§20.2 Definitions and Periodic Properties

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

§20.2(i) Fourier Series
§20.2(ii) Periodicity and Quasi-Periodicity
§20.2(iii) Translation of the Argument by Half-Periods
§20.2(iv) $z$ -Zeros

§20.2(i) Fourier Series

Notes:: See Whittaker and Watson (1927, p. 464) or Lawden (1989, p. 5).
Defines:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function
Keywords:: Fourier series, Jacobi’s theta functions, classical theta functions, theta functions
Referenced by:: §20.14, §21.2(iii), ¶ ‣ §22.16(ii), §22.2, ¶ ‣ §23.15(ii), §23.17(ii), §25.5(ii), §27.13(iv)
Permalink:: http://dlmf.nist.gov/20.2.i

20.2.1 $\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{1}}\/}\nolimits\!\left(z,q\right)=2\sum\limits _{{n=0}}^{{\infty}}(-1)^{n}q^{{(n+\frac{1}{2})^{2}}}\mathop{\sin\/}\nolimits\!\left((2n+1)z\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.27.1
Referenced by:: §20.14, §20.2(i)
Permalink:: http://dlmf.nist.gov/20.2.E1
Encodings:: TeX, pMML, png

20.2.2 $\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q\right)=2\sum\limits _{{n=0}}^{{\infty}}q^{{(n+\frac{1}{2})^{2}}}\mathop{\cos\/}\nolimits\!\left((2n+1)z\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.27.2
Permalink:: http://dlmf.nist.gov/20.2.E2
Encodings:: TeX, pMML, png

20.2.3 $\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q\right)=1+2\sum\limits _{{n=1}}^{{\infty}}q^{{n^{2}}}\mathop{\cos\/}\nolimits\!\left(2nz\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.27.3
Referenced by:: §20.10(i), §20.13, §20.14, §27.13(iv)
Permalink:: http://dlmf.nist.gov/20.2.E3
Encodings:: TeX, pMML, png

20.2.4 $\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{4}}\/}\nolimits\!\left(z,q\right)=1+2\sum\limits _{{n=1}}^{{\infty}}(-1)^{n}q^{{n^{2}}}\mathop{\cos\/}\nolimits\!\left(2nz\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.27.4
Referenced by:: §20.13, §20.2(i)
Permalink:: http://dlmf.nist.gov/20.2.E4
Encodings:: TeX, pMML, png

Corresponding expansions for ${\mathop{\theta _{{j}}\/}\nolimits^{{\prime}}}\!\left(z\middle|\tau\right)$ , $j=1,2,3,4$ , can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$ .

§20.2(ii) Periodicity and Quasi-Periodicity

Notes:: See Whittaker and Watson (1927, p. 463) and Lawden (1989, pp. 5, 6).
Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.2.ii

For fixed $\tau$ , each $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ is an entire function of $z$ with period $2\pi$ ; $\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)$ is odd in $z$ and the others are even. For fixed $z$ , each of $\ifrac{\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)}{\mathop{\sin\/}\nolimits z}$ , $\ifrac{\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right)}{\mathop{\cos\/}\nolimits z}$ , $\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)$ , and $\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\imagpart{\tau}>0$ , with a natural boundary $\imagpart{\tau}=0$ , and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ with a natural boundary $\left|q\right|=1$ .

The four points $(0,\pi,\pi+\tau\pi,\tau\pi)$ are the vertices of the fundamental parallelogram in the $z$ -plane; see Figure 20.2.1. The points

20.2.5 $z_{{m,n}}=(m+n\tau)\pi,$ $m,n\in\Integer$ ,

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter
A&S Ref:: 16.33.1 in different notation.
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.2.E5
Encodings:: TeX, pMML, png

are the lattice points. The theta functions are quasi-periodic on the lattice:

20.2.6 $\mathop{\theta _{{1}}\/}\nolimits\!\left(z+(m+n\tau)\pi\middle|\tau\right)=(-1)^{{m+n}}q^{{-n^{2}}}e^{{-2inz}}\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.33.2 in different notation.
Referenced by:: §23.6(i)
Permalink:: http://dlmf.nist.gov/20.2.E6
Encodings:: TeX, pMML, png

