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21: 27.10 Periodic Number-Theoretic Functions

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27.10.10 $$G(n,\chi )=\overline{\chi }(n)G(1,\chi ).$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\overline{z}$: complex conjugate, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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27.10.12 $$\chi \left(n\right)=\frac{G(1,\chi )}{k}\sum _{m=1}^k\overline{\chi }(m){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

22: 12.1 Special Notation

… ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …

23: 25.10 Zeros

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25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

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24: 7.13 Zeros

… ►The other zeros of $\mathrm{erf}z$ are $-z_n$, ${\overline{z}}_n$, $-{\overline{z}}_n$. … ►The other zeros of $\mathrm{erfc}z$ are ${\overline{z}}_n$. The zeros of $w\left(z\right)$ are $\mathrm{i}{z}_n$ and $\mathrm{i}{\overline{z}}_n$. … ►Then $-z_n$, ${\overline{z}}_n$, $-{\overline{z}}_n$, $\mathrm{i}{z}_n$, $-\mathrm{i}{z}_n$, $\mathrm{i}{\overline{z}}_n$, $-\mathrm{i}{\overline{z}}_n$ are also zeros of the same integral. …

25: 12.14 The Function $W(a,x)$

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12.14.34 $$W(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{l(\mu )}{{(t^2+1)}^{\frac{1}{4}}}\left(\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}-\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{u}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

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12.14.35 $$W^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim -\frac{\mu }{\sqrt{2}}l(\mu ){(t^2+1)}^{\frac{1}{4}}\left(\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{v}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}+\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{v}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{v}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

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12.14.36 $$\overline{\sigma }={\mu }^2\overline{\xi }+\frac{1}{4}\pi ,$$

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Defines:

$\overline{\sigma }$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\overline{\xi }$

Permalink:

http://dlmf.nist.gov/12.14.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

►and $\overline{\xi }$ and the coefficients ${\overline{u}}_s(t)$ and ${\overline{v}}_s(t)$ as in §12.10(v). …

26: 25.15 Dirichlet $L$-functions

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25.15.5 $$L(1-s,\chi )=\frac{k^{s-1}\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}+\chi (-1){\mathrm{e}}^{\pi \mathrm{i}s/2}\right)G(\chi )L(s,\overline{\chi }),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

connection formula

Source:

Apostol (1976, (16), p. 263)

Referenced by:

§25.15(ii)

Permalink:

http://dlmf.nist.gov/25.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

►where $\overline{\chi }$ is the complex conjugate of $\chi$, and …

27: 1.8 Fourier Series

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1.8.6_1 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\frac{1}{2}{a}_0\overline{a_0^{\prime }}+\sum _{n=1}^{\mathrm{\infty }}(a_n\overline{a_n^{\prime }}+b_n\overline{b_n^{\prime }}),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

(1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_1

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.2.0):

This equation, which was originally (1.8.13), was moved here.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.6_2 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n\overline{c_n^{\prime }}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.8(i), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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28: 14.30 Spherical and Spheroidal Harmonics

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14.30.6 $$Y_{l,-m}(\theta ,\varphi )={(-1)}^m\overline{Y_{l,m}(\theta ,\varphi )}.$$

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Symbols:

$\overline{z}$: complex conjugate, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E6

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the spherical harmonic on the right-hand side has been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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14.30.8 $${\int }_0^{2\pi }{\int }_0^{\pi }\overline{Y_{l_1,m_1}(\theta ,\varphi )}{Y}_{l_2,m_2}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l_1,l_2}{\delta }_{m_1,m_2}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E8

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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14.30.9 $${𝖯}_l\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \left({\varphi }_1-{\varphi }_2\right)\right)=\frac{4\pi }{2l+1}\sum _{m=-l}^l\overline{Y_{l,m}({\theta }_1,{\varphi }_1)}{Y}_{l,m}({\theta }_2,{\varphi }_2).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§18.18(ii)

Permalink:

http://dlmf.nist.gov/14.30.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

…

29: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►Here bars do not denote complex conjugates; instead ►

12.10.26 $$\overline{\xi }=\frac{1}{2}t\sqrt{t^2+1}+\frac{1}{2}\ln \left(t+\sqrt{t^2+1}\right),$$

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Defines:

$\overline{\xi }$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/12.10.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

►

12.10.27 $${\overline{u}}_s(t)={\mathrm{i}}^s{u}_s(-\mathrm{i}t),$$

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Defines:

${\overline{u}}_s(t)$: solution (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $u_s(t)$: solution

Permalink:

http://dlmf.nist.gov/12.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

►and the function $\overline{g}(\mu )$ has the asymptotic expansion … ►

12.10.30 $${\overline{v}}_s(t)={\mathrm{i}}^s{v}_s(-\mathrm{i}t).$$

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Defines:

${\overline{v}}_s(t)$: solution (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $v_s(t)$: solution

Permalink:

http://dlmf.nist.gov/12.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

…

30: 10.42 Zeros

… ►The zeros in the sector $-\frac{1}{2}\pi \le \mathrm{ph}z\le \frac{3}{2}\pi$ are their conjugates. …