??????? ?????? — ?????????????? ??????? , ??????? ???????? ??????? ???????? ????????????? ????????? ??????? :

${\displaystyle z^{2}{\frac {d^{2}f}{dz^{2}}}+z{\frac {df}{dz}}+(z^{2}-\nu ^{2})f=-{\frac {(z+\nu )+(z-\nu )\cos(\pi z)}{\pi }}}$

???????????? ????????? ??????? ??????:

${\displaystyle {\mathsf {E}}_{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\cdot \sin \theta )d\theta }$

??????????? ? ???????? ?????? :

${\displaystyle {\mathsf {E}}_{\nu }(z)\cdot \sin(\pi \nu )={\mathsf {J}}_{-\nu }(z)-{\mathsf {J}}_{\nu }(z)\cdot \cos(\pi \nu )}$

??? ????? ${\displaystyle n\geqslant 0}$ ??????? ?????? ??????? ? ???????? ?????? ?????????????:

${\displaystyle {\mathsf {E}}_{n}(z)={\frac {1}{\pi }}\sum _{k=0}^{\left[{\frac {n-1}{2}}\right]}{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\cdot (z/2)^{n-2k-1}}{\Gamma \left(n+{\frac {1}{2}}-k\right)}}-{\mathsf {H}}_{-n}(z)}$

??????????

• — ?????? ?? ?????????????? ????????????

??????

• Weisstein, Eric W. (????.) ?? ????? Wolfram MathWorld .