# §10.7 Limiting Forms

## §10.7(i) $z\to 0$

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

 10.7.1 $\displaystyle J_{0}\left(z\right)$ $\displaystyle\to 1,$ $\displaystyle Y_{0}\left(z\right)$ $\displaystyle\sim(2/\pi)\ln z,$
 10.7.2 ${H^{(1)}_{0}}\left(z\right)\sim-{H^{(2)}_{0}}\left(z\right)\sim(2i/\pi)\ln z,$
 10.7.3 $J_{\nu}\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\Gamma\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$: gamma function, $\sim$: asymptotic equality, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.7 Referenced by: §10.7(i), §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E3 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10
 10.7.4 $\displaystyle Y_{\nu}\left(z\right)$ $\displaystyle\sim-(1/\pi)\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$ $\Re\nu>0$ or $\nu=-\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{5}{2},\dotsc$, 10.7.5 $\displaystyle Y_{-\nu}\left(z\right)$ $\displaystyle\sim-(1/\pi)\cos\left(\nu\pi\right)\Gamma\left(\nu\right)(\tfrac{% 1}{2}z)^{-\nu},$ $\Re\nu>0$, $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dotsc$,
 10.7.6 $Y_{i\nu}\left(z\right)=\frac{i\operatorname{csch}\left(\nu\pi\right)}{\Gamma% \left(1-i\nu\right)}(\tfrac{1}{2}z)^{-i\nu}-\frac{i\coth\left(\nu\pi\right)}{% \Gamma\left(1+i\nu\right)}(\tfrac{1}{2}z)^{i\nu}+e^{|\nu\operatorname{ph}z|}o% \left(1\right),$ $\nu\in\mathbb{R}$ and $\nu\neq 0$.

See also §10.24 when $z=x$ $(>0)$.

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

For ${H^{(1)}_{-\nu}}\left(z\right)$ and ${H^{(2)}_{-\nu}}\left(z\right)$ when $\Re\nu>0$ combine (10.4.6) and (10.7.7). For ${H^{(1)}_{i\nu}}\left(z\right)$ and ${H^{(2)}_{i\nu}}\left(z\right)$ 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 $\displaystyle J_{\nu}\left(z\right)$ $\displaystyle=\sqrt{2/(\pi z)}\left(\cos\left(z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4% }\pi\right)+e^{|\Im z|}o\left(1\right)\right),$ $\displaystyle Y_{\nu}\left(z\right)$ $\displaystyle=\sqrt{2/(\pi z)}\left(\sin\left(z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4% }\pi\right)+e^{|\Im z|}o\left(1\right)\right),$ $|\operatorname{ph}z|\leq\pi-\delta(<\pi)$.

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