Модифицированные функции Бесселя

Так как ${\displaystyle I_{-\nu }(z)=I_{\nu }(z)}$ при целом ${\displaystyle \nu }$ в качестве фундаментальной системы решений уравнения ${\displaystyle (1)}$ выбирают ${\displaystyle I_{n}(z)}$ и ${\displaystyle K_{n}(z),}$ где

${\displaystyle K_{n}(z)=\lim \limits _{\nu \to n}K_{\nu }(z).}$

${\displaystyle \left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }I_{\nu }(z){\Bigr ]}=z^{\nu -m}I_{\nu -m}(z).}$

${\displaystyle \left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }I_{\nu }(z){\Bigr ]}=z^{-\nu -m}I_{\nu +m}(z).}$

${\displaystyle I_{\nu -1}(z)-I_{\nu +1}(z)=2\nu z^{-1}I_{\nu }(z).}$

${\displaystyle I_{\nu -1}(z)+I_{\nu +1}(z)=2I'_{\nu }(z).}$

${\displaystyle \left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{\nu -m}K_{\nu -m}(z).}$

${\displaystyle \left({\frac {d}{zdz}}\right)^{m}{\Bigl [}z^{-\nu }K_{\nu }(z){\Bigr ]}=(-1)^{m}z^{-\nu -m}K_{\nu +m}(z).}$

${\displaystyle K_{\nu -1}(z)-K_{\nu +1}(z)=-2\nu z^{-1}K_{\nu }(z).}$

${\displaystyle K_{\nu -1}(z)+K_{\nu +1}(z)=-2K'_{\nu }(z).}$

${\displaystyle W\left[I_{\nu }(z),I_{-\nu }(z)\right]=-{\frac {2\sin(\nu \pi )}{\pi z}}.}$

${\displaystyle W\left[I_{\nu }(z),K_{\nu }(z)\right]=-z^{-1}.}$

${\displaystyle I_{\nu }(z)={\frac {2^{-\nu }z^{\nu }}{{\sqrt {\pi }}\Gamma (\nu +{\frac {1}{2}})}}\int _{0}^{\pi }e^{z\cos t}\left(\sin t\right)^{2\nu }dt,\qquad Re(\nu )>-{\frac {1}{2}},\Gamma (z)}$ — гамма-функция.

${\displaystyle I_{\nu }(z)={\frac {2^{1-\nu }z^{\nu }}{{\sqrt {\pi }}\Gamma (\nu +{\frac {1}{2}})}}\int _{0}^{1}(1-t^{2})^{\nu -{\frac {1}{2}}}\operatorname {ch} (zt)dt,\qquad Re(\nu )>-{\frac {1}{2}}.}$

${\displaystyle I_{\nu }(z)={\frac {2^{-\nu }z^{\nu }}{{\sqrt {\pi }}\Gamma (\nu +{\frac {1}{2}})}}\int _{-1}^{1}(1-t^{2})^{\nu -{\frac {1}{2}}}e^{-zt}dt,\qquad Re(\nu )>-{\frac {1}{2}}.}$

${\displaystyle I_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }e^{z\cos t}\cos(nt)dt,\qquad n\in \mathbb {Z} ,Re(z)>0.}$

${\displaystyle {{K}_{\nu }}(z)=\int _{0}^{\infty }{{e}^{-z\operatorname {ch} t}}\operatorname {ch} (\nu t)dt,\qquad |Arg(z)|<{\frac {\pi }{2}}.}$

${\displaystyle {{K}_{\nu }}(z)={\frac {{\sqrt {\pi }}{{\left({\frac {z}{2}}\right)}^{\nu }}}{\Gamma (\nu +{\frac {1}{2}})}}\int _{1}^{\infty }{{({{t}^{2}}-1)}^{\nu -{\frac {1}{2}}}}{{e}^{-zt}}dt,\qquad Re(\nu )>-{\frac {1}{2}},|Arg(z)|<{\frac {\pi }{2}}.}$

${\displaystyle {{K}_{\nu }}(z)={\frac {{\sqrt {\pi }}{{\left({\frac {z}{2}}\right)}^{\nu }}}{\Gamma (\nu +{\frac {1}{2}})}}\int _{0}^{\infty }{{e}^{-z\operatorname {ch} t}}{{\left(\operatorname {sh} t\right)}^{2\nu }}dt,\qquad Re(\nu )>-{\frac {1}{2}},|Arg(z)|<{\frac {\pi }{2}}.}$

${\displaystyle I_{\nu }(z)\varpropto {\frac {e^{z}}{\sqrt {2\pi z}}}\left(1+O\left({\frac {1}{z}}\right)\right),\qquad \left|Arg(z)\right|<{\frac {\pi }{2}},\left|z\right|\to \infty .}$

${\displaystyle K_{\nu }(z)\varpropto {\sqrt {\frac {\pi }{2}}}{\frac {e^{-z}}{\sqrt {z}}}\left(1+O\left({\frac {1}{z}}\right)\right),\qquad \left|z\right|\to \infty .}$

Частный случай:

${\displaystyle K_{0}(z)={\sqrt {\frac {\pi }{2z}}}e^{-z}\sum \limits _{m=0}^{\infty }{\frac {\left[\left({2m-1}\right)!!\right]^{2}}{m!\left(-{8z}\right)^{m}}},\qquad \left|z\right|\to \infty ,\quad |\arg z|<2\pi }$

• Существует классический ряд Шлёмильха по j-функциям Бесселя ^{[1] [2]}