$\begin{array}{l} F(\omega) = \displaystyle\int\limits_{ - \infty }^\infty {f(t){{\rm{e}}^{ - i\omega t}}{\rm{d}}t} \\ f(t) = \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{ - \infty }^\infty {F(\omega){{\rm{e}}^{i\omega t}}{\rm{d}}\omega } \\ \end{array} $

$s(t) = {\cal{H}}{\rm{ }}[s(x)] = \dfrac{1}{{\rm{\text{π} }}}{\rm{P}}{\rm{.V}}.\int\limits_{ - \infty }^{ + \infty } {\dfrac{{s(x)}}{{t - x}}{\rm{d}}\tau } $

$S(t) = s(t) + i{\cal{H}} [s(t)]$

${C_{{\rm{fg}}}}(\tau) = \int\limits_{ - \infty }^{ + \infty } {{f^{\rm{*}}}(t)g(t + \tau){\rm{d}}t} $

$g(t + \tau) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {G(\omega '){{\rm{e}}^{i\omega '(t + \tau)}}{\rm{d}}\omega } '$

$\begin{split} {C_{{\rm{fg}}}}(\tau)&= \dfrac{1}{{4{{\rm{\text{π} }}^2}}}\int\limits_{ - \infty }^{ + \infty } {\left[ {\int\limits_{ - \infty }^{ + \infty } {{F^*}(\omega){{\rm{e}}^{ - i\omega t}}{\rm{d}}\omega } } \right]}.\\&\qquad\qquad \left[ {\int\limits_{ - \infty }^{ + \infty } {G(\omega '){{\rm{e}}^{i\omega '(t + \tau)}}{\rm{d}}\omega '} } \right] {\rm{d}}t \\& = \dfrac{1}{{4{{\rm{\text{π} }}^2}}}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {{F^*}(\omega)} } } G(\omega '){{\rm{e}}^{i(\omega ' - \omega)t}}{{\rm{e}}^{i\omega '\tau }}{\rm{d}}t{\rm{d}}\omega {\rm{d}}\omega ' \\& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {{F^*}(\omega)} } G(\omega ')\delta (\omega - \omega '){{\rm{e}}^{i\omega '\tau }}{\rm{d}}\omega {\rm{d}}\omega ' \\& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{F^*}(\omega)} G(\omega){{\rm{e}}^{i\omega \tau }}{\rm{d}}\omega \\[-11pt] \end{split} $

$\delta (x - {x_0}) = \left\{ {\begin{array}{*{20}{c}} \infty &{x = {x_0}} \\ 0&{x \ne {x_0}} \\ {\displaystyle\int_{ - \infty }^{ + \infty } {\delta (x - {x_0}){\rm{d}}x = 1} }&{} \end{array}} \right.$

$\int\limits_{ - \infty }^{ + \infty } {f(x)\delta (x - {x_0}){\rm{d}}x = f({x_0})} $

${C_{{\rm{ab}}}}(r,\tau) = \dfrac{1}{{2T}}\int\limits_{ - T}^T {{u^*}({{{r}}_{\rm{a}}},t)u({{{r}}_{\rm{b}}},t + \tau){\rm{d}}t} $

${R_{{\rm{ab}}}}({{\xi }},t) = \dfrac{1}{A}\int\limits_A {{u^*}({{r}},t)u({{r}} + {{\xi }},t){\rm{d}}{{r}}} $

${\phi _{{\rm{ab}}}}({{\xi }}) = \dfrac{1}{{2T}}\int\limits_{ - T}^T {{u^*}({{r}},t)u({{r}} + {{\xi }},t){\rm{d}}t} $

${u_{\rm{a}}}({{{r}}_{\rm{a}}},\omega,t) = A{{\rm{e}}^{i(\omega t + {\phi _0})}}$

${u_{\rm{b}}}({{{r}}_{\rm{b}}},\omega,t) = A{{\rm{e}}^{i\omega [t - \dfrac{r}{c}\cos (\theta - \varphi) + {\phi _0}]}}$

$\begin{split} {\phi _{{\rm{ab}}}}(r,\omega) &= \dfrac{1}{{2T}}\int\limits_{ - T}^T {u_{\rm{a}}^*({{{r}}_{\rm{a}}},\omega,t){u_{\rm{b}}}({{{r}}_{\rm{b}}},\omega,t){\rm{d}}t} \\&= \dfrac{1}{{2T}}\int\limits_{ - T}^T {{A^2}{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}t} = {A^2}{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}} \end{split}$

$\begin{split} {\phi _{{\rm{aa}}}} = {\phi _{{\rm{bb}}}} &= \dfrac{1}{{2T}}\int\limits_{ - T}^T {u_{\rm{a}}^{\rm{*}}({{{r}}_{\rm{a}}},\omega,t){u_{\rm{a}}}({{{r}}_{\rm{a}}},\omega,t){\rm{d}}t} \\&= \dfrac{1}{{2T}}\int\limits_{ - T}^T {u_{\rm{b}}^*({{{r}}_{\rm{b}}},\omega,t){u_{\rm{b}}}({{{r}}_{\rm{b}}},\omega,t){\rm{d}}t} = {A^2} \end{split}$

${\widehat \phi _{{\rm{ab}}}}(r,\omega) = \dfrac{{{\phi _{{\rm{ab}}}}}}{{\sqrt {{\phi _{{\rm{aa}}}}{\phi _{{\rm{bb}}}}} }} = {{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{sta}}}}(\theta){{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}\theta } $

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{wave}}}}(\varphi){{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}\varphi } $

$\hat LG({{r}},{{r}}') = \delta ({{r}} - {{r}}')$

$\hat Lu = f({{r}})$

$u = \int\limits_{ - \infty }^{ + \infty } {G({{r}},{{r}}')f({{r}}'){{\rm{d}}^3}{{r}}'} $

$\begin{split} \hat Lu &= \hat L\left({\int\limits_{ - \infty }^{ + \infty } {G({{r}},{{r}}')f({{r}}'){{\rm{d}}^3}{{r}}'} } \right){\rm{ = }}\int\limits_{ - \infty }^{ + \infty } {\hat L\left({G({{r}},{{r}}')} \right)f({{r}}'){{\rm{d}}^3}{{r}}'} \\& = \int\limits_{ - \infty }^{ + \infty } {\delta ({{r}} - {{r}}')f({{r}}'){{\rm{d}}^3}{{r}}'} = f({{r}}) \\[-11pt] \end{split} $

${\nabla ^2}u - \dfrac{1}{{{c^2}}}\dfrac{{{\partial ^2}u}}{{\partial {t^2}}} = 0$

$\begin{split}& {\rm{1D: }}{\nabla ^2} = \dfrac{{{{\rm{d}}^2}}}{{{\rm{d}}{x^2}}} \\& {\rm{2D: }}{\nabla ^2} = \dfrac{{{\partial ^2}}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}}}{{\partial {y^2}}} \\& {\rm{3D: }}{\nabla ^2} = \dfrac{{{\partial ^2}}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}}}{{\partial {y^2}}} + \dfrac{{{\partial ^2}}}{{\partial {z^2}}} \\ \end{split} $

${\nabla ^2}G({{r}},t;{{r}}',t') - \dfrac{1}{{{c^2}}}\dfrac{{{\partial ^2}G({{r}},t;{{r}}',t')}}{{\partial {t^2}}} = \delta ({{r}} - {{r}}')\delta (t - t')$

$\begin{split}& G({{r}},t;{{r}}',t') = G({{r}}', - t';{{r}}, - t) \\& G({{r}},{{r}}') = G({{r}}',{{r}}) \end{split} $

$1{\rm{D}}:\; G(x,t) = \dfrac{c}{2}H\left(t - \dfrac{x}{c}\right)\;\;\;\;\;\;\;\;x = |x - x'|$

$\begin{split} 2{\rm{D}}:\;& G(r,t) = - \dfrac{1}{{2{\rm{\text{π} }}\sqrt {{t^2} - \dfrac{{{r^2}}}{{{c^2}}}} }}H\left(t - \dfrac{r}{c}\right)\\& r = \sqrt {{{(x - x')}^2} + {{(y - y')}^2}} \end{split}\quad\;$

$\begin{split} 3 {\rm{D}}: \;&G(r,t) = - \dfrac{1}{{4{\rm{\text{π} }}r}}\delta \left[t - \dfrac{r}{c}\right]\\& r = \sqrt {{{(x - x')}^2} + {{(y - y')}^2} + {{(z - z')}^2}} \end{split} $

$H(x) = \left\{ {\begin{array}{*{20}{c}} 1&{x > 0} \\ {{1 / 2}}&{x = 0} \\ 0&{x < 0} \end{array}} \right.$

${\nabla ^2}G({{r}},\omega ;{{r}}') + {k^2}G({{r}},\omega ;{{r}}') = \delta ({{r}} - {{r}}')$

$1{\rm{D}}:\;G(x,\omega) = \dfrac{i}{{2k}}{{\rm{e}}^{ - ikx}}\;\;\;\;\;x = |x - x'| \quad\; $

$\begin{split} 2{\rm{D}}:\;& G(r,\omega) = \dfrac{i}{4}H_0^{(2)}(kr)\\& r = \sqrt {{{(x - x')}^2} + {{(y - y')}^2}} \end{split}\qquad\qquad $

$\begin{split} 3{\rm{D}}:\;& G(r,\omega) = - \dfrac{1}{{4{\rm{\text{π} }}}}\dfrac{{{{\rm{e}}^{ - ikr}}}}{r}\\& r = \sqrt {{{(x - x')}^2} + {{(y - y')}^2} + {{(z - z')}^2}} \end{split}$

${u_{\rm{a}}} = {A_0}{{\rm{e}}^{i(\omega t + {\phi _0})}}$

${u_{\rm{b}}} = {A_0}{{\rm{e}}^{i(\omega t + {\phi _0})}}{{\rm{e}}^{ - ikx}} = {A_0}{{\rm{e}}^{i\omega (t - \textstyle\frac{x}{c})}}{{\rm{e}}^{i{\phi _0}}}$

$\begin{split} C_{{\rm{ab}}}^{\rm{ + }}(x,\omega,\tau) &= \dfrac{1}{{2T}}\int\limits_{ - T}^T {u_{\rm{a}}^{\rm{*}}(t){u_{\rm{b}}}(t + \tau){\rm{d}}t}\\& = \dfrac{1}{{2T}}\int\limits_{ - T}^T {A_0^2{{\rm{e}}^{i\omega \tau }}{{\rm{e}}^{ - ikx}}{\rm{d}}t} = A_0^2{{\rm{e}}^{i\omega (\tau - \textstyle\frac{x}{c})}} \end{split}$

$\begin{split}& {u_{\rm{a}}} = {A_{\rm{\text{π} }}}{{\rm e}^{i(\omega t + {\phi _{\rm{\text{π} }}})}} \\& {u_{\rm{b}}} = {A_{\rm{\text{π} }}}{{\rm e}^{i(\omega t + {\phi _{\rm{\text{π} }}})}}{{\rm e}^{ikx}} = {A_{\rm{\text{π} }}}{{\rm{e}}^{i\omega (t{\rm{ + }}\textstyle\frac{x}{c})}}{{\rm{e}}^{i{\phi _{\rm{\text{π} }}}}} \\ \end{split} $

$C_{{\rm{ab}}}^{\rm{ - }}(x,\omega,\tau) = A_{\rm{\text{π} }}^{\rm{2}}{{\rm{e}}^{i\omega (\tau + \textstyle\frac{x}{c})}}$

${C_{{\rm{ab}}}}(x,\omega,\tau) = \dfrac{{{{\rm{e}}^{i\omega \tau }}(A_0^2{{\rm{e}}^{ - ikx}} + A_{\rm{\text{π} }}^{\rm{2}}{{\rm{e}}^{ikx}})}}{2}$

${C_{{\rm{ab}}}}(x,\omega,\tau) = \dfrac{{{{\rm{e}}^{i\omega \tau }}({{\rm{e}}^{ikx}} + {{\rm{e}}^{ - ikx}})}}{2} = {{\rm{e}}^{i\omega \tau }}\cos kx$

${\rm{Re}} \{ {C_{{\rm{ab}}}}(x,\omega,\tau)\} = \cos \omega \tau \cos kx$

$\begin{split}& {u_{\rm{a}}} = {A_0}{{\rm{e}}^{i(\omega t + {\phi _0})}} + {A_{\rm{\text{π} }}}{{\rm{e}}^{i(\omega t + {\phi _{\rm{\text{π} }}})}} \\& {u_{\rm{b}}} = {A_0}{{\rm{e}}^{i(\omega t + {\phi _0})}}{{\rm{e}}^{ - ikx}} + {A_{\rm{\text{π} }}}{{\rm{e}}^{i(\omega t + {\phi _{\rm{\text{π} }}})}}{{\rm{e}}^{ikx}} \end{split} $

