20 Theta Functions Properties 20.6 Power Series 20.8 Watson’s Expansions

§20.7 Identities

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Permalink:: http://dlmf.nist.gov/20.7
See also:: Annotations for Ch.20

§20.7(i) Sums of Squares

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Keywords:: sums of squares, theta functions
Notes:: See Lawden (1989, p. 11).
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/20.7.i
Addition (effective with 1.0.5):: The reference to Carlson (2011) was added at the end of this subsection.
See also:: Annotations for §20.7 and Ch.20

20.7.1		${\theta_{3}}^{2}\left(0,q\right){\theta_{3}}^{2}\left(z,q\right)={\theta_{4}}^% {2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)+{\theta_{2}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome A&S Ref: 16.28.3 Permalink: http://dlmf.nist.gov/20.7.E1 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

20.7.2		${\theta_{3}}^{2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)={\theta_{2}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{4}}^{2}\left(0,q% \right){\theta_{3}}^{2}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome A&S Ref: 16.28.2 Permalink: http://dlmf.nist.gov/20.7.E2 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

20.7.3		${\theta_{2}}^{2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)={\theta_{3}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{4}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome A&S Ref: 16.28.1 Permalink: http://dlmf.nist.gov/20.7.E3 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

20.7.4		${\theta_{2}}^{2}\left(0,q\right){\theta_{3}}^{2}\left(z,q\right)={\theta_{4}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{3}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome A&S Ref: 16.28.4 Permalink: http://dlmf.nist.gov/20.7.E4 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

Also

20.7.5		${\theta_{3}}^{4}\left(0,q\right)={\theta_{2}}^{4}\left(0,q\right)+{\theta_{4}}% ^{4}\left(0,q\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome A&S Ref: 16.28.5 Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/20.7.E5 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

§20.7(ii) Addition Formulas

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Keywords:: addition formulas, theta functions
Referenced by:: §21.6(ii), Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/20.7.ii
Addition (effective with 1.0.5):: The references to Carlson (2011) and Fukushima (2012), with explanatory text, were added in this subsection.
See also:: Annotations for §20.7 and Ch.20

20.7.6		${\theta_{4}}^{2}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Referenced by: §20.7(ii) Permalink: http://dlmf.nist.gov/20.7.E6 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20

20.7.7		${\theta_{4}}^{2}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Permalink: http://dlmf.nist.gov/20.7.E7 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20

20.7.8		${\theta_{4}}^{2}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Permalink: http://dlmf.nist.gov/20.7.E8 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20

20.7.9		${\theta_{4}}^{2}\left(0,q\right)\theta_{4}\left(w+z,q\right)\theta_{4}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Referenced by: §20.7(ii) Permalink: http://dlmf.nist.gov/20.7.E9 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20

For these and similar formulas see Lawden (1989, §1.4), Whittaker and Watson (1927, pp. 487–488), and Carlson (2011, §5).

Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$ . The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012).

§20.7(iii) Duplication Formula

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Keywords:: duplication formula, theta functions
Notes:: See McKean and Moll (1999, p. 129).
Referenced by:: §20.7(ii), Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/20.7.iii
Addition (effective with 1.0.5):: The reference to Carlson (2011) was added at the end of this subsection.
See also:: Annotations for §20.7 and Ch.20

20.7.10		$\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Permalink: http://dlmf.nist.gov/20.7.E10 Encodings: TeX, pMML, png See also: Annotations for §20.7(iii), §20.7 and Ch.20

§20.7(iv) Reduction Formulas for Products

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Keywords:: nome, theta functions, transformation of
Notes:: See Lawden (1989, pp. 17, 18) and Whittaker and Watson (1927, p. 477). The first equalities in (20.7.11) and (20.7.12) follow by translation of $z$ by $\tfrac{1}{2}\pi$ as in (20.2.11)–(20.2.14). The second equalities follow from (20.5.5)–(20.5.8) and the identity $\prod_{n=1}^{\infty}(1+q^{n})(1-q^{2n-1})=1$ (Walker (1996, p. 90)).
Referenced by:: §20.7(ix), Erratum (V1.0.5) for Clarifications
Permalink:: http://dlmf.nist.gov/20.7.iv
Clarification (effective with 1.0.5):: The title of this subsection was changed to be more specific.
See also:: Annotations for §20.7 and Ch.20

