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21: 12.4 Power-Series Expansions

… ►

12.4.1 $$U(a,z)=U(a,0)u_1(a,z)+U^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E1

Encodings:

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Annotations for §12.4 and Ch.12

►

12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E2

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Annotations for §12.4 and Ch.12

►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by … ►

12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

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Annotations for §12.4 and Ch.12

… ►These series converge for all values of $z$.

22: 10.12 Generating Function and Associated Series

… ►For $z\in ℂ$ and $t\in ℂ\setminus \{0\}$, … ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, … ►

$\sin z=2J_1\left(z\right)-2J_3\left(z\right)+2J_5\left(z\right)-\mathrm{\cdots },$

►

$\frac{1}{2}z\cos z=J_1\left(z\right)-9J_3\left(z\right)+25J_5\left(z\right)-49J_7\left(z\right)+\mathrm{\cdots },$

►

$\frac{1}{2}z\sin z=4J_2\left(z\right)-16J_4\left(z\right)+36J_6\left(z\right)-\mathrm{\cdots }.$

23: 4.25 Continued Fractions

… ►

4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

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Annotations for §4.25 and Ch.4

►valid when $z$ lies in the open cut plane shown in Figure 4.23.1(i). ►

4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

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Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

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Annotations for §4.25 and Ch.4

►valid when $z$ lies in the open cut plane shown in Figure 4.23.1(ii). …valid when $z$ lies in the open cut plane shown in Figure 4.23.1(iv). …

24: 10.13 Other Differential Equations

… ►

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

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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

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Annotations for §10.13 and Ch.10

… ►

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

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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

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

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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:

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Annotations for §10.13 and Ch.10

►

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

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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:

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Annotations for §10.13 and Ch.10

…

25: 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:

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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:

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Annotations for §4.36 and Ch.4

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

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

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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:

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See also:

Annotations for §4.36 and Ch.4

… ►

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

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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:

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See also:

Annotations for §4.36 and Ch.4

26: 4.35 Identities

… ►

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

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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:

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

►

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

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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:

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

►

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

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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:

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

… ►

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

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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:

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

… ►With $z=x+\mathrm{i}y$ …

27: 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 $n$. … ►

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

…

28: 7.9 Continued Fractions

… ►

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

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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:

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See also:

Annotations for §7.9 and Ch.7

►

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

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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:

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See also:

Annotations for §7.9 and Ch.7

►

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

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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:

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See also:

Annotations for §7.9 and Ch.7

…

29: 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:

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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:

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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:

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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:

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See also:

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

…

30: 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 $\alpha =a$. …