$\begin{split} C_{{\rm{ab}}}^{}(x,\omega,\tau) = &\dfrac{1}{{2T}}\int\limits_{ - T}^T {u_{\rm{a}}^*(t){u_{\rm{b}}}(t+ \tau){\rm{d}}t}= \left({A_0^2{{\rm{e}}^{ - ikx}} + A_{\rm{\text{π} }}^{\rm{2}}{{\rm{e}}^{ikx}}} \right){{\rm{e}}^{i\omega \tau }} + \\ & {A_0}{A_{\rm{\text{π} }}}{{\rm{e}}^{i({\phi _0} - {\phi _{\rm{\text{π} }}})}}{{\rm{e}}^{ikx}}{{\rm{e}}^{i\omega \tau }} + {A_0}{A_{\rm{\text{π} }}}{{\rm{e}}^{i({\phi _\text{π} } - {\phi _0})}}{{\rm{e}}^{ - ikx}}{{\rm{e}}^{i\omega \tau }} \\ \end{split} $

$\left\langle {{A_0}{A_{\rm{\text{π} }}}{{\rm{e}}^{i({\phi _0} - {\phi _{\rm{\text{π} }}})}}} \right\rangle = 0$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(x,\omega) = \dfrac{{{{\rm{e}}^{ikx\cos \varphi }} + {{\rm{e}}^{ - ikx\cos \varphi }}}}{2}$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(x,\omega) = \dfrac{1}{2}\left({{\rm{e}}^{ikx}} + {{\rm{e}}^{ - ikx}}\right) = \cos kx$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(x,\omega) = \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(x,\omega) = \dfrac{1}{2}\left({{\rm{e}}^{ikx}} + {{\rm{e}}^{ - ikx}}\right) = \cos kx$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(x,\omega) = \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(x,\omega) = {C_{{\rm{ab}}}}(x,\omega,\tau = 0)$

$\begin{split} {C_{{\rm{ab}}}}(x,t)\! &=\!\! \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {{C_{{\rm{ab}}}}(x,\omega,t)} {\rm{d}}\omega \!=\! \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {\dfrac{{({{\rm{e}}^{ikx}} + {{\rm{e}}^{ - ikx}})}}{2}{{\rm{e}}^{i\omega t}}} {\rm{d}}\omega \\&= \dfrac{1}{2}\left[ {\delta \left(t - \dfrac{x}{c}\right) + \delta \left(t + \dfrac{x}{c}\right)} \right]\\[-10pt] \end{split}$

$\dfrac{{\rm{d}}}{{{\rm{d}}x}}H(x) = \delta (x)$

${C_{{\rm{ab}}}}(x,t) = \dfrac{1}{c}\left[ {\dfrac{{\rm{d}}}{{{\rm{d}}t}}{G_{1{\rm{D}}}}(x,t) - \dfrac{{\rm{d}}}{{{\rm{d}}t}}{G_{1{\rm{D}}}}(x, - t)} \right]$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}\theta } $

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \varphi }^{2{\rm{\text{π} }} - \varphi } {{{\rm{e}}^{ - ikr\cos \alpha }}{\rm{d}}\alpha } $

${{\rm{e}}^{ - ikr\cos \alpha }} = \sum\limits_{n = 0}^\infty {{\varepsilon _n}{{(- i)}^n}{J_n}(kr)\cos n\alpha } $

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}\theta } = {J_0}(kr)$

$ \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}\varphi } = {J_0}(kr) $

$\begin{split} {C_{{\rm{ab}}}}(r,\omega,\tau) & =\dfrac{1}{{2T}}\int\limits_{ - T}^T {{{\rm{e}}^{{\rm{ - }}i(\omega t + {\phi _0})}}{{\rm{e}}^{i\omega [t{\rm{ + }}\tau - \dfrac{r}{c}\cos (\theta - \varphi) + {\phi _0}]}}{\rm{d}}t} \\ & = {{\rm{e}}^{i\omega \tau }}{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}} \end{split}$

$\begin{split} {C_{{\rm{ab}}}}(r,\omega,\tau) &= {\rm{Re}} \left[ {\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{{\rm{e}}^{i\omega \tau }}{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rm{d}}\varphi } } \right]\\& = {\rm{Re}} \left[ {{J_0}(kr){{\rm{e}}^{i\omega \tau }}} \right] = {J_0}(kr)\cos \omega \tau \end{split} $

${\rm{ }}{C_{{\rm{ab}}}}(r,t) = \dfrac{{\rm{1}}}{{{\rm{2\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{C_{{\rm{ab}}}}(r,\omega,t)} {\rm{d}}\omega = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{J_0}(kr){{\rm{e}}^{i\omega t}}} {\rm{d}}\omega $

$\int\limits_{ - \infty }^\infty {{J_0}(x){{\rm{e}}^{iax}}{\rm{d}}x} = \left\{ {\begin{array}{*{20}{c}} {\dfrac{2}{{\sqrt {1 - {a^2}} }}}&{|a| < 1} \\ 0&{|a| > 1} \end{array}} \right.$

${C_{{\rm{ab}}}}(r,t) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {{J_0}(kr){{\rm{e}}^{i\omega t}}} {\rm{d}}\omega = \left\{ {\begin{array}{*{20}{c}} {\dfrac{{\rm{1}}}{{\rm{\text{π} }}}\dfrac{1}{{\sqrt {\dfrac{{{r^2}}}{{{c^2}}} - {t^2}} }}}&{|t| < \dfrac{r}{c}} \\ 0&{|t| > \dfrac{r}{c}} \end{array}} \right.$

$f(x) = \left\{ {\begin{array}{*{20}{c}} {\dfrac{1}{{\sqrt {{a^2} - {x^2}} }}}&{x < |a|} \\ 0&{x > |a|} \end{array}} \right.$

$\hat f(x) = \left\{ {\begin{array}{*{20}{c}} {\dfrac{1}{{\sqrt {{x^2} - {a^2}} }}}&{x < - a} \\ 0&{ - a < x < a} \\ {\dfrac{{ - 1}}{{\sqrt {{x^2} - {a^2}} }}}&{x > a} \end{array}} \right.$

${\cal H} [{C_{{\rm{ab}}}}(r,t)] = 2\left[ {G(r,t) - G(r, - t)} \right]$

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\sum\limits_{n = 0}^\infty {{\varepsilon _n}{{(- i)}^n}{J_n}(kr)\cos n(\theta - \varphi)} \rho (\varphi){\rm{d}}\varphi } \\ & = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\left[ {{J_0}(kr)\rho (\varphi) + 2\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\cos n(\theta - \varphi)} \rho (\varphi)} \right]{\rm{d}}\varphi } \\ & = {J_0}(kr) + \dfrac{1}{{\rm{\text{π} }}}\int\limits_0^{2{\rm{\text{π} }}} {\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\cos n(\theta - \varphi)} \rho (\varphi){\rm{d}}\varphi } \\ \end{split} $

$\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\rho (\varphi){\rm{d}}\varphi } = 1$

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega)& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{J_0}(kr)\rho (\varphi){\rm{d}}\varphi } + \dfrac{1}{{\rm{\text{π} }}}\int\limits_0^{2{\rm{\text{π} }}} {\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\left[ {\cos n\theta \cos n\varphi + \sin n\theta \sin n\varphi } \right]} \rho (\varphi){\rm{d}}\varphi } \\& = \dfrac{{{J_0}(kr)}}{2}\dfrac{1}{{\rm{\text{π} }}}\int\limits_0^{2{\rm{\text{π} }}} {\rho (\varphi){\rm{d}}\varphi } + \sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\left[ {\cos n\theta \dfrac{1}{{\rm{\text{π} }}}\int\limits_0^{2{\rm{\text{π} }}} {\cos n\varphi \rho (\varphi){\rm{d}}\varphi } + \sin n\theta \dfrac{1}{{\rm{\text{π} }}}\int\limits_0^{2{\rm{\text{π} }}} {\sin n\varphi \rho (\varphi){\rm{d}}\varphi } } \right]} \\& = \dfrac{{{a_0}}}{2}{J_0}(kr) + \sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\left[ {{a_n}\cos n\theta + {b_n}\sin n\theta } \right]} \\[-5pt] \end{split} $

$\begin{array}{l} {a_n} = \dfrac{1}{{\rm{\text{π} }}}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {\cos n\varphi \rho (\varphi){\rm{d}}\varphi } \\ {b_n} = \dfrac{1}{{\rm{\text{π} }}}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {\sin n\varphi \rho (\varphi){\rm{d}}\varphi } \\ \end{array} $

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) =& \dfrac{{{a_0}}}{2}{J_0}(kr) +\\& \sum\limits_{m = 1}^\infty {{{(- 1)}^m}{J_{2m}}(kr)\left[ {{a_{2m}}\cos 2m\theta + {b_{2m}}\sin 2m\theta } \right]} \end{split} $

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) =& - i\sum\limits_{m = 1}^\infty {{(- 1)}^m}{J_{2m - 1}}(kr)\cdot \\& \left[ {{a_{2m - 1}}\cos (2m - 1)\theta + {b_{2m - 1}}\sin (2m - 1)\theta } \right] \end{split} $

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = {J_0}(kr) - {a_2}{J_2}(kr)\cos 2\theta - {b_2}{J_2}(kr)\sin 2\theta $

$\phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(r,\omega) = {J_0}(kr) - {a_2}{J_2}(kr)\cos 2\varphi - {b_2}{J_2}(kr)\sin 2\varphi $

${C_{{\rm{ani}}}} = {C_{{\rm{iso}}}} + {A_{\rm{c}}}\cos 2\theta + {A_{\rm{s}}}\sin 2\theta $

$ \begin{split} {\widehat \phi _{{\rm{ab}}}}(r,\omega)& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}{\rho ^{{\rm{sta}}}}(\theta){\rm{d}}\theta \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{wave}}}}(\varphi){\rm{d}}\varphi } } \\& = {J_0}(kr) + \dfrac{1}{{4{{\rm{\text{π} }}^2}}}\int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{sta}}}}(\theta){\rm{d}}\theta } \int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{wave}}}}(\varphi)\left[ {\sum\limits_{n = 1}^\infty {2{{(- i)}^n}{J_n}(kr)\cos n(\theta - \varphi)} } \right]{\rm{d}}\varphi } \\ \end{split} $

$\begin{split} {{k}} \cdot {{r}} &= kr(\sin \theta \cos \varphi \sin \gamma \cos \xi + \sin \theta \sin \varphi \sin \gamma \sin \xi +\\ & \quad \cos \theta \cos \gamma) \\ & = kr\left[ {\sin \theta \sin \gamma \cos (\varphi - \xi) + \cos \theta \cos \gamma } \right] \\[-10pt] \end{split} $

${\hat \phi _{{\rm{ab}}}}(r,\omega) = {{\rm{e}}^{{\rm{ - }}i{{k}} \cdot {{r}}}}$

$ \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{{\rm{ - }}i{{k}} \cdot {{r}}}}{\rm{d}}\varphi } } $

$ \begin{split} {{\rm{e}}^{ - i{{k}} \cdot {{r}}}} &= 4{\rm{\text{π} }}\sum\limits_{l = 0}^\infty {{{(- i)}^l}{j_l}(kr)\dfrac{{2l + 1}}{{4{\rm{\text{π} }}}}} {P_l}({{\hat k}} \cdot {{\hat r}})\\& = \sum\limits_{l = 0}^\infty {{{(- i)}^l}(2l + 1){j_l}(kr)} {P_l}(\cos \psi) \end{split} $

$\cos \psi = \sin \theta \sin \gamma \cos (\varphi - \xi) + \cos \theta \cos \gamma $

$\begin{split} {P_l}(\cos \psi) &\!=\! {P_l}\left[ {\sin \theta \sin \gamma \cos (\varphi - \xi) + \cos \theta \cos \gamma } \right] \\ & \!=\!\! \sum\limits_{m = 0}^l \!{{\varepsilon _m}\dfrac{{(l - m)!}}{{(l + m)!}}\!\cos\! \left[ {m(\varphi - \xi)} \right]P_l^m(\cos \theta)P_l^m(\cos \gamma)} \\ & \!=\! {P_l}(\cos \theta){P_l}(\cos \gamma) + \\ &\quad 2\sum\limits_{m = 1}^l {\dfrac{{(l - m)!}}{{(l + m)!}}P_l^m(\cos \theta)P_l^m(\cos \gamma)\cos \left[ {m(\varphi - \xi)} \right]} \\ \end{split} $

$\begin{split} & \int\limits_0^{\rm{\text{π} }} {{P_l}(\cos \theta){P_l}(\cos \gamma)\sin \theta {\rm{d}}\theta \int_0^{2{\rm{\text{π} }}} {{\rm{d}}\varphi } } \\&\quad= 2{\rm{\text{π} }}{P_l}(\cos \gamma)\int\limits_0^{\rm{\text{π} }} {{P_l}(\cos \theta)\sin \theta {\rm{d}}\theta } \\ &\quad = 2{\rm{\text{π} }}{P_0}(\cos \gamma)\int\limits_0^{\rm{\text{π} }} {{P_0}(\cos \theta)\sin \theta {\rm{d}}\theta } = 4{\rm{\text{π} }} \\[-3pt] \end{split} $

