§20.7 Identities

Permalink:: http://dlmf.nist.gov/20.7

§20.7(i) Sums of Squares
§20.7(ii) Addition Formulas
§20.7(iii) Duplication Formula
§20.7(iv) Reduction Formulas for Products
§20.7(v) Watson’s Identities
§20.7(vi) Landen Transformations
§20.7(vii) Derivatives of Ratios of Theta Functions
§20.7(viii) Transformations of Lattice Parameter
§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

§20.7(i) Sums of Squares

Notes:: See Lawden (1989, p. 11).
Keywords:: theta functions
Referenced by:: Version 1.0.5 (October 1, 2012)
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.
Reported 2012-04-02

20.7.1 ${\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right)={\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(z,q\right)+{\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

20.7.2 ${\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(z,q\right)={\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{1}}\/}\nolimits^{{2}}}\!\left(z,q\right)+{\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

20.7.3 ${\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(z,q\right)={\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{1}}\/}\nolimits^{{2}}}\!\left(z,q\right)+{\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

20.7.4 ${\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right)={\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{1}}\/}\nolimits^{{2}}}\!\left(z,q\right)+{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

Also

20.7.5 ${\mathop{\theta _{{3}}\/}\nolimits^{{4}}}\!\left(0,q\right)={\mathop{\theta _{{2}}\/}\nolimits^{{4}}}\!\left(0,q\right)+{\mathop{\theta _{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

§20.7(ii) Addition Formulas

Keywords:: theta functions
Referenced by:: §21.6(ii), Version 1.0.5 (October 1, 2012)
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.
Reported 2012-04-02

20.7.6 ${\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(w+z,q\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(w-z,q\right)={\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right)-{\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E6
Encodings:: TeX, pMML, png

20.7.7 ${\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(w+z,q\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(w-z,q\right)={\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right)-{\mathop{\theta _{{1}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E7
Encodings:: TeX, pMML, png

20.7.8 ${\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(w+z,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(w-z,q\right)={\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right)-{\mathop{\theta _{{1}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E8
Encodings:: TeX, pMML, png

20.7.9 ${\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(w+z,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(w-z,q\right)={\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right)-{\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(w,q\right){\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(z,q\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E9
Encodings:: TeX, pMML, png

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

Notes:: See McKean and Moll (1999, p. 129).
Keywords:: theta functions
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.iii

20.7.10 $\mathop{\theta _{{1}}\/}\nolimits\!\left(2z,q\right)=2\frac{\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z,q\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E10
Encodings:: TeX, pMML, png

§20.7(iv) Reduction Formulas for Products

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)).
Keywords:: nome, theta functions
Referenced by:: §20.7(ix), Version 1.0.5 (October 1, 2012)
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.
Reported 2012-04-02

20.7.11 $\frac{\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q\right)}{\mathop{\theta _{{1}}\/}\nolimits\!\left(2z,q^{2}\right)}=\frac{\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z,q\right)}{\mathop{\theta _{{4}}\/}\nolimits\!\left(2z,q^{2}\right)}=\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q^{2}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

20.7.12 $\frac{\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q^{2}\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z,q^{2}\right)}{\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right)}=\frac{\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q^{2}\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q^{2}\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q\right)}=\tfrac{1}{2}\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

Addendum: For a companion equation see (20.7.34).

§20.7(v) Watson’s Identities

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

20.7.13 $\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(w,q\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(z+w,q^{2}\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z-w,q^{2}\right)-\mathop{\theta _{{2}}\/}\nolimits\!\left(z+w,q^{2}\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z-w,q^{2}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E13
Encodings:: TeX, pMML, png

20.7.14 $\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(w,q\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(z+w,q^{2}\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z-w,q^{2}\right)+\mathop{\theta _{{2}}\/}\nolimits\!\left(z+w,q^{2}\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z-w,q^{2}\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E14
Encodings:: TeX, pMML, png

§20.7(vi) Landen Transformations

Notes:: See Lawden (1989, pp. 17–18).
Keywords:: Jacobian elliptic functions, Landen transformations, theta functions
Permalink:: http://dlmf.nist.gov/20.7.vi

With

20.7.15 $A\equiv A(\tau)=\ifrac{1}{\mathop{\theta _{{4}}\/}\nolimits\!\left(0\middle|2\tau\right)},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E15
Encodings:: TeX, pMML, png

20.7.16 $\mathop{\theta _{{1}}\/}\nolimits\!\left(2z\middle|2\tau\right)=A\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E16
Encodings:: TeX, pMML, png

20.7.17 $\mathop{\theta _{{2}}\/}\nolimits\!\left(2z\middle|2\tau\right)=A\mathop{\theta _{{1}}\/}\nolimits\!\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(\tfrac{1}{4}\pi+z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E17
Encodings:: TeX, pMML, png

20.7.18 $\mathop{\theta _{{3}}\/}\nolimits\!\left(2z\middle|2\tau\right)=A\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{1}{4}\pi+z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E18
Encodings:: TeX, pMML, png

20.7.19 $\mathop{\theta _{{4}}\/}\nolimits\!\left(2z\middle|2\tau\right)=A\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E19
Encodings:: TeX, pMML, png

Next, with

20.7.20 $B\equiv B(\tau)=\ifrac{1}{\left(\mathop{\theta _{{3}}\/}\nolimits\!\left(0\middle|\tau\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0\middle|\tau\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{1}{4}\pi\middle|\tau\right)\right)},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\tau$ : lattice parameter and $B$
Permalink:: http://dlmf.nist.gov/20.7.E20
Encodings:: TeX, pMML, png

