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31: 10.13 Other Differential Equations

… ►

10.13.5 $$z^2{w}^{\prime \prime }+(1-2r)z{w}^{\prime }+({\lambda }^2{q}^2{z}^{2q}+r^2-{\nu }^2{q}^2)w=0,$$ $w=z^r{𝒞}_{\nu }\left(\lambda z^q\right)$,

ⓘ

Symbols:

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

A&S Ref:

9.1.53

Permalink:

http://dlmf.nist.gov/10.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

… ►

10.13.7 $$z^2(z^2-{\nu }^2)w^{\prime \prime }+z(z^2-3{\nu }^2)w^{\prime }+({(z^2-{\nu }^2)}^2-(z^2+{\nu }^2))w=0,$$ $w={𝒞}_{\nu }^{\prime }\left(z\right)$,

ⓘ

Symbols:

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

A&S Ref:

9.1.55

Permalink:

http://dlmf.nist.gov/10.13.E7

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

…

32: 4.36 Infinite Products and Partial Fractions

… ►

4.36.1 $$\sinh z=z\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.68

Permalink:

http://dlmf.nist.gov/4.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►

4.36.2 $$\cosh z=\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.69

Permalink:

http://dlmf.nist.gov/4.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►When $z\ne n\pi \mathrm{i}$, $n\in \mathbb{Z}$, ►

4.36.3 $$\coth z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2+n^2{\pi }^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

… ►

4.36.5 $$\mathrm{csch}z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2+n^2{\pi }^2}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

33: 4.35 Identities

… ►

4.35.11 $${\cosh }^{2}z-{\sinh }^{2}z=1,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.16

Permalink:

http://dlmf.nist.gov/4.35.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(ii), §4.35 and Ch.4

►

4.35.12 $${\mathrm{sech}}^{2}z=1-{\tanh }^{2}z,$$

ⓘ

Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.17

Permalink:

http://dlmf.nist.gov/4.35.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(ii), §4.35 and Ch.4

►

4.35.13 $${\mathrm{csch}}^{2}z={\coth }^{2}z-1.$$

ⓘ

Symbols:

$\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function and $z$: complex variable

A&S Ref:

4.5.18

Permalink:

http://dlmf.nist.gov/4.35.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(ii), §4.35 and Ch.4

… ►

4.35.26 $$\sinh \left(2z\right)=2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh }^{2}z},$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.31

Permalink:

http://dlmf.nist.gov/4.35.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(iii), §4.35 and Ch.4

… ►With $z=x+\mathrm{i}y$ …

34: 10.49 Explicit Formulas

… ►

${𝗃}_0\left(z\right)={\displaystyle \frac{\sin z}{z}},$

… ►

${𝗃}_2\left(z\right)=\left(-{\displaystyle \frac{1}{z}}+{\displaystyle \frac{3}{z^3}}\right)\sin z-{\displaystyle \frac{3}{z^2}}\cos z.$

… ►

${𝗒}_2\left(z\right)=\left({\displaystyle \frac{1}{z}}-{\displaystyle \frac{3}{z^3}}\right)\cos z-{\displaystyle \frac{3}{z^2}}\sin z.$

… ► ${\sum }_{k=0}^n{a}_k(n+\frac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree $n$. … ►

$${𝗃}_n\left(z\right)=z^n{\left(-\frac{1}{z}\frac{d}{dz}\right)}^n\frac{\sin z}{z},$$

…

35: 12.8 Recurrence Relations and Derivatives

… ►

12.8.1 $$zU(a,z)-U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.4

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.2 $$U^{\prime }(a,z)+\frac{1}{2}zU(a,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.1

Permalink:

http://dlmf.nist.gov/12.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.3 $$U^{\prime }(a,z)-\frac{1}{2}zU(a,z)+U(a-1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.2

Permalink:

http://dlmf.nist.gov/12.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

… ►(12.8.1)–(12.8.4) are also satisfied by $\overline{U}(a,z)$. ►

12.8.5 $$zV(a,z)-V(a+1,z)+(a-\frac{1}{2})V(a-1,z)=0,$$

ⓘ

Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.8

Permalink:

http://dlmf.nist.gov/12.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

…

36: 7.9 Continued Fractions

… ►

7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

►

7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

►

7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (in different form)

Permalink:

http://dlmf.nist.gov/7.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

…

37: 10.72 Mathematical Applications

… ►The number $m$ can also be replaced by any real constant $\lambda$ $(>-2)$ in the sense that ${(z-z_0)}^{-\lambda }$ $f(z)$ is analytic and nonvanishing at $z_0$; moreover, $g(z)$ is permitted to have a single or double pole at $z_0$. … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. … ►In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha$, $f(z,\alpha )$ has a simple zero $z=z_0(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where $z_0(a)=0$. Assume that whether or not $\alpha =a$, $z^{2}g(z,\alpha )$ is analytic at $z=0$. …These approximations are uniform with respect to both $z$ and $\alpha$, including $z=z_0(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. …

38: 7.1 Special Notation

… ► ►►►

$x$	real variable.
$z$	complex variable.
…

… ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. ►The notations $P(z)$, $Q(z)$, and $\mathrm{\Phi }(z)$ are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions. …

39: 13.3 Recurrence Relations and Derivatives

… ►

13.3.7 $$U(a-1,b,z)+(b-2a-z)U(a,b,z)+a(a-b+1)U(a+1,b,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.15

Permalink:

http://dlmf.nist.gov/13.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.8 $$(b-a-1)U(a,b-1,z)+(1-b-z)U(a,b,z)+zU(a,b+1,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.16

Permalink:

http://dlmf.nist.gov/13.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

… ►

13.3.10 $$(b-a)U(a,b,z)+U(a-1,b,z)-zU(a,b+1,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.18

Permalink:

http://dlmf.nist.gov/13.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.11 $$(a+z)U(a,b,z)-zU(a,b+1,z)+a(b-a-1)U(a+1,b,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.19

Permalink:

http://dlmf.nist.gov/13.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.12 $$(a-1+z)U(a,b,z)-U(a-1,b,z)+(a-b+1)U(a,b-1,z)=0.$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.20

Permalink:

http://dlmf.nist.gov/13.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

…

40: 15.5 Derivatives and Contiguous Functions

… ►

15.5.5 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{c-a-1}{(1-z)}^{a+b-c}F(a,b;c;z)\right)={\left(c-a\right)}_n{z}^{c-a+n-1}{(1-z)}^{a-n+b-c}F(a-n,b;c;z).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/15.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(i), §15.5 and Ch.15

… ►

15.5.10 $${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $z$: complex variable

Referenced by:

§15.5(i)

Permalink:

http://dlmf.nist.gov/15.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(i), §15.5 and Ch.15

… ►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. … ►

15.5.20 $$z(1-z)\left(dF(a,b;c;z)/dz\right)=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

…