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{k}} \cdot {{r}}}}{\rm{d}}\varphi } } = {j_0}(kr) = \dfrac{{\sin kr}}{{kr}}$

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{sta}}} }(r,\omega)& = \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \gamma {\rm{d}}\gamma \int\limits_0^{2\text{π} } {{{\rm{e}}^{ - i{{k}} \cdot {{r}}}}{\rm{d}}\xi } } = \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) \\&= {j_0}(kr) = \dfrac{{\sin kr}}{{kr}} \end{split}$

$\begin{split} {C_{{\rm{ab}}}}(r,\omega,\tau) &= {\rm{Re}} \left[ {\dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{i\omega \tau }}{{\rm{e}}^{ - i{{k}} \cdot {{r}}}}{\rm{d}}\varphi } } } \right] \\&= {\rm{Re}} \left[ {{j_0}(kr){{\rm{e}}^{i\omega \tau }}} \right] = {j_0}(kr)\cos \omega \tau \end{split}$

$\begin{split} {C_{{\rm{ab}}}}(r,\omega,t) &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{C_{{\rm{ab}}}}(r,\omega,\tau){\rm{d}}\omega } = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{j_0}(kr)} {{\rm{e}}^{i\omega t}}{\rm{d}}\omega \\&= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\dfrac{{{{\rm{e}}^{ikr}} - {{\rm{e}}^{ - ikr}}}}{{2ikr}}} {e^{i\omega t}}{\rm{d}}\omega \\&= \dfrac{c}{{2r}}\left[ {H(t + \dfrac{r}{c}) - H(t - \dfrac{r}{c})} \right] \end{split}$

$\begin{split} \dfrac{{{\rm{d}}{C_{{\rm{ab}}}}(r,t)}}{{{\rm{d}}t}} &= \dfrac{c}{{2r}}\left[ {\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{{\rm{e}}^{ikr}}{{\rm{e}}^{i\omega t}}} {\rm{d}}\omega - \dfrac{1}{{2{\text{π} }}}\int\limits_{ - \infty }^{ + \infty } {{{\rm{e}}^{ - ikr}}{{\rm{e}}^{i\omega t}}{\rm{d}}\omega } } \right] \\& = \dfrac{c}{{2r}}\left[ {\delta \left({t + \dfrac{r}{c}} \right) - \delta \left({t - \dfrac{r}{c}} \right)} \right] \end{split} $

$\dfrac{{{\rm{d}}{C_{{\rm{ab}}}}(r,t)}}{{{\rm{d}}t}} = 2{\rm{\text{π} }}c\left[ {G(r,t) - G(r, - t)} \right]$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{k}} \cdot {{r}}}}\rho (\theta,\varphi){\rm{d}}\varphi } } $

$\begin{array}{l} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{4{\rm{\text{π} }}}}\displaystyle\int\limits_0^{\rm{\text{π} }} {\sin \theta d\theta \displaystyle\int\limits_0^{2{\rm{\text{π} }}} {\rho (\theta,\varphi)\displaystyle\sum\limits_{l = 0}^\infty {{{(- i)}^l}(2l + 1){j_l}(kr)} \displaystyle\sum\limits_{m = 0}^l {{\varepsilon _m}\dfrac{{(l - m)!}}{{(l + m)!}}\cos \left[ {m(\varphi - \xi)} \right]P_l^m(\cos \theta)P_l^m(\cos \gamma)} {\rm{d}}\varphi } } \\ \\ \end{array} $

$ \begin{split}& \widehat \phi _{{\rm{ab}}}^{{\overline {wave}}}(r,\omega) = {j_0}(kr) +\\&\quad 2\sum\limits_{l = 0}^\infty {\sum\limits_{m = 1}^l {{{(- i)}^l}{j_l}(kr)P_l^m(\cos \gamma)\left[ {a_l^m\cos m\xi + b_l^m\sin m\xi } \right]} } \end{split} $

$\begin{split}& a_l^m = \dfrac{{2l + 1}}{{4{\rm{\text{π} }}}}\dfrac{{(l - m)!}}{{(l + m)}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {\rho (\theta,\varphi)\cos m\varphi {\rm{d}}\varphi } } \\& b_l^m = \dfrac{{2l + 1}}{{4{\rm{\text{π} }}}}\dfrac{{(l - m)!}}{{(l + m)}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {\rho (\theta,\varphi)\sin m\varphi {\rm{d}}\varphi } } \\[-3pt] \end{split} $

$\dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {\rho (\omega ;\theta,\varphi){\rm{d}}\varphi } } = 1$

$ \begin{split} {\phi _{{\rm{ab}}}}(r,\omega) &= \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{wave}}}}(\theta,\varphi){\rm{d}}\varphi } \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \gamma {\rm{d}}\gamma \int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{sta}}}}(\xi,\gamma)} {{\rm{e}}^{ - i{{k}} \cdot {{r}}}}{\rm{d}}\xi } } \\& = {j_0}(kr) + \dfrac{1}{{16{{\rm{\text{π} }}^2}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{wave}}}}(\theta,\varphi){\rm{d}}\varphi } \int\limits_0^{\rm{\text{π} }} {\sin \gamma {\rm{d}}\gamma \int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{sta}}}}(\xi,\gamma)\left[ {\sum\limits_{l = 1}^\infty {{{(- i)}^l}(2l + 1){j_l}(kr){P_l}(\cos \psi)} } \right]} {\rm{d}}\xi } } \\ \end{split} $

$\begin{split} 1{\rm{D}}:\;{\hat \phi _{a{\rm{b}}}}(x,\omega) & = \cos kx = - ik\left[ {G(x,\omega) - {G^*}(x,\omega)} \right] \\&= 2k{\rm{Im}} \left[ {G(x,\omega)} \right] \end{split}$

$\begin{split} 2{\rm{D}}:\;{\hat \phi _{{\rm{ab}}}}(r,\omega) &= {J_0}(kr) = - 2i\left[ {G(r,\omega) - {G^*}(r,\omega)} \right] \\&= 4{\rm{Im}} \left[ {G(r,\omega)} \right] \end{split}$

$\begin{split} 3{\rm{D}}:\;{\hat \phi _{{\rm{ab}}}}(r,\omega) &= {j_0}(kr) = \dfrac{{\sin kr}}{{kr}} \!=\! - \dfrac{{2{\rm{\text{π} }}i}}{k}\left[ {G(r,\omega) - {G^*}(r,\omega)} \right]\\ &= \dfrac{{4{\rm{\text{π} }}}}{k}{\rm{Im}} \left[ {G(r,\omega)} \right]\\[-10pt] \end{split}$

$\widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rho ^{{\rm{wave}}}}(\varphi){\rm{d}}\varphi } $

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rho ^{{\rm{wave}}}}(\varphi){\rm{d}}\varphi } \\& = \dfrac{{{J_0}(kr)}}{{2{\rm{\text{π} }}}}\left[ {\int\limits_{ - \alpha }^\alpha {{\rho _0}{\rm{d}}\varphi } \!+\! \int\limits_{{\rm{\text{π} }} - \alpha }^{{\rm{\text{π} }} +\alpha } {{\rho _0}{\rm{d}}\varphi } } \right] +\\&\quad \dfrac{1}{{\rm{\text{π} }}}\sum\limits_{n = 1}^\infty {{(- i)}^n}{J_n}(kr) \left[ \int\limits_{ - \alpha }^\alpha {{\rho _0}\cos n\varphi {\rm{d}}\varphi } +\right. \\&\qquad\qquad\qquad \left. \int\limits_{{\rm{\text{π} }} - \alpha }^{{\rm{\text{π} }} +\alpha } {{\rho _0}\cos n\varphi {\rm{d}}\varphi } \right] \\ \end{split} $

$\begin{split}& \displaystyle\int\limits_{ - \alpha }^\alpha {\cos n\varphi {\rm{d}}\varphi } = \dfrac{2}{n}\sin n\alpha \\& \displaystyle\int\limits_{{\rm{\text{π} }} - \alpha }^{{\rm{\text{π} }} + \alpha } {\cos n\varphi {\rm{d}}\varphi } = \left\{ {\begin{array}{*{20}{c}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\dfrac{1}{m}\sin 2m\alpha }&{n = 2m} \\ { - \dfrac{2}{{2m - 1}}\sin (2m - 1)\alpha }&{n = 2m - 1} \end{array}} \right. \\ \end{split} $

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) \!&= \dfrac{{{\rho _0}{J_0}(kr)}}{{2{\rm{\text{π} }}}}\int\limits_{ - \alpha }^\alpha {{\rm{d}}\varphi } \!+\! \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\!\sum\limits_{n = 1}^\infty \!{{{(- i)}^n}{J_n}(kr)\int\limits_{ - \alpha }^\alpha \!{\cos n\varphi {\rm{d}}\varphi } } \\ & = \dfrac{{\alpha {\rho _0}}}{{\rm{\text{π} }}}{J_0}(kr) + \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\dfrac{2}{n}\sin n\alpha } \\ & = \dfrac{{\alpha {\rho _0}}}{{\rm{\text{π} }}}{J_0}(kr) + \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{{{{(- 1)}^m}}}{m}{J_{2m}}(kr)\sin 2m\alpha + }\\ &\quad \dfrac{{2{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{{{{(- i)}^{2m - 1}}}}{{2m - 1}}{J_{2m - 1}}(kr)\sin (2m - 1)\alpha } \\ \end{split} $

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) &= \dfrac{{{\rho _0}{J_0}(kr)}}{{2{\rm{\text{π} }}}}\int\limits_{{\rm{\text{π} }} - \alpha }^{{\rm{\text{π} }} + \alpha } {{\rm{d}}\varphi } +\\&\quad \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\int\limits_{{\rm{\text{π} }} - \alpha }^{{\rm{\text{π} }} + \alpha } {\cos n\varphi {\rm{d}}\varphi } } \\& = \dfrac{{\alpha {\rho _0}}}{{\rm{\text{π} }}}{J_0}(kr) + \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{{{{(- 1)}^m}}}{m}{J_{2m}}(kr)\sin 2m\alpha - }\\&\quad \dfrac{{2{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{{{{(- i)}^{2m - 1}}}}{{2m - 1}}{J_{2m - 1}}(kr)\sin (2m - 1)\alpha } \\ \end{split} $

$\begin{split} {\widehat {\phi }} _{_{{\rm{ab}}}}^{\rm{ + }}(r,\omega) &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rho _0}{\rm{d}}\varphi } \\&= \dfrac{{{\rho _0}}}{2}\left[ {{J_0}(kr) - i\dfrac{4}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{1}{{2m - 1}}{J_{2m - 1}}(kr)} } \right] \end{split}$

${{\rm H}_0}(z) = \dfrac{4}{{\rm{\text{π} }}}\sum\limits_{k = 1}^\infty {\dfrac{1}{{2k - 1}}{J_{2k - 1}}(z)} $

${\widehat {\phi }} _{_{{\rm{ab}}}}^ + (r,\omega) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rho _0}{\rm{d}}\varphi } = \dfrac{{{\rho _0}}}{2}\left[ {{J_0}(kr) - i{{\rm H}_0}(kr)} \right]$

$\begin{split} {\widehat {\phi }} _{_{{\rm{ab}}}}^{\rm{ - }}(r,\omega)& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rho _0}{\rm{d}}\varphi } \\&= \dfrac{{\alpha {\rho _0}}}{{\rm{\text{π} }}}{J_0}(kr) + i\dfrac{{2{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{1}{{2m - 1}}{J_{2m - 1}}(kr)} \\& = \dfrac{{{\rho _0}}}{2}\left[ {{J_0}(kr) + i\dfrac{4}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{1}{{2m - 1}}{J_{2m - 1}}(kr)} } \right] \\& = \dfrac{{{\rho _0}}}{2}\left[ {{J_0}(kr) + i{{\rm H}_0}(kr)} \right] \\ \end{split} $

${C_{{\rm{ab}}}}(r,\omega,t) = {\rm{Re}} \left\{ {{{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - ikr\cos (\theta - \varphi)}}} \right\} = \cos \left[ {\omega (t - {\tau _{{\rm{ab}}}})} \right]$

$\begin{split}& {J_0}(kr) \approx \sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} \cos (kr - \dfrac{{\rm{\text{π} }}}{4}) \\& {{\rm{H}}_0}(kr) \approx \sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} \sin (kr - \dfrac{{\rm{\text{π} }}}{4}) \\ \end{split} $

