt{a}x\right) <2-6> \int_{-\infty}^{\infty}e^{-ax^2}dx =\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}e^{-x^{\prime2}}dx^\prime <2-7> \int_{-\infty}^{\infty}e^{-ax^2}dx=\frac{1}{\sqrt{a}}\sqrt{\pi}=\sqrt{\frac{\pi}{a}} <2-8> \begin{array}{rl} \displaystyle{\frac{d}{da}\left(\int_{-\infty}^{\infty}e^{-ax^2}dx\right)}&=\displaystyle{\int_{-\infty}^{\infty}\frac{d\left(e^{-ax^2}\right)}{da}dx} \\ \\ &=-\displaystyle{\int_{-\infty}^{\infty}x^2e ^{-ax^2}dx} \end{array} <2-8> \begin{array}{rl} \displaystyle{\frac{d}{da}\left(\int_{-\infty}^{\infty}e^{-ax^2}dx\right)}&=\displaystyle{\int_{-\infty}^{\infty}\frac{d\left(e^{-ax^2}\right)}{da}dx} \\ \\ &=-\displaystyle{\int_{-\infty}^{\infty}x^2e ^{-ax^2}dx} \end{array} <2-9> \begin{array}{rl} \displaystyle{\frac{d}{da}\left( \sqrt{\frac{\pi}{a}} \right)}&=\displaystyle{\sqrt{\pi}\frac{d\left(a^{-1/2}\right)}{da}} \\ \\ &=\displaystyle{-\sqrt{\pi}\frac{a^{-3/2}}{2}} \end{array} <2-10> \int_{-\infty}^{\infty}x^2e ^{-ax^2}dx=\frac{\sqrt{\pi}}{2}a^{-3/2} <2-11> \begin{array}{rl} \displaystyle{\int_{-\infty}^{\infty}x^4e ^{-ax^2}dx} &=\displaystyle{\left(\frac{\sqrt{\pi}}{2}\cdot\frac{3}{2}\right)a^{-5/2}} \\ \\ &=\displaystyle{\frac{1\cdot3\sqrt{\pi}}{2^2}a^{-5/2}} \end{array} <2-12> \begin{array}{rl} \displaystyle{\int_{-\infty}^{\infty}x^6e ^{-ax^2}dx} &=\displaystyle{\left(\frac{\sqrt{\pi}}{2}\cdot\frac{3}{2}\cdot\frac{5}{2}\right)a^{-7/2}} \\ \\ &=\displaystyle{\frac{1\cdot3\cdot5\sqrt{\pi}}{2^3}a^{-7/2}} \end{array} <2-13> \int_{-\infty}^{\infty}x^{2n}e ^{-ax^2}dx =\frac{1\cdot3\cdot5\cdots(2n-1)\sqrt{\pi}}{2^n}a^{-(2n+1)/2} \quad (n=0,1,2,\cdots) <2-14> (2.2.3) \int_{-\infty}^{\infty}x^{2n}e ^{-x^2}dx =\frac{1\cdot3\cdot5\cdots(2n-1)\sqrt{\pi}}{2^n} \quad (n=0,1,2,\cdots) <2-15> x^{2n}e ^{-x^2}\equiv f(x) <2-16> f(-x)=f(x) <2-17> \begin{array}{rl} \displaystyle{\int_{-\infty}^{\infty}x^{2n}e ^{-x^2}dx} &=\displaystyle{\int_{-\infty}^{0}x^{2n}e ^{-x^2}dx+\int_{0}^{\infty}x^{2n}e ^{-x^2}dx} \\ \\ &=\displaystyle{2\int_{0}^{\infty}x^{2n}e ^{-x^2}dx} \end{array} <2-18> (2.2.4) \int_{0}^{\infty}x^{2n}e ^{-x^2}dx =\frac{1\cdot3\cdot5\cdots(2n-1)\sqrt{\pi}}{2^{n+1}} \quad (n=0,1,2,\cdots) <2-19> xe^{-ax^2}\equiv g(x) <2-20> g(-x)=-g(x) <2-21> \int_{-\infty}^{\infty}xe ^{-ax^2}dx=0 <2-22> (2.2.5) \int_{-\infty}^{\infty}x^{2n+1}e ^{-x^2}dx=0\quad(n=0,1,2,\cdots) <2-23> \int_{0}^{\infty}xe ^{-ax^2}dx <2-24> x=\sqrt{\frac{y}{a}} <2-25> dx=\frac{1}{2\sqrt{ay}}dy <2-26> \begin{array}{rl} \displaystyle{\int_{0}^{\infty}xe ^{-ax^2}dx} &=\displaystyle{\int_{0}^{\infty}\sqrt{\frac{y}{a}}e ^{-y}\frac{1}{2\sqrt{ay}}dy} \\ \\ &=\displaystyle{\frac{1}{2a}\int_{0}^{\infty} e ^{-y}dy} \end{array} <2-27> \int_{0}^{\infty} e ^{-y}dy=1 <2-28> \int_{0}^{\infty}xe ^{-ax^2}dx=\frac{1}{2a} <2-29> \int_{0}^{\infty}x^3e ^{-ax^2}dx=\frac{1}{2a^2} <2-30> \int_{0}^{\infty}x^{2n+1}e ^{-ax^2}dx=\frac{n!}{2a^{n+1}} <2-31> (2.2.6) \int_{0}^{\infty}x^{2n+1}e ^{-x^2}dx=\frac{n!}{2}\quad(n=0,1,2,\cdots) <2-32> \left\{\begin{array}{ll} \displaystyle{\int_{-\infty}^{\infty}x^{2n}e ^{-x^2}dx=\frac{1\cdot3\cdot5\cdots\sqrt{\pi}}{2^n}} &(2.2.3) \\ \\ \displaystyle{\int_{0}^{\infty}x^{2n}e ^{-x^2}dx=\frac{1\cdot3\cdot5\cdots\sqrt{\pi}}{2^{n+1}}} &(2.2.4) \\ \\ \displaystyle{\int_{-\infty}^{\infty}x^{2n+1}e ^{-x^2}dx=0}} &(2.2.5) \\ \\ \displaystyle{\int_{0}^{\infty}x^{2n+1}e ^{-x^2}dx=\frac{n!}{2}} &(2.2.6) \end{array}\right.\quad(n=0,1,2,\ldots) <2-33> dv_xdv_ydv_z\equiv d\Gamma <2-34> (2.2.7) \begin{array}{rl} P(v_x,v_y,v_z)d\Gamma&=\displaystyle{\left(\frac{m}{2\pi kT}\right)^{3/2}e^{-E/{kT}}d\Gamma} \\ \\ &=\displaystyle\left(\frac{m}{2\pi kT}\right)^{3/2}\exp\left[-\frac{mv_x^2}{2kT}\right] \exp\left[-\frac{mv_y^2}{2kT}\right] \exp\left[-\frac{mv_z^2}{2kT}\right]d\Gamma} \end{array} <2-35> (2.2.8) E=\frac{m}{2}\left(v_x^2+v_y^2+v_z^2\right) <2-36> \left\{\begin{array}{l} -\infty\le v_x\le\infty \\ \\ -\infty\le v_y\le\infty \\ \\ -\infty\le v_z\le\infty \end{array}\right. <2-37> \iiint P(v_x,v_y,v_z)d\Gamma=1 <2-38> \langle K\rangle=\frac{3kT}{2} <2-39> K=\frac{m}{2}\left(v_x^2+v_y^2+v_z^2\right) <2-40> \left(v_x^2,v_y^2,v_z^2\right) <2-41> (2.2.9) \begin{array}{rl} \displaystyle{\langle K\rangle} &=\displaystyle{\iiint P(v_x,v_y,v_z)\frac{mv_x^2}{2}d\Gamma} \\ \\ &=\displaystyle{3\left(\frac{m}{2\pi kT}\right)^{3/2}\left(\int_{-\infty}^{\infty} \frac{mv_x^2}{2}e^{-(mv_x^2/2)/kT}dv_x\right)} \\ \\ &\displaystyle{\times\left(\int_{-\infty}^{\infty}e^{-(mv_y^2/2)/kT}dv_y\right)} \\ \\ &\displaystyle{\times\left(\int_{-\infty}^{\infty}e^{-(mv_z^2/2)/kT}dv_z\right)} \end{array} <2-42> (2.2.10) \left\{\begin{array}{l} \displaystyle{v_x=\left(\frac{2kT}{m}\right)^{1/2}p}\\ \\ \displaystyle{v_y=\left(\frac{2kT}{m}\right)^{1/2}q}\\ \\ \displaystyle{v_z=\left(\frac{2kT}{m}\right)^{1/2}r} \end{array}\right. <2-43> (2.2.11) \left\{\begin{array}{l} \displaystyle{p=\left(\frac{m}{2kT}\right)^{1/2}v_x}\\ \\ \displaystyle{q=\left(\frac{m}{2kT}\right)^{1/2}v_y}\\ \\ \displaystyle{r=\left(\frac{m}{2kT}\right)^{1/2}v_z} \end{array}\right. <2-44> (2.2.12) \begin{array}{rl} \displaystyle{\langle K\rangle} &=\displaystyle{3\left(\frac{m}{2\pi kT}\right)^{3/2}\times\left(\frac{2kT}{m}\right)^{1/2} \left[\int_{-\infty}^{\infty}\left(\frac{m}{2}\cdot\frac{2kT}{m}p^2\right) e^{-p^2}dp\right] \\ \\ &\qquad\qquad\qquad\times\displaystyle{\left(\frac{2kT}{m}\right)^{1/2}\left[\int_{-\infty}^{\infty}e^{-q^2}dq\right] \times\left(\frac{2kT}{m}\right)^{1/2}\left[\int_{-\infty}^{\infty}e^{-r^2}dr\right]} \\ \\ &=\displaystyle{3\left(\frac{kT}{\pi^{3/2}}\right) \left[\int_{-\infty}^{\infty}p^2e^{-p^2}dp\right] \times\left[\int_{-\infty}^{\infty}e^{-q^2}dq\right] \times\left[\int_{-\infty}^{\infty}e^{-r^2}dr\right]} \end{array} <2-45> \int_{-\infty}^{\infty}p^2e^{-p^2}dp=\frac{\sqrt{\pi}}{2} <2-46> \int_{-\infty}^{\infty}e^{-q^2}dq=\int_{-\infty}^{\infty}e^{-r^2}dr=\sqrt{\pi} <2-47> (2.2.13) \begin{array}{rl} \langle K\rangle &=\displaystyle{3\left(\frac{kT}{\pi ^{3/2}}\right)\times\left(\frac{\sqrt{\pi}}{2}\right) \times\left(\sqrt{\pi}\right)^2} \\ \\ &=\displaystyle{\frac{3kT}{2}} \end{array} <2-48> \vec{v}=\vec{i}v_x+\vec{j}v_y+\vec{k}v_z <2-49> (2.2.14) \begin{array}{rl} \displaystyle{\langle\vec{v}\rangle} &=\displaystyle{\vec{i}\left(\frac{m}{2\pi kT}\right)^{3/2} \left[\int_{-\infty}^{\infty}v_xe^{-(mv_x^2/2)/kT}dv_x\right)} \\ \\ &\quad\displaystyle{\times\left[\int_{-\infty}^{\infty}e^{-(mv_y^2/2)/kT}dv_y\right] \times\left[\int_{-\infty}^{\infty}e^{-(mv_z^2/2)/kT}dv_z\right]} \\ \\ &+\displaystyle{\vec{j}\left(\frac{m}{2\pi kT}\right)^{3/2} \left[\int_{-\infty}^{\infty}e^{-(mv_x^2/2)/kT}dv_x\right)} \\ \\ &\quad\displaystyle{\times\left[\int_{-\infty}^{\infty}v_ye^{-(mv_y^2/2)/kT}dv_y\right] \times\left[\int_{-\infty}^{\infty}e^{-(mv_z^2/2)/kT}dv_z\right]} \\ \\ &+\displaystyle{\vec{k}\left(\frac{m}{2\pi kT}\right)^{3/2} \left[\int_{-\infty}^{\infty}e^{-(mv_x^2/2)/kT}dv_x\right)} \\ \\ &\quad\displaystyle{\times\left[\int_{-\infty}^{\infty}e^{-(mv_y^2/2)/kT}dv_y\right] \times\left[\int_{-\infty}^{\infty}v_ze^{-(mv_z^2/2)/kT}dv_z\right]} \end{array} <2-50> (2.2.15) \langle\vec{v}\rangle=0 <2-51> v=\sqrt{v_x^2+ v_y^2+ v_z^2} <2-52> (2.2.16) \begin{array}{rl} \displaystyle{\langle v\rangle} &=\displaystyle{\left(\frac{m}{2\pi kT}\right)^{3/2}\int_{-\infty}^{\infty} e^{-(mv_x^2/2)/kT}dv_x\int_{-\infty}^{\infty}e^{-(mv_y^2/2)/kT}dv_y} \\�@\\ &\qquad\times\displaystyle{\int_{-\infty}^{\infty}\sqrt{v_x^2+ v_y^2+ v_z^2} e^{-(mv_z^2/2)/kT}dv_z}} \end{array} <2-53> \left\{\begin{array}{l} x=r\sin\theta\cos\phi \\ \\ y=r\sin\theta\sin\phi \\ \\ z=r\cos\theta \end{array}\right. <2-54> (2.2.17) \left\{\begin{array}{l} v_x=v\sin\theta\cos\phi \\ \\ v_y=v\sin\theta\sin\phi \\ \\ v_z=v\cos\theta \end{array}\right. <2-55> (2.2.18) \begin{array}{rl} \displaystyle{\langle v\rangle} &=\displaystyle{\left(\frac{m}{2\pi kT}\right)^{3/2}\int_{0}^{\infty}v^2dv \int_{-\pi}^{\pi}\sin\theta d\theta\int_{0}^{2\pi}ve^{-(mv^2/2)/kT}d\phi} \\\\ &=\displaystyle{\left(\frac{m}{2\pi kT}\right)^{3/2}\int_{0}^{\infty}v^3 e^{-(mv^2/2)/kT}dv\int_{-\pi}^{\pi}\sin\theta d\theta\int_{0}^{2\pi}d\phi} \\ \\ &=\displaystyle{\left(\frac{m}{2\pi kT}\right)^{3/2}4\pi\int_{0}^{\infty}v^3 e^{-(mv^2/2)/kT}dv} \\ \\ &=\displaystyle{\sqrt{\frac{8kT}{\pi m}}} \end{array} <2-56> K=\frac{m}{2}\left(v_x^2+ v_y^2+ v_z^2\right)=\frac{m}{2}v^2 <2-57> \langle v^2\rangle=\frac{3kT}{m} <2-58> \langle v\rangle^2=\frac{8kT}{m}\simeq\frac{2,55kT}{m} <2-59> \sqrt{\langle v^2\rangle-\langle v\rangle^2} <2-60> (2.3.1) P(\vec{v};T)d\vec{v}=\left(\frac{m}{2\pi kT}\right)^{3/2} e^{-\beta\epsilon(\vec{v})} d\vec{v} <2-61> (2.3.2) \beta=\frac{1}{kT} <2-62> (2.3.3) \epsilon(\vec{v})=\frac{m}{2}\vec{v}^2 =\frac{m}{2}v_x^2+\frac{m}{2}v_y^2+\frac{m}{2}v_z^2 <2-63> (2.3.4) \iiint P(\vec{v};T)d\vec{v}=1 <2-64> (2.3.5) P({\vec{v}};T)d\vec{v}_1d\vec{v}_1\ldotas d\vec{v}_N\proptoe^{-\beta E({\vec{v}}) d\vec{v}_1\ldotas d\vec{v}_N <2-65> (2.3.6) E(\{\vec{v}\})=\frac{m}{2}\left(v_1^2+v__2^2+v_N^2) <2-66> (2.3.7) P(\{\vec{v}\};T)d\vec{v}_1 d\vec{v}_2\ldots d\vec{v}_N=CW\left[E(\{\vec{v}\})\right] e^{-\beta E(\{\vec{v}\})} d\vec{v}_1 d\vec{v}_2\ldots d\vec{v}_N <2-67> (2.3.8) \iiint P(\{\vec{v}\})d\vec{v}_1 d\vec{v}_2\ldots d\vec{v}_N=1 <2-68> (2.3.9) P[\{\vec{r}\},\{\vec{v}\};T]d\Gamma = CW[E(\{\vec{r}\},\{\vec{v}\})]e^{-\beta E(\{\vec{r}\},\{\vec{v}\})} d\Gamma <2-69> (2.3.10) d\vec{r}_1 d\vec{r}_2\ldots d\vec{r}_Nd\vec{v}_1 d\vec{v}_2\ldots d\vec{v}_N \equiv d\Gamma <2-70> (2.3.11) \iiint_VP[\{\vec{r}\},\{\vec{v}\};T]d\Gamma= 1 <2-71> (2.3.12) W[E(\{\vec{r}\},\{\vec{v}\})]e^{-\beta E(\{\vec{r}\},\{\vec{v}\})} <2-72> (2.3.13) x=e^{\ln x} <2-73> (2.3.14) \begin{array}{rl} \displaystyle{We^{-\beta E}&=\displaystyle{e^{\ln W}e^{-\beta E}}} \\ \\ &=\displaystyle{e^{-\beta E -kT\ln W}} \end{array} <2-74> (2.3.15) \begin{array}{rl} C&=\displaystyle{\frac{1}{\displaystyle{\iiint_V\exp\left[-\beta \left\{E(\{\vec{r}\},\{\vec{v}\})-kT\ln W(\{\vec{r}\},\{\vec{v}\})\right\} \right]d\Gamma} \\ \\ &=\displaystyle{\frac{1}{Z(T)}} \end{array} <2-75> (2.3.16) Z(T)\equiv\iiint_V e^{-\beta[E(\{\vec{r}\},\{\vec{v}\})-kT\ln W(\{\vec{r}\},\{\vec{v}\})]} d\Gamma <2-76> (2.3.17) \iiint_VE(\{\vec{r}\},\{\vec{v}\})P[\{\vec{r}\},\{\vec{v}\};T]d\Gamma\equiv U <2-77> (2.3.18) k\iiint_V\ln W(\{\vec{r}\},\{\vec{v}\})P[\{\vec{r}\},\{\vec{v}\};T]d\Gamma\equiv S <2-78> E-kT\ln W <2-79> U-ST\equiv F(T,V) <2-80> (2.3.20) \frac{de^{ax}}{dx}=ae^{ax} <2-81> (2.3.21) \frac{dZ(T)}{d\beta}=-\iiint_V E(\{\vec{r}\},\{\vec{v}\}) e^{-\beta E(\{\vec{r}\},\{\vec{v}\})}d\Gamma <2-82> (2.3.22) \begin{array}{rl} \displaystyle{\frac{1}{Z(T)}\frac{dZ(T)}{d\beta}} &=-\displaystyle{\iiint_V E(\{\vec{r}\},\{\vec{v}\}) \frac{e^{-\beta E(\{\vec{r}\},\{\vec{v}\})}}{Z(T)}d\Gamma} \\ \\ &=-\displaystyle{\iiint_V E(\{\vec{r}\},\{\vec{v}\})P\left[\{\vec{r}\},\{\vec{v}\};T\right]d\Gamma} \end{array} <2-83> (2.3.23) \begin{array}{rl} U&=-\displaystyle{\frac{1}{Z(T)}\frac{dZ(T)}{d\beta}} \\ \\ &=-\displaystyle{ \frac{d\ln Z(T)}{d\beta}} \end{array} <2-84> (2.3.24) \frac{d\ln y(x)}{dx}=\frac{1}{y}\frac{dy}{dx} <2-85> f(-x)=f(x) <2-86> I=\int_{-\infty}^{\infty}f(x)dx <2-87> I=\int_{-\infty}^{0}f(x)dx+\int_{0}^{\infty}f(x)dx <2-88> dx=-dx^\prime <2-89> \int_{-\infty}^{0}f(x)dx=\int_{+\infty}^{0}f(-x^\prime)(-dx^\prime) =-\int_{+\infty}^{0}f(x^\prime)dx^\prime +\int_{0}^{\infty}f(x)dx <2-90> \int_{a}^{b}F(x)dx=-\int_{b}^{a}F(x)dx <2-91> -\int_{-\infty}^{0}f(x)dx=\int_{0}^{+\infty}f(x^\prime)dx^\prime=\int_{0}^{+\infty}f(x)dx <2-92> \int_{-\infty}^{0}f(x)dx=2\int_{0}^{\infty}f(x)dx <2-93> \begin{array}{l} \displaystyle{\left(\frac{m}{2kT}\right)^{3/2}\left(\int_{-\infty}^{\infty}e^{-(mv_x^2/2)/kT}dv_x\right) \times\left(\int_{-\infty}^{\infty}e^{-(mv_y^2/2)/kT}dv_y\right)} \\ \\ &\qquad\times\displaystyle{\left(\int_{-\infty}^{\infty}e^{-(mv_z^2/2)/kT}dv_z\right)}=1 \end{array} <2-94> \left(\frac{m}{2kT}\right)^{3/2} \left(\int_{-\infty}^{\infty}e^{-(mv_x^2/2)/kT}dv_x\right)^3=1 <2-95> q=\left(\frac{m}{2kT}\right)^{1/2}v_x <2-96> dv_x=\left(\frac{2kT}{m}\right)^{1/2}dq <2-97> \mbox{the left side}=\left(\frac{m}{2\pi kT}\right)^{3/2} \left[\left(\frac{ {2kT}{m}\right)^{1/2}\int_{-\infty}^{\infty}e^{-q^2}dq\right]^3 <2-98> \mbox{the left side}=\left(\frac{m}{2\pi}\right)^{3/2} \left[\left(\frac{2kT}{m}\right)^{1/2}\sqrt{\pi}\right]^3=1 <2-99> \left(\frac{mN^{2/3}}{2\pi kT}\right)^{3/2}