20.2.7 $\mathop{\theta _{{2}}\/}\nolimits\!\left(z+(m+n\tau)\pi\middle|\tau\right)=(-1)^{m}q^{{-n^{2}}}e^{{-2inz}}\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.33.3 in different notation.
Permalink:: http://dlmf.nist.gov/20.2.E7
Encodings:: TeX, pMML, png

20.2.8 $\mathop{\theta _{{3}}\/}\nolimits\!\left(z+(m+n\tau)\pi\middle|\tau\right)=q^{{-n^{2}}}e^{{-2inz}}\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.33.4 in different notation.
Permalink:: http://dlmf.nist.gov/20.2.E8
Encodings:: TeX, pMML, png

20.2.9 $\mathop{\theta _{{4}}\/}\nolimits\!\left(z+(m+n\tau)\pi\middle|\tau\right)=(-1)^{n}q^{{-n^{2}}}e^{{-2inz}}\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.2.E9
Encodings:: TeX, pMML, png

Figure 20.2.1: $z$ -plane. Fundamental parallelogram. Left-hand diagram is the rectangular case ( $\tau$ purely imaginary); right-hand diagram is the general case. $\bullet$ zeros of $\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)$ , $\blacksquare$ zeros of $\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right)$ , $\blacktriangle$ zeros of $\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)$ , $\blacklozenge$ zeros of $\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)$ .

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter
Referenced by:: §20.2(ii)
Permalink:: http://dlmf.nist.gov/20.2.F1
Encodings:: pdf, pdf, png, png

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

Notes:: See Whittaker and Watson (1927, p. 464) and Lawden (1989, pp. 5, 6).
Keywords:: theta functions
Referenced by:: §21.3(ii)
Permalink:: http://dlmf.nist.gov/20.2.iii

With

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

Symbols:: $e$ : base of exponential function, $z$ : complex, $\tau$ : lattice parameter and $M$
A&S Ref:: 16.33.1 in different notation.
Permalink:: http://dlmf.nist.gov/20.2.E10
Encodings:: TeX, pMML, png

20.2.11 $\mathop{\theta _{{1}}\/}\nolimits\!\left(z\middle|\tau\right)=-\mathop{\theta _{{2}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=-iM\mathop{\theta _{{4}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=-iM\mathop{\theta _{{3}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : 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:: TeX, pMML, png

20.2.12 $\mathop{\theta _{{2}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{1}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=M\mathop{\theta _{{3}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\mathop{\theta _{{4}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $M$
A&S Ref:: 16.33.3 in different notation.
Permalink:: http://dlmf.nist.gov/20.2.E12
Encodings:: TeX, pMML, png

20.2.13 $\mathop{\theta _{{3}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{4}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=M\mathop{\theta _{{2}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\mathop{\theta _{{1}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $M$
A&S Ref:: 16.33.4 in different notation.
Permalink:: http://dlmf.nist.gov/20.2.E13
Encodings:: TeX, pMML, png

20.2.14 $\mathop{\theta _{{4}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=-iM\mathop{\theta _{{1}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=iM\mathop{\theta _{{2}}\/}\nolimits\!\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $M$
Referenced by:: §20.7(iv)
Permalink:: http://dlmf.nist.gov/20.2.E14
Encodings:: TeX, pMML, png

§20.2(iv) $z$ -Zeros

Notes:: See Whittaker and Watson (1927, p. 465) and Lawden (1989, p. 6).
Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.2.iv

For $m,n\in\Integer$ , the $z$ -zeros of $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ , $j=1,2,3,4$ , are $(m+n\tau)\pi$ , $(m+\tfrac{1}{2}+n\tau)\pi$ , $(m+\tfrac{1}{2}+(n+\tfrac{1}{2})\tau)\pi$ , $(m+(n+\tfrac{1}{2})\tau)\pi$ respectively.

§20.2 Definitions and Periodic Properties

Contents

§20.2(i) Fourier Series

§20.2(ii) Periodicity and Quasi-Periodicity

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

§20.2(iv) -Zeros

§20.2(iv) $z$ -Zeros