$\begin{split} {\rm{Re}} \left\{ {{{\rm{e}}^{i\omega t}}{\widehat {\phi }} _{_{ab}}^ + } \right\} \!\!&= \dfrac{{{\rho _0}}}{2}\!\sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} {\rm{Re}} \left\{ \!{{{\rm{e}}^{i\omega t}}\left[\! {\cos (kr \!-\! \dfrac{{\rm{\text{π} }}}{4}) \!-\! i\sin (kr \!-\! \dfrac{{\rm{\text{π} }}}{4})} \right]} \right\} \\& = \dfrac{{{\rho _0}}}{2}\sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} \cos \left[ {\omega \left({t - ({\tau _{{\rm{ab}}}} - \dfrac{{\rm{\text{π} }}}{{4\omega }})} \right)} \right] \\& = \dfrac{{{\rho _0}}}{2}\sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} \cos \left[ {\omega (t - \tau _{{\rm{ab}}}^{\rm{ + }})} \right] \\ \end{split} $

$\tau _{{\rm{ab}}}^{\rm{ + }} = {\tau _{{\rm{ab}}}} - \dfrac{{\rm{\text{π} }}}{{4\omega }}$

$\begin{split} {\rm{Re}}\! \left\{ \! {{{\rm{e}}^{i\omega t}}{\widehat {\phi }} _{_{{\rm{ab}}}}^{\rm{ - }}} \right\}\!\! &=\! \dfrac{{{\rho _0}}}{2}\!\sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} {\rm{Re}}\! \left\{ \!{{{\rm{e}}^{i\omega t}}\left[ {\cos \left(kr \!-\! \dfrac{{\rm{\text{π} }}}{4}\right)\! +\! i\sin \left(kr \!-\! \dfrac{{\rm{\text{π} }}}{4}\right)} \right]} \right\} \\& = \dfrac{{{\rho _0}}}{2}\sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} \cos \left[ {\omega \left({t + \left({\tau _{{\rm{ab}}}} - \dfrac{{\rm{\text{π} }}}{{4\omega }}\right)} \right)} \right] \\ & = \dfrac{{{\rho _0}}}{2}\sqrt {\dfrac{2}{{{\rm{\text{π} }}kr}}} \cos \left[ {\omega (t - \tau _{{\rm{ab}}}^{\rm{ - }})} \right] \\ \end{split} $

$\tau _{{\rm{ab}}}^{\rm{ - }} = - \left({{\tau _{{\rm{ab}}}} - \dfrac{{\rm{\text{π} }}}{{4\omega }}} \right)$

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{{\rm{ - }}ikr\cos \varphi }}{\rho ^{{\rm{wave}}}}(\varphi){\rm{d}}\varphi } \\ &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\left[ {{J_0}(kr) + \sum\limits_{n = 1}^\infty {{\varepsilon _n}{{(- i)}^n}{J_n}(kr)\cos n\varphi } } \right]{\rho _0}{\rm{d}}\varphi } \\ & = \dfrac{{{J_0}(kr)}}{{2{\rm{\text{π} }}}}\left[ {\int\limits_\alpha ^{{\rm{\text{π} }} - \alpha } {{\rho _0}{\rm{d}}\varphi } {\rm{ + }}\int\limits_{{\rm{\text{π} }} + \alpha }^{2{\rm{\text{π} }} - \alpha } {{\rho _0}{\rm{d}}\varphi } } \right] + \\ &\quad \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\int\limits_\alpha ^{{\rm{\text{π} }} - \alpha } \sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\cos n\varphi } {\rm{d}}\varphi + \\ &\quad\dfrac{{{\rho _0}}}{{\rm{\text{π} }}} \int\limits_{{\rm{\text{π} }} + \alpha }^{2{\rm{\text{π} }} - \alpha } {\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\cos n\varphi } {\rm{d}}\varphi } \\[-15pt] \end{split} $

$\int\limits_\alpha ^{{\rm{\text{π} }} - \alpha } {\cos n\varphi {\rm{d}}\varphi } = \int\limits_{{\rm{\text{π} }} + \alpha }^{2{\rm{\text{π} }} - \alpha } {\cos n\varphi {\rm{d}}\varphi } {\rm{ = }}\left\{ {\begin{array}{*{20}{l}} { - \dfrac{1}{m}\sin 2m\alpha }&{n = 2m} \\ \qquad 0&{n = 2m - 1} \end{array}} \right.$

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(r,\omega) &= \dfrac{{{\rho _0}{J_0}(kr)}}{{2{\rm{\text{π} }}}}\int\limits_\alpha ^{{\rm{\text{π} }} - \alpha } {{\rm{d}}\varphi } + \\ &\quad\dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\int\limits_\alpha ^{{\rm{\text{π} }} - \alpha } {\cos n\varphi {\rm{d}}\varphi } } \\ & = \dfrac{{({\rm{\text{π} }} - 2\alpha){\rho _0}}}{{2{\rm{\text{π} }}}}{J_0}(kr) -\\ &\quad \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{{{{(- 1)}^m}}}{m}{J_{2m}}(kr)\sin 2m\alpha } \\[-15pt] \end{split} $

$\begin{split} \widehat \phi _{{\rm{ab}}}^{\overline {{\rm{wave}}} }(\omega) &= \dfrac{{{\rho _0}{J_0}(kr)}}{{2{\rm{\text{π} }}}}\int\limits_{{\rm{\text{π} }} + \alpha }^{2{\rm{\text{π} }} - \alpha } {{\rm{d}}\varphi } +\\&\quad \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{n = 1}^\infty {{{(- i)}^n}{J_n}(kr)\int\limits_{{\rm{\text{π} }} + \alpha }^{2{\rm{\text{π} }} - \alpha } {\cos n\varphi {\rm{d}}\varphi } } \\& = \dfrac{{({\rm{\text{π} }} - 2\alpha){\rho _0}}}{{2{\rm{\text{π} }}}}{J_0}(kr) - \\&\quad \dfrac{{{\rho _0}}}{{\rm{\text{π} }}}\sum\limits_{m = 1}^\infty {\dfrac{{{{(- 1)}^m}}}{m}{J_{2m}}(kr)\sin 2m\alpha } \end{split} $

${\widehat \phi _{{\rm{ab}}}}(r,\omega) = \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{\rho ^{{\rm{wave}}}}(\omega ;\theta,\varphi){{\rm{e}}^{{\rm{ - }}ikr\sin \theta \cos \varphi }}{\rm{d}}\varphi } } $

$\begin{split} {\widehat \phi _{{\rm{ab}}}}(r,\omega) &= \dfrac{1}{2}\int\limits_0^{\rm{\text{π} }} {\rho (\theta)\sin \theta {\rm{d}}\theta } \cdot \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{{\rm{ - }}ikr\sin \theta \cos \varphi }}{\rm{d}}\varphi } \\& = \dfrac{1}{2}\int\limits_0^{\rm{\text{π} }} {\rho (\theta){J_0}(kr\sin \theta)\sin \theta {\rm{d}}\theta } \\[-15pt] \end{split} $

$\begin{split} {J_0}(x) &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ix\sin \theta }}{\rm{d}}\theta = } \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ix\cos \theta }}{\rm{d}}\theta } \\ & = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ix\sin \theta }}{\rm{d}}\theta = } \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ix\cos \theta }}{\rm{d}}\theta } \end{split} $

$\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\rho (\varphi){\rm{d}}\varphi = 1} $

$\dfrac{1}{\text{π} }\int\limits_0^\text{π} {{\rho _0}\sin \theta d\theta = 1} $

$\begin{split} {\widehat \phi _{{\rm{ab}}}}(r,\omega) &= \dfrac{1}{{2(1 - \cos \alpha)}}\int\limits_0^{\rm{\text{π} }} {{J_0}(kr\sin \theta)\sin \theta {\rm{d}}\theta } \\ & = \dfrac{1}{{2(1 - \cos \alpha)}}\left[ \int\limits_0^\alpha {{J_0}(kr\sin \theta)\sin \theta {\rm{d}}\theta } +\right.\\ &\qquad\qquad \left. \int\limits_{\text{π} - \alpha }^\text{π} {{J_0}(kr\sin \theta)\sin \theta {\rm{d}}\theta } \right] \\ & = \dfrac{1}{{(1 - \cos \alpha)}}\int\limits_0^\alpha {{J_0}(kr\sin \theta)\sin \theta {\rm{d}}\theta } \end{split} $

${\widehat \phi _{{\rm{ab}}}}(r,\omega) = \int\limits_0^{{{\rm{\text{π} }} / {\rm{2}}}} {{J_0}(kr\sin \theta)\sin \theta {\rm{d}}\theta } {\rm{ = }}\dfrac{{\sin kr}}{{kr}} = {j_0}(kr)$

${\widehat \phi _{{\rm{ab}}}}(r,\omega) = \dfrac{1}{{4{\rm{\text{π} }}}}\int\limits_0^{\rm{\text{π} }} {{{\rm{e}}^{{\rm{ - }}ikr\cos \theta }}\sin \theta {\rm{d}}\theta \int\limits_0^{2{\rm{\text{π} }}} {{\rm{d}}\varphi } } $

$\begin{split} {\widehat \phi _{{\rm{ab}}}}(r,\omega) &= \dfrac{1}{2}{\rho _0}\left[ {\int\limits_0^\alpha {{{\rm{e}}^{ikr\cos \theta }}\sin \theta {\rm{d}}\theta } + \int\limits_{{\rm{\text{π} }} - \alpha }^{\rm{\text{π} }} {{{\rm{e}}^{ikr\cos \theta }}\sin \theta {\rm{d}}\theta } } \right] \\& = \dfrac{{\sin (kr) - \sin (kr\cos \alpha)}}{{kr(1 - \cos \alpha)}} \\[-10pt] \end{split} $

$\begin{split} G_{ij}^{^{{\rm{Rayleigh}}}} &= \left[ {\begin{array}{*{20}{c}} {{G_{NN}}}&{{G_{NE}}}&{{G_{NZ}}} \\ {{G_{EN}}}&{{G_{EE}}}&{{G_{EZ}}} \\ {{G_{ZN}}}&{{G_{ZE}}}&{{G_{ZZ}}} \end{array}} \right] \\ & = \sum\limits_n {\dfrac{{ - i}}{{8cuI}}\left[ {\begin{array}{*{20}{c}} {V(z)V(z')\left[ {\dfrac{1}{{kr}}{H_1} - \cos {}^2\varphi {H_2}} \right]}&{ - V(z)V(z')\sin \varphi \cos \varphi {H_2}}&{V(z)U(z')\cos \varphi {H_1}} \\ { - V(z)V(z')\sin \varphi \cos \varphi {H_2}}&{V(z)V(z')\left[ {\dfrac{1}{{kr}}{H_1} - \sin {}^2\varphi {H_2}} \right]}&{V(z)U(z')\sin \varphi {H_1}} \\ { - U(z)V(z')\cos \varphi {H_1}}&{ - U(z)V(z')\sin \varphi {H_1}}&{U(z)U(z'){H_0}} \end{array}} \right]} \\ \end{split} $

$G_{ij}^{{\rm{Love}}} = \left[ {\begin{array}{*{20}{c}} {{G_{NN}}}&{{G_{NE}}}&{{G_{NZ}}} \\ {{G_{EN}}}&{{G_{EE}}}&{{G_{EZ}}} \\ {{G_{ZN}}}&{{G_{ZE}}}&{{G_{ZZ}}} \end{array}} \right] = \sum\limits_n {\dfrac{{ - i}}{{8cuI}}\left[ {\begin{array}{*{20}{c}} {W(z)W(z')\left[ {\dfrac{1}{{kr}}{H_1} - \sin {}^2\varphi {H_2}} \right]}&{W(z)W(z')\sin \varphi \cos \varphi {H_2}}&0 \\ {W(z)W(z')\sin \varphi \cos \varphi {H_2}}&{W(z)W(z')\left[ {\dfrac{1}{{kr}}{H_1} - \cos {}^2\varphi {H_2}} \right]}&0 \\ 0&0&0 \end{array}} \right]} $

$\left[ {\begin{array}{*{20}{c}} {{G_{ZZ}}}&{{G_{ZR}}}&{{G_{ZT}}} \\ {{G_{RZ}}}&{{G_{RR}}}&{{G_{RT}}} \\ {{G_{TZ}}}&{{G_{TR}}}&{{G_{TT}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&0&1 \\ {\cos \varphi }&{\sin \varphi }&0 \\ { - \sin \varphi }&{\cos \varphi }&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{G_{NN}}}&{{G_{NE}}}&{{G_{NZ}}} \\ {{G_{EN}}}&{{G_{EE}}}&{{G_{EZ}}} \\ {{G_{ZN}}}&{{G_{ZE}}}&{{G_{ZZ}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&{\cos \varphi }&{ - \sin \varphi } \\ 0&{\sin \varphi }&{\cos \varphi } \\ 1&0&0 \end{array}} \right]$

$\begin{split} G_{ij}^{{\rm{Rayleigh}}} = \left[ {\begin{array}{*{20}{c}} {{G_{ZZ}}}&{{G_{ZR}}}&{{G_{ZT}}} \\ {{G_{RZ}}}&{{G_{RR}}}&{{G_{RT}}} \\ {{G_{TZ}}}&{{G_{TR}}}&{{G_{TT}}} \end{array}} \right] = \sum\limits_n {\dfrac{{ - i}}{{8cuI}}\left[ {\begin{array}{*{20}{c}} {U(z)U(z'){H_0}}&{ - U(z)V(z'){H_1}}&0 \\ {V(z)U(z'){H_1}}&{V(z)V(z')\left[ {\dfrac{{{H_0} - {H_2}}}{2}} \right]}&0 \\ 0&0&{V(z)V(z')\left[ {\dfrac{{{H_0} + {H_2}}}{2}} \right]} \end{array}} \right]} \end{split} $

