# §10.7 Limiting Forms

## §10.7(i) $z\to 0$

When $\nu$ is fixed and $z\to 0$,

 10.7.1 $\displaystyle\mathop{J_{0}\/}\nolimits\!\left(z\right)$ $\displaystyle\to 1,$ $\displaystyle\mathop{Y_{0}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim(2/\pi)\mathop{\ln\/}\nolimits z,$
 10.7.2 $\mathop{{H^{(1)}_{0}}\/}\nolimits\!\left(z\right)\sim-\mathop{{H^{(2)}_{0}}\/}% \nolimits\!\left(z\right)\sim(2i/\pi)\mathop{\ln\/}\nolimits z,$
 10.7.3 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\mathop{% \Gamma\/}\nolimits\!\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$,
 10.7.4 $\displaystyle\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim-(1/\pi)\mathop{\Gamma\/}\nolimits\!\left(\nu\right)(\tfrac{1% }{2}z)^{-\nu},$ $\Re{\nu}>0$ or $\nu=-\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{5}{2},\ldots$, 10.7.5 $\displaystyle\mathop{Y_{-\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim-(1/\pi)\mathop{\cos\/}\nolimits(\nu\pi)\mathop{\Gamma\/}% \nolimits\!\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$ $\Re{\nu}>0$, $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\ldots$,
 10.7.6 $\mathop{Y_{i\nu}\/}\nolimits\!\left(z\right)=\frac{i\mathop{\mathrm{csch}\/}% \nolimits(\nu\pi)}{\mathop{\Gamma\/}\nolimits\!\left(1-i\nu\right)}(\tfrac{1}{% 2}z)^{-i\nu}-\frac{i\mathop{\coth\/}\nolimits(\nu\pi)}{\mathop{\Gamma\/}% \nolimits\!\left(1+i\nu\right)}(\tfrac{1}{2}z)^{i\nu}+e^{|\nu\mathop{\mathrm{% ph}\/}\nolimits z|}\mathop{o\/}\nolimits\!\left(1\right),$ $\nu\in\mathbb{R}$ and $\nu\neq 0$.

See also §10.24 when $z=x$ $(>0)$.

 10.7.7 $\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)\sim-\mathop{{H^{(2)}_{\nu}% }\/}\nolimits\!\left(z\right)\sim-(i/\pi)\mathop{\Gamma\/}\nolimits\!\left(\nu% \right)(\tfrac{1}{2}z)^{-\nu},$ $\Re{\nu}>0$.

For $\mathop{{H^{(1)}_{-\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{(2)}_{-\nu}}\/}\nolimits\!\left(z\right)$ when $\Re{\nu}>0$ combine (10.4.6) and (10.7.7). For $\mathop{{H^{(1)}_{i\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{(2)}_{i\nu}}\/}\nolimits\!\left(z\right)$ when $\nu\in\mathbb{R}$ and $\nu\neq 0$ combine (10.4.3), (10.7.3), and (10.7.6).

## §10.7(ii) $z\to\infty$

When $\nu$ is fixed and $z\to\infty$,

 10.7.8 $\displaystyle\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sqrt{2/(\pi z)}\left(\mathop{\cos\/}\nolimits\!\left(z-\tfrac{1% }{2}\nu\pi-\tfrac{1}{4}\pi\right)+e^{|\Im{z}|}\mathop{o\/}\nolimits\!\left(1% \right)\right),$ $\displaystyle\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sqrt{2/(\pi z)}\left(\mathop{\sin\/}\nolimits\!\left(z-\tfrac{1% }{2}\nu\pi-\tfrac{1}{4}\pi\right)+e^{|\Im{z}|}\mathop{o\/}\nolimits\!\left(1% \right)\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta(<\pi)$.

For the corresponding results for $\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)$ see (10.2.5) and (10.2.6).