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20 Theta FunctionsProperties

§20.2 Definitions and Periodic Properties

Contents

§20.2(i) Fourier Series

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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

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

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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

20.2.5z_{{m,n}}=(m+n\tau)\pi,m,n\in\Integer,
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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),
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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),
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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),
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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
See accompanying textSee accompanying text
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). Magnify
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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

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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

20.2.10M\equiv M(z|\tau)=e^{{iz+(i\pi\tau/4)}},
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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),
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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),
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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),
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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).
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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:
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§20.2(iv) z-Zeros

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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.

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