$G_{ij}^{{\rm{Love}}} = \left[ {\begin{array}{*{20}{c}} {{G_{ZZ}}}&{{G_{ZR}}}&{{G_{ZT}}} \\ {{G_{RZ}}}&{{G_{RR}}}&{{G_{RT}}} \\ {{G_{TZ}}}&{{G_{TR}}}&{{G_{TT}}} \end{array}} \right] = \sum\limits_n {\dfrac{{ - i}}{{8cuI}}\left[ {\begin{array}{*{20}{c}} 0&0&0 \\ 0&{W(z)W(z')\left[ {\dfrac{{{H_0} + {H_2}}}{2}} \right]}&0 \\ 0&0&{W(z)W(z')\left[ {\dfrac{{{H_0} - {H_2}}}{2}} \right]} \end{array}} \right]} $

$\left\langle {u_{_i}^{\rm{*}}(a;\omega){u_j}(b;\omega)} \right\rangle = - s(\omega)\left[ {{G_{ij}}(b,a;\omega) - G_{_{ji}}^*(a,b;\omega)} \right]$

$\left\langle {u_i^*(a;\omega){u_j}(b;\omega)} \right\rangle = - 2s(\omega){\rm{Im}} \left[ {{G_{ij}}(b,a;\omega)} \right]$

${\overline \phi _{ii}}(a,b;\omega) = \dfrac{{\left\langle {u_i^*(a;\omega){u_i}(b;\omega)} \right\rangle }}{{\sqrt {\left\langle {u_i^*(a;\omega){u_i}(a;\omega)} \right\rangle } \sqrt {\left\langle {u_i^*(b;\omega){u_i}(b;\omega)} \right\rangle } }}$

$\left\langle {u_i^*(a;\omega){u_i}(a;\omega)} \right\rangle = \left\langle {u_i^*(b;\omega){u_i}(b;\omega)} \right\rangle $

${\overline \phi _{ii}}(a,b;\omega) = \dfrac{{\left\langle {u_i^*(a;\omega){u_i}(b;\omega)} \right\rangle }}{{\left\langle {u_i^*(a;\omega){u_i}(a;\omega)} \right\rangle }}$

${\overline \phi _{ii}}(a,b;\omega) = \dfrac{{{\rm{Im}} \left[ {{G_{ii}}(b,a;\omega)} \right]}}{{{\rm{Im}} \left[ {{G_{ii}}(a,a;\omega)} \right]}}$

$ \begin{split} G_{ZZ}^{\rm{Ray}}(b,a;\omega) &= - \dfrac{i}{{8cuI}}U(b)U(a){H_0} \\ &= - \dfrac{i}{{8cuI}}U(b)U(a)\left[ {{J_0}(kr) - i{Y_0}(kr)} \right] \end{split} $

$ \begin{split}& {\rm{Im}} \left[ {G_{ZZ}^{\rm{Ray}}(b,a;\omega)} \right] = - \dfrac{1}{{8cuI}}U(b)U(a){J_0}(kr) \\& {\rm{Im}} \left[ {G_{ZZ}^{\rm{Ray}}(a,a;\omega)} \right] =- \dfrac{1}{{8cuI}}U(a)U(a){J_0}(0) \end{split} $

$ \overline \phi _{ZZ}^{\rm{Ray}}(a,b;\omega) = {J_0}(kr) $

$ \begin{split}& \overline \phi _{RR}^{\rm{Ray}}(a,b;\omega) = {J_0}(kr) - {J_2}(kr) \\& \overline \phi _{TT}^{\rm{Ray}}(a,b;\omega) = {J_0}(kr) + {J_2}(kr) \\ \end{split} $

$\begin{array}{*{20}{c}} {{\phi _{ij}}(a,b;\omega) = \left\langle {u_i^*(a;\omega){u_j}(b;\omega)} \right\rangle }&{(i \ne j)} \end{array}$

$ \begin{split} \phi _{ZR}^{\rm{Ray}}(a,b;\omega) &\!=\! \left\langle {u_Z^*(a;\omega){u_R}(b;\omega)} \right\rangle \!=\! -\! 2s(\omega){\rm{Im}} \left[ {{G_{ZR}}(b,a;\omega)} \right] \\& \!=\! -\! 2s(\omega){\rm{Im}} \left[ {\dfrac{i}{{8cuI}}U(b)V(a)\left[ {{J_1}(kr) + i{Y_1}(kr)} \right]} \right] \\& \! =\! -\! \dfrac{{s(\omega)}}{{4cuI}}U(b)V(a){J_1}(kr) = - S(\omega){J_1}(kr) \\ \end{split} $

$\phi _{RZ}^{\rm{Ray}}(a,b;\omega) = \left\langle {u_{{R}}^*(a;\omega){u_Z}(b;\omega)} \right\rangle = S(\omega){J_1}(kr)$

$S(\omega) = \dfrac{{s(\omega)}}{{4cuI}}UV$

$ \begin{split} {\rm{SPA}}{{\rm{C}}^{{\rm{Rayleigh}}}} = \left[ {\begin{array}{*{20}{c}} {\bar \phi _{ZZ}^{\rm{Ray}}}&{\phi _{{\rm{Z}}R}^{\rm{Ray}}}&{\phi _{ZT}^{\rm{Ray}}} \\ {\phi _{RZ}^{\rm{Ray}}}&{\bar \phi _{RR}^{\rm{Ray}}}&{\phi _{RT}^{\rm{Ray}}} \\ {\phi _{TZ}^{\rm{Ray}}}&{\phi _{TR}^{\rm{Ray}}}&{\bar \phi _{TT}^{\rm{Ray}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{J_0}(kr)}&{ - S(\omega){J_1}(kr)}&0 \\ {S(\omega){J_1}(kr)}&{{J_0}(kr) - {J_2}(kr)}&0 \\ 0&0&{{J_0}(kr) + {J_2}(kr)} \end{array}} \right] \end{split} $

$ {\rm{SPA}}{{\rm{C}}^{{\rm{Love}}}} = \left[ {\begin{array}{*{20}{c}} {\bar \phi _{ZZ}^{\rm{Lov}}}&{\phi _{{\rm{Z}}R}^{\rm{Lov}}}&{\phi _{ZT}^{\rm{Lov}}} \\ {\phi _{RZ}^{\rm{Lov}}}&{\bar \phi _{RR}^{\rm{Lov}}}&{\phi _{RT}^{\rm{Lov}}} \\ {\phi _{TZ}^{\rm{Lov}}}&{\phi _{TR}^{\rm{Lov}}}&{\bar \phi _{TT}^{\rm{Lov}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&0&0 \\ 0&{{J_0}(kr) + {J_2}(kr)}&0 \\ 0&0&{{J_0}(kr) - {J_2}(kr)} \end{array}} \right] $

$ \phi _{ZR}^{\rm{Ray}}(a,b;\omega) = - \dfrac{V}{U}{J_1}(kr) $

$ \phi _{RZ}^{\rm{Ray}}(a,b;\omega) = \dfrac{V}{U}{J_1}(kr) $

$ {\rm{SPA}}{{\rm{C}}^{{\rm{Rayleigh}}}} = \left[ {\begin{array}{*{20}{c}} {\phi _{ZZ}^{\rm{Ray}}}&{\phi _{{\rm{Z}}R}^{\rm{Ray}}}&{\phi _{ZT}^{\rm{Ray}}} \\ {\phi _{RZ}^{\rm{Ray}}}&{\phi _{RR}^{\rm{Ray}}}&{\phi _{RT}^{\rm{Ray}}} \\ {\phi _{TZ}^{\rm{Ray}}}&{\phi _{TR}^{\rm{Ray}}}&{\phi _{TT}^{\rm{Ray}}} \end{array}} \right] = \dfrac{{s(\omega)}}{{4cuI}}UU\left[ {\begin{array}{*{20}{c}} {{J_0}(kr)}&{ - \dfrac{V}{U}{J_1}(kr)}&0 \\ {\dfrac{V}{U}{J_1}(kr)}&{\dfrac{{{V^2}}}{{{U^2}}}\dfrac{1}{2}\left[ {{J_0}(kr) - {J_2}(kr)} \right]}&0 \\ 0&0&{\dfrac{{{V^2}}}{{{U^2}}}\dfrac{1}{2}\left[ {{J_0}(kr) + {J_2}(kr)} \right]} \end{array}} \right] $

$ {\rm{SPA}}{{\rm{C}}^{{\rm{Love}}}} = \left[ {\begin{array}{*{20}{c}} {\phi _{ZZ}^{\rm{Lov}}}&{\phi _{{\rm{Z}}R}^{\rm{Lov}}}&{\phi _{ZT}^{\rm{Lov}}} \\ {\phi _{RZ}^{\rm{Lov}}}&{\phi _{RR}^{\rm{Lov}}}&{\phi _{RT}^{\rm{Lov}}} \\ {\phi _{TZ}^{\rm{Lov}}}&{\phi _{TR}^{\rm{Lov}}}&{\phi _{TT}^{\rm{Lov}}} \end{array}} \right] = \dfrac{{s(\omega)}}{{4cuI}}WW\left[ {\begin{array}{*{20}{c}} 0&0&0 \\ 0&{\dfrac{1}{2}\left[ {{J_0}(kr) + {J_2}(kr)} \right]}&0 \\ 0&0&{\dfrac{1}{2}\left[ {{J_0}(kr) - {J_2}(kr)} \right]} \end{array}} \right] $

$u(\omega,r) = \int\limits_0^\infty {\dfrac{{{N^{{\rm{Ray}}}}(k,\omega)}}{{{D_R}(k,\omega)}}{J_n}(kr)k{\rm{d}}k} + \int\limits_0^\infty {\dfrac{{{N^{{\rm{Lov}}}}(k,\omega)}}{{{D_L}(k,\omega)}}{J_n}(kr)k{\rm{d}}k} $

$I(k,\omega) = \dfrac{{N(k,\omega)}}{{D(k,\omega)}} = \int\limits_0^\infty {u(\omega,r){J_n}(kr)rdr} $

${G_{ij}}(\omega,r) = \int\limits_0^\infty {{g_{ij}}(k,\omega){J_{ij}}(k,r)k{\rm{d}}k} $

$ \begin{split} g_{ij}^{{\rm{Rayleigh}}}(k,\omega) &\!= \!\left[ {\begin{array}{*{20}{c}} {{g_{{\rm{Z}}Z}}}&{{g_{{\rm{ZR}}}}}&{{g_{{\rm{ZT}}}}} \\ {{g_{{\rm{RZ}}}}}&{{g_{{\rm{RR}}}}}&{{g_{{\rm{RT}}}}} \\ {{g_{{\rm{TZ}}}}}&{{g_{{\rm{TR}}}}}&{{g_{{\rm{TT}}}}} \end{array}} \right] \\&\!=\! \left[ {\begin{array}{*{20}{c}} {\dfrac{{N_{{\rm{ZZ}}}^{{\rm{Ray}}}(k,\omega)}}{{{D_{\rm{R}}}(k,\omega)}}}&{\dfrac{{N_{{\rm{ZR}}}^{{\rm{Ray}}}(k,\omega)}}{{{D_{\rm{R}}}(k,\omega)}}}&0 \\ {\dfrac{{N_{{\rm{RZ}}}^{{\rm{Ray}}}(k,\omega)}}{{{D_{\rm{R}}}(k,\omega)}}}&{\dfrac{{N_{{\rm{RR}}}^{{\rm{Ray}}}(k,\omega)}}{{{D_{\rm{R}}}(k,\omega)}}}&0 \\ 0&0&{\dfrac{{N_{{\rm{TT}}}^{{\rm{Ray}}}(k,\omega)}}{{{D_{\rm{R}}}(k,\omega)}}} \end{array}} \right] \end{split} $

$ \begin{split} J_{ij}^{{\rm{Rayleigh}}}(k,r) &\!\!=\!\! \left[\!\! {\begin{array}{*{20}{c}}\!\! {{J_{{\rm{ZZ}}}}}&\!{{J_{{\rm{ZR}}}}}&\!{{J_{{\rm{ZT}}}}} \\ \!\!{{J_{{\rm{RZ}}}}}&\!{{J_{{\rm{RR}}}}}&\!{{J_{{\rm{RT}}}}} \\ \!\!{{J_{{\rm{TZ}}}}}&\!{{J_{{\rm{TR}}}}}&\!{{J_{{\rm{TT}}}}} \end{array}} \right] \\&\!\!=\!\! \left[\!\! {\begin{array}{*{20}{c}}\!\! {{J_0}(kr)}&\!\!{{J_1}(kr)}&\!\!\!\!0 \\ \!\!\!{{J_1}(kr)}&\!\!\!{[{J_0}(kr) - {J_2}(kr)]}&\!\!\!\!0 \\ \!\!\!0&\!\!\!0&\!\!\!\!{[{J_0}(kr) + {J_2}(kr)]} \end{array}} \!\!\!\!\right] \end{split} $