20.7.21 $\mathop{\theta _{{1}}\/}\nolimits\!\left(4z\middle|4\tau\right)=B\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\*\mathop{\theta _{{1}}\/}\nolimits\!\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $B$
Permalink:: http://dlmf.nist.gov/20.7.E21
Encodings:: TeX, pMML, png

20.7.22 $\mathop{\theta _{{2}}\/}\nolimits\!\left(4z\middle|4\tau\right)=B\mathop{\theta _{{2}}\/}\nolimits\!\left(\tfrac{1}{8}\pi-z\middle|\tau\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(\tfrac{1}{8}\pi+z\middle|\tau\right)\*\mathop{\theta _{{2}}\/}\nolimits\!\left(\tfrac{3}{8}\pi-z\middle|\tau\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(\tfrac{3}{8}\pi+z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $B$
Permalink:: http://dlmf.nist.gov/20.7.E22
Encodings:: TeX, pMML, png

20.7.23 $\mathop{\theta _{{3}}\/}\nolimits\!\left(4z\middle|4\tau\right)=B\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{1}{8}\pi-z\middle|\tau\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{1}{8}\pi+z\middle|\tau\right)\*\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{3}{8}\pi-z\middle|\tau\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(\tfrac{3}{8}\pi+z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $B$
Permalink:: http://dlmf.nist.gov/20.7.E23
Encodings:: TeX, pMML, png

20.7.24 $\mathop{\theta _{{4}}\/}\nolimits\!\left(4z\middle|4\tau\right)=B\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\*\mathop{\theta _{{4}}\/}\nolimits\!\left(\tfrac{1}{4}\pi+z\middle|\tau\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $B$
Permalink:: http://dlmf.nist.gov/20.7.E24
Encodings:: TeX, pMML, png

§20.7(vii) Derivatives of Ratios of Theta Functions

Keywords:: derivatives, theta functions
Referenced by:: §20.7(ii), Version 1.0.3 (Aug 29, 2011)
Permalink:: http://dlmf.nist.gov/20.7.vii
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).
Reported 2011-08-21

20.7.25 $\frac{d}{dz}\left(\frac{\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right)}{\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)}\right)=-\frac{{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0\middle|\tau\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)}{{\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(z\middle|\tau\right)}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\frac{df}{dx}$ : 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 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

Notes:: See Lawden (1989, pp. 16–17), Bellman (1961, p. 61), Serre (1973, p. 109), and Whittaker and Watson (1927, pp. 474–475).
Keywords:: lattice parameter, theta functions
Referenced by:: §20.10(i), §20.14, §21.5(ii)
Permalink:: http://dlmf.nist.gov/20.7.viii

20.7.26 $\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau+1\right)=e^{{i\pi/4}}\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E26
Encodings:: TeX, pMML, png

20.7.27 $\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau+1\right)=e^{{i\pi/4}}\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E27
Encodings:: TeX, pMML, png

20.7.28 $\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau+1\right)=\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E28
Encodings:: TeX, pMML, png

20.7.29 $\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau+1\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E29
Encodings:: TeX, pMML, png

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

20.7.30 $(-i\tau)^{{1/2}}\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)=-i\mathop{\exp\/}\nolimits\!\left(i\tau^{{\prime}}z^{2}/\pi\right)\mathop{\theta _{{1}}\/}\nolimits\!\left(z\tau^{{\prime}}\middle|\tau^{{\prime}}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E30
Encodings:: TeX, pMML, png

20.7.31 $(-i\tau)^{{1/2}}\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\exp\/}\nolimits\!\left(i\tau^{{\prime}}z^{2}/\pi\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z\tau^{{\prime}}\middle|\tau^{{\prime}}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E31
Encodings:: TeX, pMML, png

20.7.32 $(-i\tau)^{{1/2}}\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\exp\/}\nolimits\!\left(i\tau^{{\prime}}z^{2}/\pi\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z\tau^{{\prime}}\middle|\tau^{{\prime}}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex and $\tau$ : lattice parameter
Referenced by:: §20.13, §20.14
Permalink:: http://dlmf.nist.gov/20.7.E32
Encodings:: TeX, pMML, png

20.7.33 $(-i\tau)^{{1/2}}\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\exp\/}\nolimits\!\left(i\tau^{{\prime}}z^{2}/\pi\right)\mathop{\theta _{{2}}\/}\nolimits\!\left(z\tau^{{\prime}}\middle|\tau^{{\prime}}\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex and $\tau$ : lattice parameter
Referenced by:: §20.13
Permalink:: http://dlmf.nist.gov/20.7.E33
Encodings:: TeX, pMML, png

These are examples of modular transformations; see §23.15.

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

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.)
Keywords:: nome, theta functions
Permalink:: http://dlmf.nist.gov/20.7.ix

20.7.34 $\frac{\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q^{2}\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q^{2}\right)}{\mathop{\theta _{{1}}\/}\nolimits\!\left(z,iq\right)}=\frac{\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q^{2}\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z,q^{2}\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(z,iq\right)}=i^{{-1/4}}\sqrt{\frac{\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q^{2}\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q^{2}\right)}{2}}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(iv), Version 1.0.5 (October 1, 2012)
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).
Reported 2012-04-02