Watson sum

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21: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. … ►The partition function $p\left(n\right)$ is tabulated in Gupta (1935, 1937), Watson (1937), and Gupta et al. (1958). Tables of the Ramanujan function $\tau \left(n\right)$ are published in Lehmer (1943) and Watson (1949). …

22: 20.5 Infinite Products and Related Results

… ►

20.5.9 $${\theta }_3\left(\pi z\right|\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{p}^{2n}{q}^{n^2}=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+q^{2n-1}{p}^2\right)\left(1+q^{2n-1}{p}^{-2}\right),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

Referenced by:

§17.8, §20.12(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5(i), §20.5 and Ch.20

… ►

20.5.10 $$\frac{{\theta }_1^{\prime }(z,q)}{{\theta }_1(z,q)}-\cot z=4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.1

Referenced by:

§20.5(ii), §20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

►

20.5.11 $$\frac{{\theta }_2^{\prime }(z,q)}{{\theta }_2(z,q)}+\tan z=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1+2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.2

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

… ►

20.5.12 $$\frac{{\theta }_3^{\prime }(z,q)}{{\theta }_3(z,q)}=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^n}{1-q^{2n}}\sin \left(2nz\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.3

Permalink:

http://dlmf.nist.gov/20.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

►

20.5.13 $$\frac{{\theta }_4^{\prime }(z,q)}{{\theta }_4(z,q)}=4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1-q^{2n}}\sin \left(2nz\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.4

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

…

23: 10.21 Zeros

… ►No two of the functions $J_0\left(z\right)$, $J_1\left(z\right)$, $J_2\left(z\right),\mathrm{\dots }$, have any common zeros other than $z=0$; see Watson (1944, §15.28). … ►

10.21.22 $${\rho }_{\nu }(t)\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\nu$: complex parameter, ${\rho }_{\nu }(t)$: zero of cylinder function and $\alpha$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.27 $${\sigma }_{\nu }(t)\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\nu$: complex parameter, ${\sigma }_{\nu }(t)$: zero of cylinder function and ${\alpha }^{\prime }$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.55 $${\sigma }_n\left(\nu \right)=\sum _{m=1}^{\mathrm{\infty }}{(j_{\nu ,m})}^{-2n},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Defines:

${\sigma }_n\left(\nu \right)$: Rayleigh function

Symbols:

$m$: integer, $n$: integer and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.21.E55

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(xiii), §10.21 and Ch.10

… ►See also Watson (1944, §§15.5, 15.51). …

24: 2.11 Remainder Terms; Stokes Phenomenon

… ►Application of Watson’s lemma (§2.4(i)) yields … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. … ►Multiplying these differences by ${(-1)}^j{2}^{-j-1}$ and summing, we obtain …Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. … ►The next column lists the partial sums $s_n=a_0+a_1+\mathrm{\cdots }+a_n$. …

25: 11.4 Basic Properties

… ►

11.4.1 $${𝐊}_{n+\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$

ⓘ

Symbols:

${𝐊}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: integer order

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

►

11.4.2 $${𝐋}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{{(-1)}^m(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $n$: integer order

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

… ►

11.4.19 $${𝐇}_{\nu }\left(z\right)={\left(\frac{z}{2\pi }\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(k+\frac{1}{2})}{J}_{k+\nu +\frac{1}{2}}\left(z\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iv), §11.4 and Ch.11

… ►

11.4.21 $${𝐇}_0\left(z\right)=\frac{4}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{J_{2k+1}\left(z\right)}{2k+1}=2\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{J}_{k+\frac{1}{2}}^2\left(\frac{1}{2}z\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: nonnegative integer

A&S Ref:

12.1.19 (First part)

Permalink:

http://dlmf.nist.gov/11.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iv), §11.4 and Ch.11

►

11.4.22 $${𝐇}_1\left(z\right)=\frac{2}{\pi }(1-J_0\left(z\right))+\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{J_{2k}\left(z\right)}{4k^2-1}=4\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+\frac{1}{2}}\left(\frac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\frac{1}{2}z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: nonnegative integer

A&S Ref:

12.1.20 (First part)

Permalink:

http://dlmf.nist.gov/11.4.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iv), §11.4 and Ch.11

…

26: 10.31 Power Series

… ►

10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

… ►

10.31.3 $$I_{\nu }\left(z\right)I_{\mu }\left(z\right)={(\frac{1}{2}z)}^{\nu +\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\nu +\mu +k+1\right)}_k{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\mu +k+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.31

Permalink:

http://dlmf.nist.gov/10.31.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §10.31 and Ch.10

27: 20.4 Values at $z$ = 0

… ►

20.4.8 $$\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{{\theta }_1^{\prime }(0,q)}=-1+24\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{{(1-q^{2n})}^2}.$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/20.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

►

20.4.9 $$\frac{{\theta }_2^{\prime \prime }(0,q)}{{\theta }_2(0,q)}=-1-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{{(1+q^{2n})}^2},$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

►

20.4.10 $$\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}=-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1+q^{2n-1})}^2},$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

►

20.4.11 $$\frac{{\theta }_4^{\prime \prime }(0,q)}{{\theta }_4(0,q)}=8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1-q^{2n-1})}^2}.$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

…

28: 2.4 Contour Integrals

… ►

§2.4(i) Watson’s Lemma

… ►

2.4.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{z^{(s+\lambda )/\mu }}$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\lambda$: constant, $a$: left endpoint, $q(t)$: analytic function and $\mu$: positive constant

Referenced by:

§2.4(i)

Permalink:

http://dlmf.nist.gov/2.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(i), §2.4 and Ch.2

… ►

2.4.4 $$Q(z)\sim \sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{z^{(s+\lambda )/\mu }},$$ $z\to \mathrm{\infty }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\lambda$: constant, $a$: left endpoint, $\mu$: positive constant and $Q(z)$: Laplace transform of $q(t)$

Referenced by:

§2.4(ii)

Permalink:

http://dlmf.nist.gov/2.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

… ►

In a neighborhood of $a$

2.4.11

$$p(t)=p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)=\sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

ⓘ

Symbols:

$\lambda$: constant, $a$: left endpoint, $p(t)$: analytic function, $q(t)$: analytic function, $p_s$: coefficients, $q_s$: coefficients and $\mu$: positive constant

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E11

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

with $\mathrm{\Re }\lambda >0$, $\mu >0$, $p_0\ne 0$, and the branches of ${(t-a)}^{\lambda }$ and ${(t-a)}^{\mu }$ continuous and constructed with $\mathrm{ph}\left(t-a\right)\to \omega$ as $t\to a$ along $𝒫$.

… ►

2.4.12 $$I(z)\sim {\mathrm{e}}^{-zp(a)}\sum _{s=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{s+\lambda }{\mu }\right)\frac{b_s}{z^{(s+\lambda )/\mu }}$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $\lambda$: constant, $a$: left endpoint, $b$: right endpoint, $I(z)$: contour integral, $p(t)$: analytic function and $\mu$: positive constant

Permalink:

http://dlmf.nist.gov/2.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

…

29: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►

11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

…

30: 12.9 Asymptotic Expansions for Large Variable

… ►

12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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