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31: 20.11 Generalizations and Analogs

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20.11.1 $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$

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Defines:

$G(m,n)$: Gauss sum (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

… ►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): ►

20.11.2 $$\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}\sum _{j=0}^{m-1}{\mathrm{e}}^{\pi \mathrm{i}{j}^{2}n/m}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}G(-n,m).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $G(m,n)$: Gauss sum

Permalink:

http://dlmf.nist.gov/20.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

… ►With the substitutions $a=q{\mathrm{e}}^{2\mathrm{i}z}$, $b=q{\mathrm{e}}^{-2\mathrm{i}z}$, with $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, we have …

32: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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Products

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§10.40(ii) Error Bounds for Real Argument and Order

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§10.40(iii) Error Bounds for Complex Argument and Order

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33: 11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

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11.6.7 $${𝐌}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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34: 20.10 Integrals

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>0$ because $1-{\theta }_4\left(0\right|\mathrm{i}{x}^2)=1-x^{-1}{\theta }_2\left(0\right|\mathrm{i}{x}^{-2})\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

… ►Let $s$, $\mathrm{\ell }$, and $\beta$ be constants such that $\mathrm{\Re }s>0$, $\mathrm{\ell }>0$, and $\left|\mathrm{\Re }\beta \right|+\left|\mathrm{\Im }\beta \right|\le \mathrm{\ell }$. … ►For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). …

35: 5.3 Graphics

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§5.3(i) Real Argument

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§5.3(ii) Complex Argument

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Figure 5.3.4: $|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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Figure 5.3.5: $1/|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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Figure 5.3.6: $|\psi \left(x+\mathrm{i}y\right)|$. Magnify 3D Help

36: 1.10 Functions of a Complex Variable

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Phase (or Argument) Principle

… ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. Similarly if $D=ℂ\setminus [0,\mathrm{\infty })$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. … ►Thus if $F(z)$ is continued along a path that circles $z=1$ $m$ times in the positive sense and returns to $z_0$ without circling $z=-1$, then $F((z_0-1){\mathrm{e}}^{2m\pi \mathrm{i}}+1)={\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}{\mathrm{e}}^{2\pi \mathrm{i}m\alpha }$. If the path also circles $z=-1$ $n$ times in the clockwise or negative sense before returning to $z_0$, then the value of $F(z_0)$ becomes ${\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}{\mathrm{e}}^{2\pi \mathrm{i}m\alpha }{\mathrm{e}}^{-2\pi \mathrm{i}n\beta }$. …

37: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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$s=\sigma +\mathrm{i}t$	complex variable.
$z=x+\mathrm{i}y$	complex variable.
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primes	on function symbols: derivatives with respect to argument.

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38: 19.3 Graphics

… ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The imaginary part is 0 for , and is antisymmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The imaginary part is 0 for $k^2\le 1$ and is antisymmetric under reflection in the real axis. … Magnify 3D Help

39: 36.2 Catastrophes and Canonical Integrals

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36.2.6 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2\sqrt{\pi /3}\exp \left(\mathrm{i}\left(\frac{4}{27}{z}^3+\frac{1}{3}xz-\frac{1}{4}\pi \right)\right){\int }_{\mathrm{\infty }\exp \left(-7\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(u^6+2z{u}^4+(z^2+x)u^2+\frac{y^2}{12u^2}\right)\right)du,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii), §36.5(iii)

Permalink:

http://dlmf.nist.gov/36.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)={\displaystyle \frac{4\pi }{3^{1/3}}}\exp \left(\mathrm{i}\left(\frac{2}{27}{z}^3-\frac{1}{3}xz\right)\right)\left(\exp \left(-\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{+}(𝐱)+\exp \left(\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{-}(𝐱)\right),$

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36.2.12 $${\mathrm{\Psi }}_0=\sqrt{\pi }\exp \left(\mathrm{i}\frac{\pi }{4}\right).$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function and $\mathrm{i}$: imaginary unit

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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${\mathrm{\Psi }}_2\left(\mathrm{𝟎}\right)=1.67481+\mathrm{i}\mathrm{\hspace{0.17em}0.69373}$

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${\mathrm{\Psi }}_4\left(\mathrm{𝟎}\right)=1.79222+\mathrm{i}\mathrm{\hspace{0.17em}0.48022}.$

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40: 20.2 Definitions and Periodic Properties

… ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. … ►

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Figure 20.2.1: $z$-plane. …Left-hand diagram is the rectangular case ($\tau$ purely imaginary); right-hand diagram is the general case. …

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§20.2(iii) Translation of the Argument by Half-Periods

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20.2.10 $$M\equiv M(z|\tau )={\mathrm{e}}^{\mathrm{i}z+(\mathrm{i}\pi \tau /4)},$$

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Defines:

$M$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

A&S Ref:

16.33.1 (in different notation)

Permalink:

http://dlmf.nist.gov/20.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(iii), §20.2 and Ch.20

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20.2.11 $${\theta }_1\left(z\right|\tau )=-{\theta }_2\left(z+\frac{1}{2}\pi \right|\tau )=-\mathrm{i}M{\theta }_4\left(z+\frac{1}{2}\pi \tau \right|\tau )=-\mathrm{i}M{\theta }_3\left(z+\frac{1}{2}\pi +\frac{1}{2}\pi \tau \right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex, $\tau$: lattice parameter and $M$

A&S Ref:

16.33.2 (in different notation)

Referenced by:

§20.7(iv)

Permalink:

http://dlmf.nist.gov/20.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(iii), §20.2 and Ch.20

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