theta functions

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31: 19.10 Relations to Other Functions

… ►

§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …

32: Peter L. Walker

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20 Theta Functions

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33: Bille C. Carlson

… ►In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …

34: 20.6 Power Series

§20.6 Power Series

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20.6.2 $${\theta }_1\left(\pi z\right|\tau )=\pi z{\theta }_1^{\prime }\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\delta }_{2j}(\tau )z^{2j}\right),$$

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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, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\delta }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.3 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

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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, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\alpha }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.4 $${\theta }_3\left(\pi z\right|\tau )={\theta }_3\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\beta }_{2j}(\tau )z^{2j}\right),$$

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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, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\beta }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.5 $${\theta }_4\left(\pi z\right|\tau )={\theta }_4\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\gamma }_{2j}(\tau )z^{2j}\right).$$

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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, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\gamma }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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35: William P. Reinhardt

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20 Theta Functions

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36: 20.14 Methods of Computation

§20.14 Methods of Computation

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37: 22.20 Methods of Computation

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§22.20(i) Via Theta Functions

►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►If either $k$ or $x$ is complex then (22.2.6) gives the definition of $\mathrm{dn}(x,k)$ as a quotient of theta functions. … ►If either $\tau$ or $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ is given, then we use $k={\theta }_2^2(0,q)/{\theta }_3^2(0,q)$, $k^{\prime }={\theta }_4^2(0,q)/{\theta }_3^2(0,q)$, $K=\frac{1}{2}\pi {\theta }_3^2(0,q)$, and $K^{\prime }=-\mathrm{i}\tau K$, obtaining the values of the theta functions as in §20.14. … ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

38: 10.68 Modulus and Phase Functions

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10.68.1 $$M_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\theta }_{\nu }\left(x\right)}={\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x,$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.68.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(i), §10.68 and Ch.10

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►

10.68.8 $${\mathrm{ber}}_{\nu }^{\prime }x=\frac{1}{2}{M}_{\nu +1}\cos \left({\theta }_{\nu +1}-\frac{1}{4}\pi \right)-\frac{1}{2}{M}_{\nu -1}\cos \left({\theta }_{\nu -1}-\frac{1}{4}\pi \right)=(\nu /x)M_{\nu }\cos {\theta }_{\nu }+M_{\nu +1}\cos \left({\theta }_{\nu +1}-\frac{1}{4}\pi \right)=-(\nu /x)M_{\nu }\cos {\theta }_{\nu }-M_{\nu -1}\cos \left({\theta }_{\nu -1}-\frac{1}{4}\pi \right),$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.11

Referenced by:

§10.68(ii)

Permalink:

http://dlmf.nist.gov/10.68.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(ii), §10.68 and Ch.10

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10.68.11 $$M_{\nu }^{\prime }=(\nu /x)M_{\nu }+M_{\nu +1}\cos \left({\theta }_{\nu +1}-{\theta }_{\nu }-\frac{1}{4}\pi \right)=-(\nu /x)M_{\nu }-M_{\nu -1}\cos \left({\theta }_{\nu -1}-{\theta }_{\nu }-\frac{1}{4}\pi \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.14

Permalink:

http://dlmf.nist.gov/10.68.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(ii), §10.68 and Ch.10

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10.68.12 $${\theta }_{\nu }^{\prime }=(M_{\nu +1}/M_{\nu })\sin \left({\theta }_{\nu +1}-{\theta }_{\nu }-\frac{1}{4}\pi \right)=-(M_{\nu -1}/M_{\nu })\sin \left({\theta }_{\nu -1}-{\theta }_{\nu }-\frac{1}{4}\pi \right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: sine function and $\nu$: complex parameter

A&S Ref:

9.10.15

Permalink:

http://dlmf.nist.gov/10.68.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.68(ii), §10.68 and Ch.10

…

39: 22.21 Tables

… ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

40: 9.8 Modulus and Phase

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9.8.1 $$\mathrm{Ai}\left(x\right)=M\left(x\right)\sin \theta \left(x\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $M\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: Airy phase function, $\sin z$: sine function and $x$: real variable

Source:

Olver (1997b, (2.01), p. 394)

Referenced by:

(9.8.13), (9.9.3)

Permalink:

http://dlmf.nist.gov/9.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.2 $$\mathrm{Bi}\left(x\right)=M\left(x\right)\cos \theta \left(x\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\cos z$: cosine function, $M\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Source:

Olver (1997b, (2.01), p. 394)

Referenced by:

(9.8.13), (9.9.3)

Permalink:

http://dlmf.nist.gov/9.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

… ►(These definitions of $\theta \left(x\right)$ and $\varphi \left(x\right)$ differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) … ►

9.8.13 $$M\left(x\right)N\left(x\right)\sin \left(\theta \left(x\right)-\varphi \left(x\right)\right)={\pi }^{-1},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function, $\sin z$: sine function and $x$: real variable

Proof sketch:

First use (4.21.2) then (9.8.1), (9.8.2), (9.8.5), (9.8.6), and finally (9.2.7).

Permalink:

http://dlmf.nist.gov/9.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(ii), §9.8 and Ch.9

… ►As $x$ increases from $-\mathrm{\infty }$ to $0$ each of the functions $M\left(x\right)$, $M^{\prime }\left(x\right)$, ${|x|}^{-1/4}N\left(x\right)$, $M\left(x\right)N\left(x\right)$, ${\theta }^{\prime }\left(x\right)$, ${\varphi }^{\prime }\left(x\right)$ is increasing, and each of the functions ${|x|}^{1/4}M\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ is decreasing. …