# §10.2 Definitions

## §10.2(i) Bessel’s Equation

 10.2.1 $z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+z\frac{\mathrm{d}w}{\mathrm{d% }z}+(z^{2}-\nu^{2})w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.1 Referenced by: §10.13, §10.15, §10.2(ii), §10.2(ii), §10.2(iii), §10.24, §10.25(i), §10.27, §10.4, §10.61(ii), §10.74(ii) Permalink: http://dlmf.nist.gov/10.2.E1 Encodings: TeX, pMML, png See also: Annotations for §10.2(i), §10.2 and Ch.10

This differential equation has a regular singularity at $z=0$ with indices $\pm\nu$, and an irregular singularity at $z=\infty$ of rank $1$; compare §§2.7(i) and 2.7(ii).

## §10.2(ii) Standard Solutions

### Bessel Function of the First Kind

 10.2.2 $J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(% \tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}.$ ⓘ Defines: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.10 Referenced by: §10.11, §10.16, §10.19(i), §10.22(ii), §10.22(iii), §10.24, §10.53, §10.65(i), §10.7(i), §10.74(ii), §10.8, §8.6(i) Permalink: http://dlmf.nist.gov/10.2.E2 Encodings: TeX, pMML, png See also: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

This solution of (10.2.1) is an analytic function of $z\in\mathbb{C}$, except for a branch point at $z=0$ when $\nu$ is not an integer. The principal branch of $J_{\nu}\left(z\right)$ corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\infty,0]$.

When $\nu=n$ $(\in\mathbb{Z})$, $J_{\nu}\left(z\right)$ is entire in $z$.

For fixed $z$ $(\neq 0)$ each branch of $J_{\nu}\left(z\right)$ is entire in $\nu$.

### Bessel Function of the Second Kind (Weber’s Function)

 10.2.3 $Y_{\nu}\left(z\right)=\frac{J_{\nu}\left(z\right)\cos\left(\nu\pi\right)-J_{-% \nu}\left(z\right)}{\sin\left(\nu\pi\right)}.$ ⓘ Defines: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.2 Referenced by: §10.11, §10.15, §10.19(i), §10.2(ii), §10.22(ii), §10.24, §10.4, §10.47(ii), §10.7(i), §10.8, §11.10(v), §11.4(i) Permalink: http://dlmf.nist.gov/10.2.E3 Encodings: TeX, pMML, png See also: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

When $\nu$ is an integer the right-hand side is replaced by its limiting value:

 10.2.4 $Y_{n}\left(z\right)=\frac{1}{\pi}\left.\frac{\partial J_{\nu}\left(z\right)}{% \partial\nu}\right|_{\nu=n}+\left.\frac{(-1)^{n}}{\pi}\frac{\partial J_{\nu}% \left(z\right)}{\partial\nu}\right|_{\nu=-n},$ $n=0,\pm 1,\pm 2,\ldots$.

Whether or not $\nu$ is an integer $Y_{\nu}\left(z\right)$ has a branch point at $z=0$. The principal branch corresponds to the principal branches of $J_{\pm\nu}\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\infty,0]$.

Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu}\left(z\right)$ and $Y_{\nu}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$; compare §4.2(i).

Both $J_{\nu}\left(z\right)$ and $Y_{\nu}\left(z\right)$ are real when $\nu$ is real and $\operatorname{ph}z=0$.

For fixed $z$ $(\neq 0)$ each branch of $Y_{\nu}\left(z\right)$ is entire in $\nu$.

### 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

 10.2.5 ${H^{(1)}_{\nu}}\left(z\right)\sim\sqrt{2/(\pi z)}e^{i(z-\frac{1}{2}\nu\pi-% \frac{1}{4}\pi)}$ ⓘ Defines: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function) Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.2.3 Referenced by: §10.2(ii), §10.7(ii) Permalink: http://dlmf.nist.gov/10.2.E5 Encodings: TeX, pMML, png See also: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

as $z\to\infty$ in $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$, and

 10.2.6 ${H^{(2)}_{\nu}}\left(z\right)\sim\sqrt{2/(\pi z)}e^{-i\left(z-\frac{1}{2}\nu% \pi-\frac{1}{4}\pi\right)}$ ⓘ Defines: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function) Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.2.4 Referenced by: §10.2(ii), §10.7(ii) Permalink: http://dlmf.nist.gov/10.2.E6 Encodings: TeX, pMML, png See also: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

as $z\to\infty$ in $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$, where $\delta$ is an arbitrary small positive constant. Each solution has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\infty,0]$.

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 $\operatorname{ph}z=\pm\pi$.

For fixed $z$ $(\neq 0)$ each branch of ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ is entire in $\nu$.

### Branch Conventions

Except where indicated otherwise, it is assumed throughout the DLMF that the symbols $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, and ${H^{(2)}_{\nu}}\left(z\right)$ denote the principal values of these functions.

### Cylinder Functions

The notation $\mathscr{C}_{\nu}\left(z\right)$ denotes $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$.

## §10.2(iii) Numerically Satisfactory Pairs of Solutions

Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\Re\nu\geq 0$. When $\Re\nu<0$, $\nu$ is replaced by $-\nu$ throughout.