20.7.11		$\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z,q\right)}{\theta_{1}\left(2z% ,q^{2}\right)}=\frac{\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{% \theta_{4}\left(2z,q^{2}\right)}=\theta_{4}\left(0,q^{2}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Referenced by: §20.7(iv), §20.7(ix) Permalink: http://dlmf.nist.gov/20.7.E11 Encodings: TeX, pMML, png See also: Annotations for §20.7(iv), §20.7 and Ch.20

20.7.12		$\frac{\theta_{1}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{1}% \left(z,q\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}% \right)}{\theta_{2}\left(z,q\right)}=\tfrac{1}{2}\theta_{2}\left(0,q\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Referenced by: §20.7(iv), §20.7(ix) Permalink: http://dlmf.nist.gov/20.7.E12 Encodings: TeX, pMML, png See also: Annotations for §20.7(iv), §20.7 and Ch.20

Addendum: For a companion equation see (20.7.34).

§20.7(v) Watson’s Identities

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Keywords:: Watson’s identities, theta functions
Notes:: See McKean and Moll (1999, p. 130) and Watson (1935a).
Permalink:: http://dlmf.nist.gov/20.7.v
See also:: Annotations for §20.7 and Ch.20

20.7.13		$\theta_{1}\left(z,q\right)\theta_{1}\left(w,q\right)=\theta_{3}\left(z+w,q^{2}% \right)\theta_{2}\left(z-w,q^{2}\right)-\theta_{2}\left(z+w,q^{2}\right)\theta% _{3}\left(z-w,q^{2}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Permalink: http://dlmf.nist.gov/20.7.E13 Encodings: TeX, pMML, png See also: Annotations for §20.7(v), §20.7 and Ch.20

20.7.14		$\theta_{3}\left(z,q\right)\theta_{3}\left(w,q\right)=\theta_{3}\left(z+w,q^{2}% \right)\theta_{3}\left(z-w,q^{2}\right)+\theta_{2}\left(z+w,q^{2}\right)\theta% _{2}\left(z-w,q^{2}\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome Permalink: http://dlmf.nist.gov/20.7.E14 Encodings: TeX, pMML, png See also: Annotations for §20.7(v), §20.7 and Ch.20

§20.7(vi) Landen Transformations

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Keywords:: Jacobian elliptic functions, Landen transformation, Landen transformations, theta functions
Notes:: See Lawden (1989, pp. 17–18).
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/20.7.vi
See also:: Annotations for §20.7 and Ch.20

With

20.7.15		$A\equiv A(\tau)=\ifrac{1}{\theta_{4}\left(0\middle\|2\tau\right)},$
ⓘ Defines: $A$ (locally) Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\equiv$ : equals by definition and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E15 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

20.7.16	$\displaystyle\theta_{1}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{1}\left(z\middle\|\tau\right)\theta_{2}\left(z\middle\|% \tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E16 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.17	$\displaystyle\theta_{2}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{1}\left(\tfrac{1}{4}\pi-z\middle\|\tau\right)\theta_{1}% \left(\tfrac{1}{4}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E17 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.18	$\displaystyle\theta_{3}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{3}\left(\tfrac{1}{4}\pi-z\middle\|\tau\right)\theta_{3}% \left(\tfrac{1}{4}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E18 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.19	$\displaystyle\theta_{4}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{3}\left(z\middle\|\tau\right)\theta_{4}\left(z\middle\|% \tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E19 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

Next, with

20.7.20		$B\equiv B(\tau)=\ifrac{1}{\left(\theta_{3}\left(0\middle\|\tau\right)\theta_{4}% \left(0\middle\|\tau\right)\theta_{3}\left(\tfrac{1}{4}\pi\middle\|\tau\right)% \right)},$
ⓘ Defines: $B$ (locally) Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\equiv$ : equals by definition and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E20 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

