# Hankel

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##### 1: 10.1 Special Notation
The main functions treated in this chapter are the Bessel functions $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$; Hankel functions ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$; modified Bessel functions $I_{\nu}\left(z\right)$, $K_{\nu}\left(z\right)$; spherical Bessel functions $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$; modified spherical Bessel functions ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, $\mathsf{k}_{n}\left(z\right)$; Kelvin functions $\operatorname{ber}_{\nu}\left(x\right)$, $\operatorname{bei}_{\nu}\left(x\right)$, $\operatorname{ker}_{\nu}\left(x\right)$, $\operatorname{kei}_{\nu}\left(x\right)$. … Abramowitz and Stegun (1964): $j_{n}(z)$, $y_{n}(z)$, $h_{n}^{(1)}(z)$, $h_{n}^{(2)}(z)$, for $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$, respectively, when $n\geq 0$. Jeffreys and Jeffreys (1956): $\mathrm{Hs}_{\nu}(z)$ for ${H^{(1)}_{\nu}}\left(z\right)$, $\mathrm{Hi}_{\nu}(z)$ for ${H^{(2)}_{\nu}}\left(z\right)$, $\mathrm{Kh}_{\nu}(z)$ for $(2/\pi)K_{\nu}\left(z\right)$. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 2: 10.4 Connection Formulas
###### §10.4 Connection Formulas
Other solutions of (10.2.1) include $J_{-\nu}\left(z\right)$, $Y_{-\nu}\left(z\right)$, ${H^{(1)}_{-\nu}}\left(z\right)$, and ${H^{(2)}_{-\nu}}\left(z\right)$. …
${H^{(1)}_{-n}}\left(z\right)=(-1)^{n}{H^{(1)}_{n}}\left(z\right),$
${H^{(2)}_{-n}}\left(z\right)=(-1)^{n}{H^{(2)}_{n}}\left(z\right).$
$J_{\nu}\left(z\right)=\frac{1}{2}\left({H^{(1)}_{\nu}}\left(z\right)+{H^{(2)}_% {\nu}}\left(z\right)\right),$
##### 3: 10.5 Wronskians and Cross-Products
###### §10.5 Wronskians and Cross-Products
10.5.3 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (1)}_{\nu+1}}\left(z\right)=2i/(\pi z),$
10.5.4 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (2)}_{\nu+1}}\left(z\right)=-2i/(\pi z),$
10.5.5 $\mathscr{W}\left\{{H^{(1)}_{\nu}}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)% \right\}={H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-{H^{(1)}% _{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-4i/(\pi z).$
##### 4: 10.11 Analytic Continuation
###### §10.11 Analytic Continuation
10.11.3 $\sin\left(\nu\pi\right){H^{(1)}_{\nu}}\left(ze^{m\pi i}\right)=-\sin\left((m-1% )\nu\pi\right){H^{(1)}_{\nu}}\left(z\right)-e^{-\nu\pi i}\sin\left(m\nu\pi% \right){H^{(2)}_{\nu}}\left(z\right),$
10.11.7 ${H^{(1)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn-1}((m-1){H^{(1)}_{n}}\left(z% \right)+m{H^{(2)}_{n}}\left(z\right)),$
10.11.8 ${H^{(2)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn}(m{H^{(1)}_{n}}\left(z\right)+(% m+1){H^{(2)}_{n}}\left(z\right)).$
$\displaystyle{H^{(1)}_{\nu}}\left(\overline{z}\right)=\overline{{H^{(2)}_{\nu}% }\left(z\right)},$ $\displaystyle{H^{(2)}_{\nu}}\left(\overline{z}\right)=\overline{{H^{(1)}_{\nu}% }\left(z\right)}.$
##### 5: 10.2 Definitions
###### Bessel Functions of the Third Kind (Hankel Functions)
These solutions of (10.2.1) are denoted by ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$, and their defining properties are given by … The principal branches of ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
##### 6: 10.47 Definitions and Basic Properties
10.47.5 ${\mathsf{h}^{(1)}_{n}}\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{n+\frac% {1}{2}}}\left(z\right)=(-1)^{n+1}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{-% n-\frac{1}{2}}}\left(z\right),$
