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There are a number of notational systems for the [[Theta function|Jacobi theta functions]]. The notations given in the Wikipedia article define the original function
:<math>
\vartheta_{00}(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)
</math>
which is equivalent to
:<math>
\vartheta_{00}(z, q) = \sum_{n=-\infty}^\infty q^{n^2} \exp (2 \pi i n z)
</math>
However, a similar notation is defined somewhat differently in Whittaker and Watson, p.&nbsp;487:
:<math>
\vartheta_{0,0}(x) = \sum_{n=-\infty}^\infty q^{n^2} \exp (2 \pi i n x/a)
</math>
This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define
:<math>
\vartheta_{1,1}(x) = \sum_{n=-\infty}^\infty (-1)^n q^{(n+1/2)^2} \exp (\pi i (2 n + 1) x/a)
</math>
This is a factor of ''i'' off from the definition of <math>\vartheta_{11}</math> as defined in the Wikipedia article. These definitions can be made at least proportional by ''x'' = ''za'', but other definitions cannot. Whittaker and Watson, Abramowitz and Stegun, and Gradshteyn and Ryzhik all follow Tannery and Molk, in which
:<math>
\vartheta_1(z) = -i \sum_{n=-\infty}^\infty (-1)^n q^{(n+1/2)^2} \exp ((2 n + 1) i z)</math>
:<math>
\vartheta_2(z) = \sum_{n=-\infty}^\infty q^{(n+1/2)^2} \exp ((2 n + 1) i z)</math>
:<math>
\vartheta_3(z) = \sum_{n=-\infty}^\infty q^{n^2} \exp (2 n i z)</math>
:<math>
\vartheta_4(z) = \sum_{n=-\infty}^\infty (-1)^n q^{n^2} \exp (2 n i z)</math>


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Note that there is no factor of π in the argument as in the previous definitions.  


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Whittaker and Watson refer to still other definitions of <math>\vartheta_j</math>. The warning in Abramowitz and Stegun, "There is a bewildering variety of notations...in consulting books caution should be exercised," may be viewed as an understatement. In any expression, an occurrence of <math>\vartheta(z)</math> should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of <math>\vartheta(z)</math> is intended.


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==References==
* Milton Abramowitz and Irene A. Stegun, ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', (1964) Dover Publications, New York. ISBN 0-486-61272-4.  ''(See section 16.27ff.)''
* I. S. Gradshteyn and I. M. Ryzhik, ''Table of Integrals, Functions, and Products'', (1980) Academic Press, London. ISBN 0-12-294760-6. ''(See section 8.18)''
* [[E. T. Whittaker]] and [[G. N. Watson]], ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1927. ''(See chapter XXI for the history of Jacobi's &theta; functions)''
 
[[Category:Theta functions| ]]
[[Category:Elliptic functions]]

Latest revision as of 06:58, 3 December 2013

There are a number of notational systems for the Jacobi theta functions. The notations given in the Wikipedia article define the original function

ϑ00(z;τ)=n=exp(πin2τ+2πinz)

which is equivalent to

ϑ00(z,q)=n=qn2exp(2πinz)

However, a similar notation is defined somewhat differently in Whittaker and Watson, p. 487:

ϑ0,0(x)=n=qn2exp(2πinx/a)

This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define

ϑ1,1(x)=n=(1)nq(n+1/2)2exp(πi(2n+1)x/a)

This is a factor of i off from the definition of ϑ11 as defined in the Wikipedia article. These definitions can be made at least proportional by x = za, but other definitions cannot. Whittaker and Watson, Abramowitz and Stegun, and Gradshteyn and Ryzhik all follow Tannery and Molk, in which

ϑ1(z)=in=(1)nq(n+1/2)2exp((2n+1)iz)
ϑ2(z)=n=q(n+1/2)2exp((2n+1)iz)
ϑ3(z)=n=qn2exp(2niz)
ϑ4(z)=n=(1)nqn2exp(2niz)

Note that there is no factor of π in the argument as in the previous definitions.

Whittaker and Watson refer to still other definitions of ϑj. The warning in Abramowitz and Stegun, "There is a bewildering variety of notations...in consulting books caution should be exercised," may be viewed as an understatement. In any expression, an occurrence of ϑ(z) should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of ϑ(z) is intended.

References

  • Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. (See section 16.27ff.)
  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Functions, and Products, (1980) Academic Press, London. ISBN 0-12-294760-6. (See section 8.18)
  • E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927. (See chapter XXI for the history of Jacobi's θ functions)