20.7.21	$\displaystyle\theta_{1}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{1}\left(z\middle\|\tau\right)\theta_{1}\left(\tfrac{1}{4% }\pi-z\middle\|\tau\right)\*\theta_{1}\left(\tfrac{1}{4}\pi+z\middle\|\tau\right% )\theta_{2}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E21 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.22	$\displaystyle\theta_{2}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{2}\left(\tfrac{1}{8}\pi-z\middle\|\tau\right)\theta_{2}% \left(\tfrac{1}{8}\pi+z\middle\|\tau\right)\*\theta_{2}\left(\tfrac{3}{8}\pi-z% \middle\|\tau\right)\theta_{2}\left(\tfrac{3}{8}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E22 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.23	$\displaystyle\theta_{3}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{3}\left(\tfrac{1}{8}\pi-z\middle\|\tau\right)\theta_{3}% \left(\tfrac{1}{8}\pi+z\middle\|\tau\right)\*\theta_{3}\left(\tfrac{3}{8}\pi-z% \middle\|\tau\right)\theta_{3}\left(\tfrac{3}{8}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E23 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.24	$\displaystyle\theta_{4}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{4}\left(z\middle\|\tau\right)\theta_{4}\left(\tfrac{1}{4% }\pi-z\middle\|\tau\right)\*\theta_{4}\left(\tfrac{1}{4}\pi+z\middle\|\tau\right% )\theta_{3}\left(z\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E24 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

§20.7(vii) Derivatives of Ratios of Theta Functions

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Keywords:: derivatives, of ratios, theta functions
Referenced by:: §20.7(ii), Erratum (V1.0.3) for References, Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/20.7.vii
Addition (effective with 1.0.5):: The reference to Carlson (2011) was added at the end of this subsection.
Clarification (effective with 1.0.3):: The last sentence of this subsection was revised to state more precisely the content of Lawden (1989, pp. 19–20).
See also:: Annotations for §20.7 and Ch.20

20.7.25		$\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\theta_{2}\left(z\middle\|\tau\right)% }{\theta_{4}\left(z\middle\|\tau\right)}\right)=-\frac{{\theta_{3}}^{2}\left(0% \middle\|\tau\right)\theta_{1}\left(z\middle\|\tau\right)\theta_{3}\left(z% \middle\|\tau\right)}{{\theta_{4}}^{2}\left(z\middle\|\tau\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E25 Encodings: TeX, pMML, png See also: Annotations for §20.7(vii), §20.7 and Ch.20

See Lawden (1989, pp. 19–20). This reference also gives the eleven additional identities for the permutations of the four theta functions.

§20.7(viii) Transformations of Lattice Parameter

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Keywords:: lattice parameter, modular transformations, theta functions, transformation of
Notes:: See Lawden (1989, pp. 16–17), Bellman (1961, p. 61), Serre (1973, p. 109), and Whittaker and Watson (1927, pp. 474–475).
Referenced by:: §20.10(i), §20.14, §20.7(viii), §21.5(ii), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/20.7.viii
Change (effective with 1.0.11):: The sentence,

These are examples of modular transformations; see §23.15.

at the end of 20.7(viii) now contains an additional reference to Rademacher (1973, pp. 181–183).
See also:: Annotations for §20.7 and Ch.20

20.7.26	$\displaystyle\theta_{1}\left(z\middle\|\tau+1\right)$	$\displaystyle=e^{i\pi/4}\theta_{1}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E26 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.27	$\displaystyle\theta_{2}\left(z\middle\|\tau+1\right)$	$\displaystyle=e^{i\pi/4}\theta_{2}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E27 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.28	$\displaystyle\theta_{3}\left(z\middle\|\tau+1\right)$	$\displaystyle=\theta_{4}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E28 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.29	$\displaystyle\theta_{4}\left(z\middle\|\tau+1\right)$	$\displaystyle=\theta_{3}\left(z\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E29 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20

In the following equations $\tau^{\prime}=-1/\tau$ , and all square roots assume their principal values.

20.7.30	$\displaystyle(-i\tau)^{1/2}\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=-i\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{1}\left(z\tau^% {\prime}\middle\|\tau^{\prime}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E30 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.31	$\displaystyle(-i\tau)^{1/2}\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{4}\left(z\tau^{% \prime}\middle\|\tau^{\prime}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Referenced by: (20.10.1) Permalink: http://dlmf.nist.gov/20.7.E31 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.32	$\displaystyle(-i\tau)^{1/2}\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{3}\left(z\tau^{% \prime}\middle\|\tau^{\prime}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Referenced by: (20.10.2), §20.13, §20.14 Permalink: http://dlmf.nist.gov/20.7.E32 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.33	$\displaystyle(-i\tau)^{1/2}\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{2}\left(z\tau^{% \prime}\middle\|\tau^{\prime}\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Referenced by: (20.10.3), §20.13 Permalink: http://dlmf.nist.gov/20.7.E33 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20

These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183).