$\mathsf{j}_{n}\left(z\right)$ and $\mathsf{y}_{n}\left(z\right)$ are the spherical Bessel functions of the first and second kinds, respectively; ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ and ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ are the spherical Bessel functions of the third kind. … Many properties of $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$, ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, and $\mathsf{k}_{n}\left(z\right)$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, $z^{-n}\mathsf{j}_{n}\left(z\right)$, $z^{n+1}\mathsf{y}_{n}\left(z\right)$, $z^{n+1}{\mathsf{h}^{(1)}_{n}}\left(z\right)$, $z^{n+1}{\mathsf{h}^{(2)}_{n}}\left(z\right)$, $z^{-n}{\mathsf{i}^{(1)}_{n}}\left(z\right)$, $z^{n+1}{\mathsf{i}^{(2)}_{n}}\left(z\right)$, and $z^{n+1}\mathsf{k}_{n}\left(z\right)$ are all entire functions of $z$. …
10.47.15 $\displaystyle{\mathsf{h}^{(1)}_{n}}\left(-z\right)=(-1)^{n}{\mathsf{h}^{(2)}_{% n}}\left(z\right),$ $\displaystyle{\mathsf{h}^{(2)}_{n}}\left(-z\right)=(-1)^{n}{\mathsf{h}^{(1)}_{% n}}\left(z\right).$
##### 7: 10.52 Limiting Forms
10.52.2 $-\mathsf{y}_{n}\left(z\right),i{\mathsf{h}^{(1)}_{n}}\left(z\right),-i{\mathsf% {h}^{(2)}_{n}}\left(z\right),(-1)^{n}{\mathsf{i}^{(2)}_{n}}\left(z\right),(2/% \pi)\mathsf{k}_{n}\left(z\right)\sim(2n-1)!!/z^{n+1}.$
${\mathsf{h}^{(1)}_{n}}\left(z\right)\sim i^{-n-1}z^{-1}e^{iz},$
${\mathsf{h}^{(2)}_{n}}\left(z\right)\sim i^{n+1}z^{-1}e^{-iz},$
##### 8: 10.16 Relations to Other Functions
${H^{(1)}_{\frac{1}{2}}}\left(z\right)=-i{H^{(1)}_{-\frac{1}{2}}}\left(z\right)% =-i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{iz},$
${H^{(2)}_{\frac{1}{2}}}\left(z\right)=i{H^{(2)}_{-\frac{1}{2}}}\left(z\right)=% i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-iz}.$
###### Confluent Hypergeometric Functions
10.16.6 $\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=\mp 2\pi^{-\frac{1}{2}}ie^{\mp\nu\pi i}(2z)^{% \nu}\*e^{\pm iz}U\left(\nu+\tfrac{1}{2},2\nu+1,\mp 2iz\right).$
10.16.8 $\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=e^{\mp(2\nu+1)\pi i/4}\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}W_{0,\nu}\left(\mp 2iz\right).$
##### 9: 9.6 Relations to Other Functions
###### §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
9.6.6 $\operatorname{Ai}\left(-z\right)=(\sqrt{z}/3)\left(J_{1/3}\left(\zeta\right)+J% _{-1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1% )}_{1/3}}\left(\zeta\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta\right)% \right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1/3}}\left(\zeta% \right)+e^{\pi i/6}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),$
9.6.7 $\operatorname{Ai}'\left(-z\right)=(z/3)\left(J_{2/3}\left(\zeta\right)-J_{-2/3% }\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/6}{H^{(1)}_% {2/3}}\left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{2/3}}\left(\zeta\right)\right)=% \tfrac{1}{2}(z/\sqrt{3})\left(e^{-5\pi i/6}{H^{(1)}_{-2/3}}\left(\zeta\right)+% e^{5\pi i/6}{H^{(2)}_{-2/3}}\left(\zeta\right)\right),$
9.6.8 $\operatorname{Bi}\left(-z\right)=\sqrt{z/3}\left(J_{-1/3}\left(\zeta\right)-J_% {1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{2\pi i/3}{H^{(1)% }_{1/3}}\left(\zeta\right)+e^{-2\pi i/3}{H^{(2)}_{1/3}}\left(\zeta\right)% \right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/3}{H^{(1)}_{-1/3}}\left(\zeta% \right)+e^{-\pi i/3}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),$
##### 10: 10.7 Limiting Forms
###### §10.7 Limiting Forms
10.7.2 ${H^{(1)}_{0}}\left(z\right)\sim-{H^{(2)}_{0}}\left(z\right)\sim(2i/\pi)\ln z,$
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). … 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).