\[ \int\limits_0^\infty \exp\left(-ax^2\right)\,{\mathrm d}x=\frac{1}{2}\sqrt\frac{\pi}{a}, \hbox{ pro } \quad a>0 \] \[ \int\limits_0^\infty x^{2n} \exp\left(-ax^2\right)\,{\mathrm d}x = \frac{(2n-1)!!\sqrt\pi}{2^{n+1} a^{(2n+1)/2}}, \quad \hbox{pro} \quad n=1,2,\ldots, \quad a>0. \] \[ n!!=n(n-2)(n-4)\ldots. \] \[ \int\limits_0^\infty x^{2n+1} \exp\left(-ax^2\right)\,{\mathrm d}x=\frac{n!}{2a^{n+1}}, \hbox{ pro} \quad n=0,1,2,\ldots, \quad a>0. \]
Boltzmann constant: \(1.380649\times 10^{-23}\) JK\(^{-1}\)
Elementary charge: \(1.602176634\times10^{-19}\) C
Electron mass: \(9.1093837015\times10^{-31}\) kg
Proton mass: \(1.67262192369\times10^{-27}\) kg
Vacuum electric permittivity: \(8.8541878128\times10^{-12}\) F m\(^{-1}\)
Vacuum mag. permeability: \(1.25663706212\times10^{-6}\) N A\(^{-2}\)
\[{\displaystyle {\begin{aligned} x & = \rho \cos \varphi \\ y & = \rho \sin \varphi \\ z & = z \end{aligned}}}\]
\[{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\\ {\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}}\]
\[{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\cos \varphi {\hat {\boldsymbol {\rho }}}-\sin \varphi {\hat {\boldsymbol {\varphi }}}\\ {\hat {\mathbf {y} }}&=\sin \varphi {\hat {\boldsymbol {\rho }}}+\cos \varphi {\hat {\boldsymbol {\varphi }}}\\ {\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}} \]
\[{\boldsymbol A} = {\displaystyle A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\varphi }{\hat {\boldsymbol {\varphi }}}+A_{z}{\hat {\mathbf {z} }}}\] \[ \nabla f = {\displaystyle {\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+ {1 \over \rho }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}} \] \[ \nabla \cdot {\bf A} = {\displaystyle {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}} \]
\[\nabla\times {\bf A} = {\displaystyle {\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\ +\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\ +{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\varphi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}} \]
\[ \nabla^2 f = {\displaystyle {1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+ {1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}} \]
\(J_0(2.405)=0,\quad J_1(2.405)=0.5191\) \[J_{-n}(x)=(-1)^n J_n(x)\] \[J_n^\prime(x)=\frac{1}{2}\left[J_{n-1}(x) - J_{n+1}(x) \right]\] \[\frac{{\mathrm d}}{{\mathrm d}x}\left[x^n J_n(x)\right] = x^n J_{n-1}(x)\] \[\frac{{\mathrm d}}{{\mathrm d}x}\left[x^{-n} J_n(x)\right] = -x^{-n} J_{n+1}(x)\]