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

ⓘ

Keywords:: nome, theta functions, transformation of
Notes:: See Walker (2012). (The material in this subsection was added in Version 1.0.5. It will be incorporated into the next printed edition.)
Permalink:: http://dlmf.nist.gov/20.7.ix
See also:: Annotations for §20.7 and Ch.20

20.7.34		$\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $q$ : nome Referenced by: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36 Permalink: http://dlmf.nist.gov/20.7.E34 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation has been added. See Walker (2012). See also: Annotations for §20.7(ix), §20.7 and Ch.20

20.7.16	$\displaystyle\theta_{1}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{1}\left(z\middle\|\tau\right)\theta_{2}\left(z\middle\|% \tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E16 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.17	$\displaystyle\theta_{2}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{1}\left(\tfrac{1}{4}\pi-z\middle\|\tau\right)\theta_{1}% \left(\tfrac{1}{4}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E17 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.18	$\displaystyle\theta_{3}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{3}\left(\tfrac{1}{4}\pi-z\middle\|\tau\right)\theta_{3}% \left(\tfrac{1}{4}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E18 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.19	$\displaystyle\theta_{4}\left(2z\middle\|2\tau\right)$	$\displaystyle=A\theta_{3}\left(z\middle\|\tau\right)\theta_{4}\left(z\middle\|% \tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E19 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

20.7.21	$\displaystyle\theta_{1}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{1}\left(z\middle\|\tau\right)\theta_{1}\left(\tfrac{1}{4% }\pi-z\middle\|\tau\right)\*\theta_{1}\left(\tfrac{1}{4}\pi+z\middle\|\tau\right% )\theta_{2}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E21 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.22	$\displaystyle\theta_{2}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{2}\left(\tfrac{1}{8}\pi-z\middle\|\tau\right)\theta_{2}% \left(\tfrac{1}{8}\pi+z\middle\|\tau\right)\*\theta_{2}\left(\tfrac{3}{8}\pi-z% \middle\|\tau\right)\theta_{2}\left(\tfrac{3}{8}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E22 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.23	$\displaystyle\theta_{3}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{3}\left(\tfrac{1}{8}\pi-z\middle\|\tau\right)\theta_{3}% \left(\tfrac{1}{8}\pi+z\middle\|\tau\right)\*\theta_{3}\left(\tfrac{3}{8}\pi-z% \middle\|\tau\right)\theta_{3}\left(\tfrac{3}{8}\pi+z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E23 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
20.7.24	$\displaystyle\theta_{4}\left(4z\middle\|4\tau\right)$	$\displaystyle=B\theta_{4}\left(z\middle\|\tau\right)\theta_{4}\left(\tfrac{1}{4% }\pi-z\middle\|\tau\right)\*\theta_{4}\left(\tfrac{1}{4}\pi+z\middle\|\tau\right% )\theta_{3}\left(z\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$ Permalink: http://dlmf.nist.gov/20.7.E24 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

20.7.26	$\displaystyle\theta_{1}\left(z\middle\|\tau+1\right)$	$\displaystyle=e^{i\pi/4}\theta_{1}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E26 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.27	$\displaystyle\theta_{2}\left(z\middle\|\tau+1\right)$	$\displaystyle=e^{i\pi/4}\theta_{2}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E27 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.28	$\displaystyle\theta_{3}\left(z\middle\|\tau+1\right)$	$\displaystyle=\theta_{4}\left(z\middle\|\tau\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E28 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.29	$\displaystyle\theta_{4}\left(z\middle\|\tau+1\right)$	$\displaystyle=\theta_{3}\left(z\middle\|\tau\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E29 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20

20.7.30	$\displaystyle(-i\tau)^{1/2}\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=-i\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{1}\left(z\tau^% {\prime}\middle\|\tau^{\prime}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.7.E30 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.31	$\displaystyle(-i\tau)^{1/2}\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{4}\left(z\tau^{% \prime}\middle\|\tau^{\prime}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Referenced by: (20.10.1) Permalink: http://dlmf.nist.gov/20.7.E31 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.32	$\displaystyle(-i\tau)^{1/2}\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{3}\left(z\tau^{% \prime}\middle\|\tau^{\prime}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Referenced by: (20.10.2), §20.13, §20.14 Permalink: http://dlmf.nist.gov/20.7.E32 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20
20.7.33	$\displaystyle(-i\tau)^{1/2}\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{2}\left(z\tau^{% \prime}\middle\|\tau^{\prime}\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $\tau$ : lattice parameter Referenced by: (20.10.3), §20.13 Permalink: http://dlmf.nist.gov/20.7.E33 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20