$ \begin{split} g_{ij}^{{\rm{Love}}}(k,\omega) &= \left[ {\begin{array}{*{20}{c}} {{g_{{\rm{ZZ}}}}}&{{g_{{\rm{ZR}}}}}&{{g_{{\rm{ZT}}}}} \\ {{g_{{\rm{RZ}}}}}&{{g_{{\rm{RR}}}}}&{{g_{{\rm{RT}}}}} \\ {{g_{{\rm{TZ}}}}}&{{g_{{\rm{TR}}}}}&{{g_{{\rm{TT}}}}} \end{array}} \right] \\&= \left[ {\begin{array}{*{20}{c}} 0&0&0 \\ 0&{\dfrac{{N_{{\rm{RR}}}^{{\rm{Lov}}}(k,\omega)}}{{{D_{\rm{L}}}(k,\omega)}}}&0 \\ 0&0&{\dfrac{{N_{{\rm{TT}}}^{{\rm{Lov}}}(k,\omega)}}{{{D_{\rm{L}}}(k,\omega)}}} \end{array}} \right] \end{split} $

$ \begin{split} J_{ij}^{{\rm{Love}}}(k,r) &= \left[ {\begin{array}{*{20}{c}} {{J_{{\rm{ZZ}}}}}&{{J_{{\rm{ZR}}}}}&{{J_{{\rm{ZT}}}}} \\ {{J_{{\rm{RZ}}}}}&{{J_{{\rm{RR}}}}}&{{J_{{\rm{RT}}}}} \\ {{J_{{\rm{TZ}}}}}&{{J_{{\rm{TR}}}}}&{{J_{{\rm{TT}}}}} \end{array}} \right] \\&= \left[ {\begin{array}{*{20}{c}} 0&0&0 \\ 0&{[{J_0}(kr) + {J_2}(kr)]}&0 \\ 0&0&{[{J_0}(kr) - {J_2}(kr)]} \end{array}} \right] \end{split}$

$\,{\phi _{ij}}(r,\omega)\! \approx \!- 2s(\omega){\rm{Im}} [{{G}}(\omega,r)] \!=\! - 2s(\omega)\int\limits_0^\infty {{\rm{Im}} [{g_{ij}}]{J_{ij}}(kr)k{\rm{d}}k} $

$ - 2s(\omega){\rm{Im}} [{g_{ij}}(k,\omega)] \approx \int\limits_0^\infty {{\phi _{ij}}(r,\omega){J_{ij}}(k,r)r{\rm{d}}r} $

$\,I_{ZZ}^{{\rm{Ray}}}(k,\omega) \!=\! - 2s(\omega){\rm{Im}} \left[ {\dfrac{{N_{ZZ}^{{\rm{Ray}}}(k,\omega)}}{{{D_R}(k,\omega)}}} \right] \!\approx\! \int\limits_0^\infty {\phi _{ZZ}^{{\rm{Ray}}}(\omega,r){J_0}(kr)r{\rm{d}}r} $

$\begin{split} I_{ZZ}^{{\rm{Ray}}}(k,\omega) &= - 2s(\omega){\rm{Im}} \left[ {\dfrac{{N_{ZZ}^{{\rm{Ray}}}(k,\omega)}}{{{D_R}(k,\omega)}}} \right] \\& \approx \dfrac{{s(\omega)}}{{4cuI}}UU\int\limits_0^\infty {{J_0}({k_R}r){J_0}(kr)r{\rm d}r} \end{split} $

$\int\limits_0^\infty {{J_m}(kr){J_m}(k'r)r{\rm{d}}r} = \dfrac{{\delta (k - k')}}{{\sqrt {kk'} }}$

$\,I_{ZZ}^{{\rm{Ray}}}(k,\omega) \!=\! - 2s(\omega){\rm{Im}} \left[ {\dfrac{{N_{ZZ}^{{\rm{Ray}}}(k,\omega)}}{{{D_R}(k,\omega)}}} \right] \!\approx\! \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}\!U\!U\!\dfrac{{\delta (k - {k_R})}}{{{k_R}}}$

$\begin{split}& I_{RZ}^{{\rm{Ray}}}(k,\omega) = \int\limits_0^\infty {\phi _{RZ}^{{\rm{Ray}}}(\omega,r){J_1}(kr)r{\rm{d}}r} \\& I_{ZR}^{{\rm{Ray}}}(k,\omega) = \int\limits_0^\infty {\phi _{ZR}^{{\rm{Ray}}}(\omega,r){J_1}(kr)r{\rm{d}}r} \end{split} $

$I_{ZR}^{{\rm{Ray}}}(k,\omega) = - I_{RZ}^{{\rm{Ray}}}(k,\omega)$

$\begin{split} I_{ZR,RZ}^{{\rm{Ray}}}(k,\omega) &= \dfrac{1}{2}\int\limits_0^\infty {\left[ {\phi _{RZ}^{{\rm{Ray}}}(\omega,r) - \phi _{ZR}^{{\rm{Ray}}}(\omega,r)} \right]{J_1}(kr)r{\rm{d}}r} \\& = \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}UV\dfrac{{\delta (k - {k_R})}}{{{k_R}}} \end{split} $

$\begin{split} {\phi _{RR}} + {\phi _{TT}} &= \phi _{_{RR}}^{{\rm{Ray}}} + \phi _{_{RR}}^{{\rm{Lov}}} + \phi _{_{TT}}^{{\rm{Ray}}} + \phi _{_{TT}}^{{\rm{Lov}}} \\&= \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}VV \cdot {J_0}({k_R}r) + \dfrac{{s(\omega)}}{{4{c_L}{u_L}{I_L}}}WW \cdot {J_0}({k_L}r) \\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\phi _{RR}}\! -\! {\phi _{TT}} = \phi _{_{RR}}^{{\rm{Ray}}} + \phi _{_{RR}}^{{\rm{Lov}}} - \phi _{_{TT}}^{{\rm{Ray}}} - \phi _{_{TT}}^{{\rm{Lov}}} \\&= - \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}VV \!\cdot\! {J_2}({k_R}r) \!+\! \dfrac{{s(\omega)}}{{4{c_L}{u_L}{I_L}}}WW \!\cdot\! {J_2}({k_L}r) \end{split} $

$\begin{split}\, & {I^ + } = \int\limits_0^\infty {\left[ {{\phi _{RR}} + {\phi _{TT}}} \right]} {J_0}(kr)r{\rm{d}}r \\&\quad= \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}VV\dfrac{1}{{{k_R}}}\delta (k - {k_R}) + \dfrac{{s(\omega)}}{{4{c_L}{u_L}{I_L}}}WW\dfrac{1}{{{k_L}}}\delta (k - {k_L}) \\& {I^ - } = \int\limits_0^\infty {\left[ {{\phi _{RR}} - {\phi _{TT}}} \right]} {J_2}(kr)r{\rm{d}}r \\&\quad= - \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}VV\dfrac{1}{{{k_R}}}\delta (k - {k_R}) + \dfrac{{s(\omega)}}{{4{c_L}{u_L}{I_L}}}WW\dfrac{1}{{{k_L}}}\delta (k - {k_L}) \end{split} $

${I_{RR}}(k,\omega) = \dfrac{1}{2}({I^ + } - {I^ - }) = \dfrac{{s(\omega)}}{{4{c_R}{u_R}{I_R}}}VV\dfrac{1}{{{k_R}}}\delta (k - {k_R})$

${I_{TT}}(k,\omega) = \dfrac{1}{2}({I^ + } + {I^ - }) = \dfrac{{s(\omega)}}{{4{c_L}{u_L}{I_L}}}WW\dfrac{1}{{{k_L}}}\delta (k - {k_L})$

${\nabla ^2}u = \dfrac{1}{{c_0^2}}\dfrac{{{\partial^2}u}}{{\partial {t^2}}} + \dfrac{{2\alpha }}{{{c_0}}}\dfrac{{\partial u}}{{\partial t}} + {\alpha ^2}u$

$\left[ {{\nabla ^2} + {{({k_0} - i\alpha)}^2}} \right]u = 0$

$({\nabla ^2} + {k^2}u) = 0$

$u = A{{\rm{e}}^{ - ikr}}{{\rm{e}}^{i\omega t}} = A{{\rm{e}}^{ - i({k_0}{\rm{ - }}i\alpha)r}}{{\rm{e}}^{i\omega t}}$

${Q_{\rm{a}}} = A{{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - i{{k}} \cdot ({{{r}}_1} - {{R}})}} = A{{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_1} - {{R}})}}$

${Q_{\rm{b}}} = A{{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_2} - {{R}})}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{R}})}}$

$\langle A(\varphi)A(\varphi ')\rangle = {A^2}\delta (\varphi - \varphi ')$

$\begin{split} {P_{\rm{a}}}& = \int\limits_0^{2{\rm{\text{π} }}} {A(\varphi){{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - i{{k}} \cdot ({{{r}}_1} - {{R}})}}{\rm{d}}\varphi }\\ & {\rm{ = }}\int\limits_0^{2{\rm{\text{π} }}} {A(\varphi){{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_1} - {{R}})}}{\rm{d}}\varphi } \end{split} $

${P_{\rm{b}}} = \int\limits_0^{2{\rm{\text{π} }}} {A(\varphi '){{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - \alpha {{\hat k'}} \cdot ({{{r}}_2} - {{R'}})}}{{\rm{e}}^{ - i{{{{{k'}}_{\!\!\! 0}}}} \cdot ({{{r}}_2} - {{R'}})}}{\rm{d}}\varphi '} $

$ \begin{split} P_{\rm{a}}^{\rm{*}}{P_{\rm{b}}} =& \int\limits_0^{2{\rm{\text{π} }}} {A(\varphi){{\rm{e}}^{ - i\omega t}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}{{\rm{e}}^{i{k_0} \cdot ({{{r}}_1} - {{R}})}}{\rm{d}}\varphi }\cdot \\& \int\limits_0^{2{\rm{\text{π} }}} {A(\varphi '){{\rm{e}}^{i\omega t}}{{\rm{e}}^{ - \alpha {{\hat k'}} \cdot ({{{r}}_2} - {{R'}})}}{{\rm{e}}^{ - i{{{{k'}}_{\!\!\!0}}} \cdot ({{{r}}_2} - {{R'}})}}{\rm{d}}\varphi '} \end{split} $

$\begin{split} \left\langle {P_{\rm{a}}^{\rm{*}}{P_{\rm{b}}}} \right\rangle &= \int\limits_0^{2{\rm{\text{π} }}} {\int\limits_0^{2{\rm{\text{π} }}} {\left\langle {A(\varphi)A(\varphi ')} \right\rangle } } {e^{ - i{{{k}}_0} \cdot {{R}} + i{{{{k'}}}_{\!\!\!0}} \cdot {{R'}}}}{e^{ - i{{{{k'}}_{\!\!\!0}}} \cdot {{{r}}_2} + i{{{k}}_0} \cdot {{{r}}_1}}}{e^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}}) - \alpha {{\hat k'}} \cdot ({{{r}}_2} - {{R'}})}}{\rm{d}}\varphi {\rm{d}}\varphi ' \\& = \int\limits_0^{2{\rm{\text{π} }}} {\int\limits_0^{2{\rm{\text{π} }}} {{A^2}\delta (\varphi - \varphi '){{\rm{e}}^{ - i{{{k}}_0} \cdot {{R}} + i{{{{k'}}_{\!\!\!0}}} \cdot {{R'}}}}{{\rm{e}}^{ - i{{{{k'}}_{\!\!\!0}}} \cdot {{{r}}_2} + i{{{k}}_0} \cdot {{{r}}_1}}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}}) - \alpha {{\hat k'}} \cdot ({{{r}}_2} - {{R'}})}}{\rm{d}}\varphi {\rm{d}}\varphi '} } \\& = \int\limits_0^{2{\rm{\text{π} }}} {{A^2}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot [({{{r}}_1} - {{R}}) + ({{{r}}_2} - {{R}})]}}} {\rm{d}}\varphi \end{split} $

$\left\langle {P_{\rm{a}}^{\rm{*}}{P_{\rm{a}}}} \right\rangle = \int\limits_0^{2{\rm{\text{π} }}} {{A^2}{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}{\rm{d}}\varphi } $

$\left\langle {P_{\rm{b}}^{\rm{*}}{P_{\rm{b}}}} \right\rangle = \int\limits_0^{2{\rm{\text{π} }}} {{A^2}{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_2} - {{R}})}}{\rm{d}}\varphi } $

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}}{\rm{ = }}\dfrac{{ < P_{\rm{a}}^{\rm{*}}{P_{\rm{b}}} > }}{{\sqrt { < P_{\rm{a}}^{\rm{*}}{P_{\rm{a}}}>} \sqrt { < P_{\rm{b}}^{\rm{*}}{P_{\rm{b}}} > } }}$

