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1: 10.29 Recurrence Relations and Derivatives

… ►With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii), ►

${𝒵}_{\nu -1}\left(z\right)-{𝒵}_{\nu +1}\left(z\right)=(2\nu /z){𝒵}_{\nu }\left(z\right),$

►

${𝒵}_{\nu -1}\left(z\right)+{𝒵}_{\nu +1}\left(z\right)=2{𝒵}_{\nu }^{\prime }\left(z\right).$

►

${𝒵}_{\nu }^{\prime }\left(z\right)={𝒵}_{\nu -1}\left(z\right)-(\nu /z){𝒵}_{\nu }\left(z\right),$

… ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). …

2: 10.13 Other Differential Equations

… ►

10.13.1 $$w^{\prime \prime }+\left({\lambda }^2-\frac{{\nu }^2-\frac{1}{4}}{z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.49

Referenced by:

§1.18(vi), §10.36

Permalink:

http://dlmf.nist.gov/10.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

… ►In (10.13.9)–(10.13.11) ${𝒞}_{\nu }\left(z\right)$, ${𝒟}_{\mu }(z)$ are any cylinder functions of orders $\nu ,\mu$, respectively, and $\mathit{\vartheta }=z(d/dz)$. ►

10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.11 $$\left({\mathit{\vartheta }}^4-2({\nu }^2+{\mu }^2){\mathit{\vartheta }}^2+{({\nu }^2-{\mu }^2)}^2\right)w+4z^2(\mathit{\vartheta }+1)(\mathit{\vartheta }+2)w=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\mu }(z).$

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter, $\mathit{\vartheta }=z(d/dz)$ and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.57

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

…

3: 10.36 Other Differential Equations

… ►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

ⓘ

Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

…

4: 23.18 Modular Transformations

… ► $\lambda \left(𝒜\tau \right)$ equals …according as the elements of $𝒜$ in (23.15.3) have the respective forms … ►

23.18.3 $$\lambda \left(𝒜\tau \right)=\lambda \left(\tau \right),$$

ⓘ

Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $\tau$: complex variable and $𝒜$: bilinear transformation

Permalink:

http://dlmf.nist.gov/23.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

… ►

23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

ⓘ

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

►where the square root has its principal value and …

5: 31.10 Integral Equations and Representations

… ►and the kernel $𝒦(z,t)$ is a solution of the partial differential equation …where ${𝒟}_z$ is Heun’s operator in the variable $z$ : … ►The kernel $𝒦$ must satisfy … ►where ${𝒟}_z$ is given by (31.10.4). … ►The kernel $𝒦$ must satisfy …

6: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. $p(𝒟k,n)$ denotes the number of partitions of $n$ into parts with difference at least $k$. …$p(𝒪,n)$ denotes the number of partitions of $n$ into odd parts. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. …

7: 28.23 Expansions in Series of Bessel Functions

… ►

${𝒞}_{\mu }^{(1)}=J_{\mu },$

►

${𝒞}_{\mu }^{(2)}=Y_{\mu },$

►

${𝒞}_{\mu }^{(3)}=H_{\mu }^{(1)},$

►

${𝒞}_{\mu }^{(4)}=H_{\mu }^{(2)};$

… ►

28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

…

8: 10.6 Recurrence Relations and Derivatives

… ►With ${𝒞}_{\nu }\left(z\right)$ defined as in §10.2(ii), ►

$${𝒞}_{\nu -1}\left(z\right)+{𝒞}_{\nu +1}\left(z\right)=(2\nu /z){𝒞}_{\nu }\left(z\right),$$

►

$${𝒞}_{\nu -1}\left(z\right)-{𝒞}_{\nu +1}\left(z\right)=2{𝒞}_{\nu }^{\prime }\left(z\right).$$

… ►If $f_{\nu }(z)=z^p{𝒞}_{\nu }\left(\lambda z^q\right)$, where $p,q$, and $\lambda$ ($\ne 0$) are real or complex constants, then … ►For results on modified quotients of the form $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). …

9: 10.28 Wronskians and Cross-Products

… ►

10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

►

10.28.2 $$𝒲\left\{K_{\nu }\left(z\right),I_{\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)K_{\nu +1}\left(z\right)+I_{\nu +1}\left(z\right)K_{\nu }\left(z\right)=1/z.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.15

Permalink:

http://dlmf.nist.gov/10.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

10: 10.5 Wronskians and Cross-Products

… ►

10.5.1 $$𝒲\left\{J_{\nu }\left(z\right),J_{-\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)J_{-\nu }\left(z\right)+J_{\nu }\left(z\right)J_{-\nu -1}\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.15

Permalink:

http://dlmf.nist.gov/10.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.2 $$𝒲\left\{J_{\nu }\left(z\right),Y_{\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)Y_{\nu }\left(z\right)-J_{\nu }\left(z\right)Y_{\nu +1}\left(z\right)=2/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.16

Permalink:

http://dlmf.nist.gov/10.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.5 $$𝒲\left\{H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=H_{\nu +1}^{(1)}\left(z\right)H_{\nu }^{(2)}\left(z\right)-H_{\nu }^{(1)}\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-4\mathrm{i}/(\pi z).$$

ⓘ

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.17

Permalink:

http://dlmf.nist.gov/10.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10