$\hat R_{{\rm{ab}}}^{{\rm{(2)}}} = \dfrac{{ < P_{\rm{a}}^{\rm{*}}{P_{\rm{b}}} > }}{{ < P_{\rm{a}}^{\rm{*}}{P_{\rm{a}}} > }}$

$\hat R_{{\rm{ab}}}^{{\rm{(3)}}} = \dfrac{{ < P_{\rm{a}}^{\rm{*}}{P_{\rm{b}}} > }}{{ < P_{\rm{b}}^{\rm{*}}{P_{\rm{b}}} > }}$

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}} = \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot [({{{r}}_1} - {{R}}) + ({{{r}}_2} - {{R}})]}}} {\rm{d}}\varphi }}{{\sqrt {\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}{\rm{d}}\varphi } } \right)\left({\displaystyle\int\limits_0^{2\text{π} } {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_2} - {{R}})}}{\rm{d}}\varphi '} } \right)} }}$

$\begin{split} \hat R_{{\rm{ab}}}^{(1)} &= \dfrac{{{{\rm{e}}^{ - 2\alpha \left| {{R}} \right|}}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{{{\rm{e}}^{ - 2\alpha \left| {{R}} \right|}}\sqrt {\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_1}}}{\rm{d}}\varphi } } \right)\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_2}}}{\rm{d}}\varphi '} } \right)} }} \\&= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\sqrt {\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_1}}}} {\rm{d}}\varphi } \right)\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_2}}}{\rm{d}}\varphi '} } \right)} }} \end{split}$

$\hat R_{{\rm{ab}}}^{{\rm{(2)}}} = \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_1}}}} {\rm{d}}\varphi }}$

$\hat R_{{\rm{ab}}}^{{\rm{(3)}}} = \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_2}}}} {\rm{d}}\varphi }}$

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}} = \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{{\rm{e}}^{ - \alpha r\cos \varphi }}} {\rm{d}}\varphi }}{{\sqrt {2{\rm{\text{π} }}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } } }} = \dfrac{{\dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } }}{{\sqrt {\dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } } }}$

${I_0}(z) = \dfrac{1}{{\rm{\text{π} }}}\int\limits_0^{\rm{\text{π} }} {{{\rm{e}}^{ \pm z\cos \theta }}{\rm{d}}\theta } $

${I_0}(z) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ \pm z\cos \varphi }}{\rm{d}}\varphi } $

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}}{\rm{ = }}\dfrac{{{J_0}(kr)}}{{\sqrt {{I_0}(2\alpha r)} }}$

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(+)}} = \dfrac{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } }}{{\sqrt {{\rm{\text{π} }}\displaystyle\int\limits_{ - {{\rm{\text{π} }} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } } }}$

${I_\nu }(ak) - {{\rm{L}}_\nu }(ak) = \dfrac{{2{{\left(\dfrac{1}{2}ak\right)}^\nu }}}{{\Gamma \left(\nu + \dfrac{1}{2}\right)\Gamma \left(\dfrac{1}{2}\right)}}\int\limits_0^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ak\cos \theta }}{{\sin }^{2\nu }}\theta {\rm{d}}\theta } $

${I_0}(ka) - {{\rm{L}}_0}(ka) = \dfrac{2}{{\rm{\text{π} }}}\int\limits_0^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ka\cos \theta }}{\rm{d}}\theta } $

$\begin{array}{l} {I_0}(ka) - {{\rm{L}}_0}(ka) = \dfrac{1}{{\rm{\text{π} }}}\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ka\cos \theta }}{\rm{d}}\theta } {\rm{ = }}\dfrac{1}{{\rm{\text{π} }}}\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{{\rm{3\text{π} }}} / 2}} {{{\rm{e}}^{ka\cos \theta }}{\rm{d}}\theta } \\ {I_0}(ka){\rm{ + }}{{\rm{L}}_0}(ka){\rm{ = }}\dfrac{1}{{\rm{\text{π} }}}\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{\rm{\text{π} }}/2} {{{\rm{e}}^{ka\cos \theta }}{\rm{d}}\theta } = \dfrac{1}{{\rm{\text{π} }}}\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{{\rm{3\text{π} }}} / 2}} {{{\rm{e}}^{{\rm{ - }}ka\cos \theta }}{\rm{d}}\theta } \end{array} $

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(+)}} = \dfrac{{{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)}}{{\sqrt {{I_0}(2\alpha r) - {{\rm{L}}_0}(2\alpha r)} }}$

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(}} - {\rm{)}} = \dfrac{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } }}{{\sqrt {{\rm{\text{π} }}\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } } }} = \dfrac{{{J_0}(kr){\rm{ + }}i{{\rm{H}}_{\rm{0}}}(kr)}}{{\sqrt {{I_0}(2\alpha r) + {{\rm{L}}_0}(2\alpha r)}\\ }}$

$\begin{split} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \hat R_{{\rm{ab}}}^{{\rm{(2)}}} &= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_1}}}} {\rm{d}}\varphi }} \\&= \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = {J_0}(kr) \end{split}\qquad\quad\;$

$\begin{split}& \hat R_{{\rm{ab}}}^{{\rm{(2)}}}(+) = \dfrac{1}{{\rm{\text{π} }}}\int\limits_{ - {\text{π} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = {J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr) \\& \hat R_{{\rm{ab}}}^{{\rm{(2)}}}(-) = \dfrac{1}{{\rm{\text{π} }}}\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = {J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr) \\ \end{split} $

$\begin{split} \!\!\!\!\!\!\!\!\!\!\!\! \hat R_{{\rm{ab}}}^{{\rm{(3)}}}& = \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_2}}}} {\rm{d}}\varphi }} \\&= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{{\rm{e}}^{ - \alpha r\cos \varphi }}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } }} = \dfrac{{{J_0}(kr)}}{{{I_0}(2\alpha r)}} \end{split}\qquad\quad$

$\begin{array}{l} \hat R_{{\rm{ab}}}^{{\rm{(3)}}}(+) = \dfrac{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{{\rm{e}}^{ - \alpha r\cos \varphi }}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_{ - {{\rm{\text{π} }} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } }} = \dfrac{{{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)}}{{{I_0}(2\alpha r) - {{\rm{L}}_0}(2\alpha r)}} \\ \hat R_{ab}^{(3)}(-) = \dfrac{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{{\rm{e}}^{ - \alpha r\cos \varphi }}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - 2\alpha r\cos \varphi }}{\rm{d}}\varphi } }} = \dfrac{{{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)}}{{{I_0}(2\alpha r) + {{\rm{L}}_0}(2\alpha r)}} \end{array} $

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}} = \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } }}{{\sqrt {\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - \alpha r\cos \varphi }}{\rm{d}}\varphi } } \right)\left({\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{\alpha r\cos \varphi '}}{\rm{d}}\varphi '} } \right)} }}$

$\hat R_{{\rm{ab}}}^{{\rm{(1)}}} = \dfrac{{{J_0}({k_0}r)}}{{{I_0}(\alpha r)}}$

$\begin{split} \hat R_{{\rm{ab}}}^{{\rm{(1)}}}({\rm{ +)}}& = \dfrac{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } }}{{\sqrt {\left({\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - \alpha r\cos \varphi }}{\rm{d}}\varphi } } \right)\left({\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{\alpha r\cos \varphi '}}{\rm{d}}\varphi '} } \right)} }}\\&= \dfrac{{{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{\sqrt {{I_0}^2(\alpha r) - {{\rm L}_0}^2(\alpha r)} }} \end{split}$

$\begin{split} \hat R_{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(}} - {\rm{)}}& = \dfrac{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } }}{{\sqrt {\left({\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - \alpha r\cos \varphi }}{\rm{d}}\varphi } } \right)\left({\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{\alpha r\cos \varphi '}}{\rm{d}}\varphi '} } \right)} }} \\&= \dfrac{{{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{\sqrt {{I_0}^2(\alpha r) - {{\rm L}_0}^2(\alpha r)} }} \end{split}$

$\begin{split} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \hat R_{{\rm{ab}}}^{{\rm{(2)}}} &= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_1}}}{\rm{d}}} \varphi }} \\&= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{\alpha r\cos \varphi }}} {\rm{d}}\varphi }} \\&= \dfrac{{{J_0}({k_0}r)}}{{{I_0}(\alpha r)}} \end{split}$

$\begin{split}& \hat R_{{\rm{ab}}}^{{\rm{(2)}}}(+) = \dfrac{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } }}{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{\alpha r\cos \varphi }}} {\rm{d}}\varphi }} = \dfrac{{{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{{I_0}(\alpha r) + {{\rm L}_0}(\alpha r)}} \\& \hat R_{{\rm{ab}}}^{{\rm{(2)}}}(-) = \dfrac{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } }}{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{\alpha r\cos \varphi }}} {\rm{d}}\varphi }} = \dfrac{{{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)}}{{{I_0}(\alpha r) - {{\rm L}_0}(\alpha r)}} \end{split} $

$ \begin{split} \hat R_{{\rm{ab}}}^{{\rm{(3)}}} &= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot ({{{r}}_1} + {{{r}}_2})}}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot {{{r}}_2}}}} {\rm{d}}\varphi }} \\[-1pt] &= \dfrac{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - \alpha r\cos \varphi }}{\rm{d}}\varphi } }} = \dfrac{{{J_0}({k_0}r)}}{{{I_0}(\alpha r)}} \end{split}\qquad\qquad\;\;\;\;\;$

$\begin{split}& \hat R_{{\rm{ab}}}^{{\rm{(3)}}}(+) = \dfrac{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - \alpha r\cos \varphi }}{\rm{d}}\varphi } }} = \dfrac{{{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)}}{{{I_0}(2\alpha r) - {{\rm{L}}_0}(2\alpha r)}} \\& \hat R_{{\rm{ab}}}^{{\rm{(3)}}}(-) = \dfrac{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}} {\rm{d}}\varphi }}{{\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - \alpha r\cos \varphi }}{\rm{d}}\varphi } }} = \dfrac{{{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)}}{{{I_0}(2\alpha r) + {{\rm{L}}_0}(2\alpha r)}} \\ \end{split} $

$Q_{\rm{a}}^{\rm{*}}{Q_{\rm{b}}} = {A^2}{{\rm{e}}^{ - \alpha {{\hat k}} \cdot \left[ {({{{r}}_1} - {{R}}) + ({{{r}}_2} - {{R}})} \right]}}{{\rm{e}}^{{\rm{ - }}i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}$

$Q_{\rm{a}}^{\rm{*}}{Q_{\rm{a}}} = {A^2}{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}$

$Q_{\rm{b}}^{\rm{*}}{Q_{\rm{b}}} = {A^2}{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_2} - {{R}})}}$

$\hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{Q_{\rm{a}}^*{Q_{\rm{b}}}}}{{\sqrt {Q_{\rm{a}}^{\rm{*}}{Q_{\rm{a}}}} \sqrt {Q_{\rm{b}}^{\rm{*}}{Q_{\rm{b}}}} }}} {\rm{d}}\varphi $

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{Q_{\rm{a}}^{\rm{*}}{Q_{\rm{b}}}}}{{Q_{\rm{a}}^{\rm{*}}{Q_{\rm{a}}}}}} {\rm{d}}\varphi $

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{Q_{\rm{a}}^{\rm{*}}{Q_{\rm{b}}}}}{{Q_{\rm{b}}^{\rm{*}}{Q_{\rm{b}}}}}} {\rm{d}}\varphi \;\;\;$

$\begin{split} \hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}} &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot \left[ {({{{r}}_1} - {{R}}) + ({{{r}}_2} - {{R}})} \right]}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{\sqrt {{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_1} - {{R}})}}{{\rm{e}}^{ - 2\alpha {{\hat k}} \cdot ({{{r}}_2} - {{R}})}}} }}{\rm{d}}\varphi } \\ &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{\rm{d}}\varphi } \end{split} $

$\hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_2}}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_1}}}}}{\rm{d}}\varphi }\qquad\qquad\; $

$\hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_1}}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_2}}}}}{\rm{d}}\varphi } \qquad\qquad\;$

$\hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}}{\rm{ = }}\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{\rm{d}}\varphi } = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{{\rm{ - }}i{k_0}r\cos \varphi }}{\rm{d}}\varphi } = J({k_0}r)$

$\hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(+)}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right]$

$\hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(}} - {\rm{)}} = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i{k_0}r\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right]$

$\,\hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}}\! =\! \dfrac{1}{{2{\rm{\text{π} }}}}\!\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_2}}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_1}}}}}{\rm{d}}\varphi } \!=\! \dfrac{1}{{2{\rm{\text{π} }}}}\!\int\limits_0^{2{\rm{\text{π} }}} \!{{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } \!=\! {J_0}(kr)$

$\begin{split}& \hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}}(+) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right] \\& \hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}}(-) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}({k_0}r) + i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right] \end{split} $

$\begin{split} \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}}& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_1}}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_2}}}}}{\rm{d}}\varphi } \\&= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i({k_0} + i\alpha)r\cos \varphi }}{\rm{d}}\varphi } = {J_0}({k_0}r + i\alpha r) \end{split}$

$\begin{split} \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}}(+)& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i({k_0} + i\alpha)r\cos \varphi }}{\rm{d}}\varphi } \\&= \dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) - i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right] \\ \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}}(-)& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i({k_0} + i\alpha)r\cos \varphi }}{\rm{d}}\varphi } \\&= \dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) + i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right] \end{split} $

$\hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}}{\rm{ = }}\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}{\rm{d}}\varphi } = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{{\rm{ - }}i{k_0}r\cos \varphi }}{\rm{d}}\varphi } = J({k_0}r)$

$\begin{array}{l} \hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(+)}} = \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{ - {{\rm{\text{π} }} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{{\rm{ - }}i{k_0}r\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}({k_0}r) - i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right] \\ \hat \gamma _{{\rm{ab}}}^{{\rm{(1)}}}{\rm{(}} - {\rm{)}} = \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{{\rm{3\text{π} }}} / 2}} {{{\rm{e}}^{{\rm{ - }}i{k_0}r\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}({k_0}r){\rm{ + }}i{{\rm{H}}_{\rm{0}}}({k_0}r)} \right] \\ \end{array} $

$\,\hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}} \!=\! \dfrac{1}{{2{\rm{\text{π} }}}}\!\int\limits_0^{2{\rm{\text{π} }}}\! {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_2}}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_1}}}}}{\rm{d}}\varphi } \! =\! \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}}\! {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } \!=\! {J_0}(kr)$

$\begin{array}{l} \hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}}(+) = \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}(kr) - i{{\rm{H}}_{\rm{0}}}(kr)} \right] \\ \hat \gamma _{{\rm{ab}}}^{{\rm{(2)}}}(-) = \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - ikr\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{2}\left[ {{J_0}(kr) + i{{\rm{H}}_{\rm{0}}}(kr)} \right] \\ \end{array} $

$\begin{split} \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}} &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {\dfrac{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_1}}}{{\rm{e}}^{ - i{{{k}}_0} \cdot ({{{r}}_2} - {{{r}}_1})}}}}{{{{\rm{e}}^{ - \alpha {{\hat k}} \cdot {{{r}}_2}}}}}{\rm{d}}\varphi } \\ &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_0^{2{\rm{\text{π} }}} {{{\rm{e}}^{ - i({k_0} + i\alpha)r\cos \varphi }}{\rm{d}}\varphi } = {J_0}({k_0}r + i\alpha r) \end{split}$

$\begin{split} \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}}(+) &= \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{{{ - {\rm{\text{π} }}} / 2}}^{{{\rm{\text{π} }} / 2}} {{{\rm{e}}^{ - i({k_0} + i\alpha)r\cos \varphi }}{\rm{d}}\varphi } \\ &= \dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) - i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right] \\&\quad \hat \gamma _{{\rm{ab}}}^{{\rm{(3)}}}(-) = \dfrac{1}{{2{\rm{\text{π} }}}}\displaystyle\int\limits_{{{\rm{\text{π} }} / 2}}^{{{3{\rm{\text{π} }}} / 2}} {{{\rm{e}}^{ - i({k_0} + i\alpha)r\cos \varphi }}{\rm{d}}\varphi }\\& = \dfrac{1}{2}\left[ {{J_0}({k_0}r + i\alpha r) + i{{\rm{H}}_{\rm{0}}}({k_0}r + i\alpha r)} \right] \\ \end{split} $

${C_{{\rm{fg}}}}(\tau) = \int\limits_{ - \infty }^{ + \infty } {{f^{\rm{*}}}(t)g(t + \tau){\rm{d}}t} \tag{A1}$

${C_{{\rm{fg}}}}(\tau) = \int\limits_{ - \infty }^{ + \infty } {{f^{\rm{*}}}(t' - \tau)g(t'){\rm{d}}t'} = \int\limits_{ - \infty }^{ + \infty } {{f^{\rm{*}}}(t - \tau)g(t){\rm{d}}t} \tag{A2}$

${f^*}(t) = f(t) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {{F^*}(\omega){{\rm{e}}^{ - i\omega t}}{\rm{d}}\omega } \tag{A3}$

$f(- t') = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {{F^*}(\omega){{\rm{e}}^{i\omega t'}}{\rm{d}}\omega } \tag{A4}$

$C_{_{{\rm{fg}}}}^{'}(\tau) = \int\limits_{ - \infty }^{ + \infty } {f(t + \tau){g^*}(t){\rm{d}}t} \tag{A5}$

$\begin{split} C_{_{{\rm{fg}}}}^{\rm{'}}(\tau) &= \dfrac{1}{{4{{\rm{\text{π} }}^2}}}\int\limits_{ - \infty }^{ + \infty } {\left[ {\int\limits_{ - \infty }^{ + \infty } {F(\omega){{\rm{e}}^{i\omega (t + \tau)}}{\rm{d}}\omega } } \right]\left[ {\int\limits_{ - \infty }^{ + \infty } {{G^*}(\omega '){{\rm{e}}^{ - i\omega 't}}{\rm{d}}\omega '} } \right]} {\rm{d}}t \\& = \dfrac{1}{{4{{\rm{\text{π} }}^2}}}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {F(\omega)} } } {G^*}(\omega '){{\rm{e}}^{i(\omega - \omega ')t}}{{\rm{e}}^{i\omega \tau }}{\rm{d}}t{\rm{d}}\omega {\rm{d}}\omega ' \\& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {F(\omega)} } {G^*}(\omega ')\delta (\omega - \omega '){{\rm{e}}^{i\omega \tau }}{\rm{d}}\omega {\rm{d}}\omega ' \\& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {F(\omega)} {G^*}(\omega){{\rm{e}}^{i\omega \tau }}{\rm{d}}\omega \end{split} \tag{A6}$

${J_n}(z) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_\alpha ^{2{\rm{\text{π} }} + \alpha } {{{\rm{e}}^{i(n\theta - z\sin \theta)}}{\rm{d}}\theta } \tag{B1}$

${J_0}(z) = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_\alpha ^{2{\rm{\text{π} }} + \alpha } {{{\rm{e}}^{ - iz\sin \theta }}{\rm{d}}\theta } \tag{B2}$

$ {J_0}(z) = - \dfrac{1}{{2\text{π} }}\int\limits_{{\text{π} / 2} - \alpha }^{{\text{π} / 2} - 2\text{π} - \alpha } {{e^{ - iz\cos \varphi }}{\rm{d}}\varphi } \tag{B3} $

${J_0}(z) = - \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{2{\rm{\text{π} }} + \beta }^\beta {{{\rm{e}}^{ - iz\cos \varphi }}{\rm{d}}\varphi } = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_\beta ^{2{\rm{\text{π} }} + \beta } {{{\rm{e}}^{ - iz\cos \varphi }}{\rm{d}}\varphi } \tag{B4}$

${J_0}(z) = \dfrac{1}{{2\text{π} }}\int\limits_\beta ^{2\text{π} + \beta } {{{\rm e}^{ \pm iz\cos \varphi }}{\rm d}\varphi } = \dfrac{1}{{2\text{π} }}\int\limits_\beta ^{2\text{π} + \beta } {{{\rm e}^{ \pm iz\sin \varphi }}{\rm d}\varphi } \tag{B5}$

$\mathcal{F}[{J_0}(kr)] = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {{J_0}(kr){{\rm{e}}^{i\omega t}}} {\rm{d}}\omega \tag{C1}$

$\begin{split} \mathcal{F}[{J_0}(kr)] &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^\infty {{J_0}(x){{\rm{e}}^{i\dfrac{c}{r}xt}}} \dfrac{c}{r}{\rm{d}}x \\& = \dfrac{1}{{2{\rm{\text{π} }}}}\dfrac{c}{r}\int\limits_{ - \infty }^\infty {{J_0}(x){{\rm{e}}^{i(\dfrac{c}{r}t)x}}} {\rm{d}}x \end{split} \tag{C2}$

$\mathcal{F}[{J_0}(kr)] = \dfrac{1}{{2{\rm{\text{π} }}}}\dfrac{c}{r}\dfrac{2}{{\sqrt {1 - {{(\dfrac{c}{r}t)}^2}} }} = \dfrac{1}{{\rm{\text{π} }}}\dfrac{2}{{\sqrt {\dfrac{{{r^2}}}{{{c^2}}} - {t^2}} }}\tag{C3}$

$\begin{split} \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{j_0}(kr)} {{\rm{e}}^{i\omega t}}{\rm{d}}\omega &= \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\dfrac{{\sin kr}}{{kr}}} {{\rm{e}}^{i\omega t}}{\rm{d}}\omega = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\dfrac{{{{\rm{e}}^{ikr}} - {{\rm{e}}^{ - ikr}}}}{{2ikr}}} {{\rm e}^{i\omega t}}{\rm{d}}\omega \\& = \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\dfrac{{{{\rm{e}}^{i\omega {t_0}}} - {{\rm{e}}^{ - i\omega {t_0}}}}}{{2i\omega {t_0}}}} {{\rm{e}}^{i\omega t}}{\rm{d}}\omega \\& = \dfrac{1}{{4{t_0}}}\left[ {\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\dfrac{2}{{i\omega }}{{\rm{e}}^{i\omega (t + {t_0})}}{\rm{d}}\omega } - \dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {\dfrac{2}{{i\omega }}{{\rm{e}}^{i\omega (t - {t_0})}}{\rm{d}}\omega } } \right] \end{split} \tag{D1}$

$\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{j_0}(kr)} {{\rm{e}}^{i\omega t}}{\rm{d}}\omega = \dfrac{1}{{4{t_0}}}\left[ {{\rm{sgn}} (t + {t_0}) - {\rm{sgn}} (t - {t_0})} \right]\tag{D2}$

${\rm{sgn}} (x) = \left\{ {\begin{array}{*{20}{c}} 1&{x > 0} \\ 0&{x = 0} \\ { - 1}&{x < 0} \end{array}} \right. \tag{D3}$

${\rm{sgn}} (x) = 2H(x) - 1 \tag{D4}$

$\dfrac{1}{{2{\rm{\text{π} }}}}\int\limits_{ - \infty }^{ + \infty } {{j_0}(kr)} {{\rm{e}}^{i\omega t}}{\rm{d}}\omega = \dfrac{c}{{2r}}\left[ {H(t + \dfrac{r}{c}) - H(t - \dfrac{r}{c})} \right] \tag{D5}$

${G_v} = \dfrac{\partial }{{\partial t}}{G_u} \tag{E1}$

${G_v} = i\omega {G_u}{\rm{ }} \Rightarrow {\rm{ }}{G_u}{\rm{ = }}\dfrac{1}{{i\omega }}{G_v}{\rm{ }} \tag{E2}$

$1{\rm{D}}:\;{\hat \phi _{{\rm{ab}}}}(x,\omega) = \cos kx = - ik\dfrac{1}{{i\omega }}\left[ {{G_v}(x,\omega) + {G_v}^*(x,\omega)} \right] = \dfrac{1}{{2c}}{\rm{Re}} \left[ {{G_v}(x,\omega)} \right] \tag{E3}$

$2{\rm{D}}:\;{\hat \phi _{{\rm{ab}}}}(r,\omega) = {J_0}(kr) = - 2i\dfrac{1}{{i\omega }}\left[ {{G_v}(r,\omega) + {G_v}^*(r,\omega)} \right] = - \dfrac{2}{\omega }{\rm{Re}} \left[ {{G_v}(r,\omega)} \right] \tag{E4}$

$3{\rm{D}}:\;{\hat \phi _{_{{\rm{ab}}}}}(r,\omega) \!=\! {j_0}(kr) \!=\! \dfrac{{\sin kr}}{{kr}} \!=\! - \dfrac{{2{\rm{\text{π} }}i}}{k}\dfrac{1}{{i\omega }}\left[ {{G_v}(r,\omega) + {G_v}^*(r,\omega)} \right] \!=\! - \dfrac{{4{\rm{\text{π} }}}}{{k\omega }}{\rm{Re}} \left[ {{G_v}(r,\omega)} \right] \tag{E5}$

$\int\limits_0^{{{\rm{\text{π} }} / {\rm{2}}}} {{J_\nu }(z\sin \theta){{(\sin \theta)}^{\nu + 1}}{{(\cos\theta)}^{ - 2\nu }}{\rm{d}}\theta } = {2^{ - \nu }}\dfrac{{{z^{\nu - 1}}}}{{\sqrt {\rm{\text{π} }} }}\Gamma \left({\dfrac{1}{2} - \nu } \right)\sin z \tag{F1}$

$\int\limits_0^{{{\rm{\text{π} }} / 2}} {{J_0}(z\sin \theta)\sin \theta {\rm{d}}\theta } = \dfrac{{\sin z}}{z} \tag{F2}$