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Jacobi theta functions (notational variations)
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<p>There are a number of notational systems for the <a href="/facts/Theta_function/g09xvoQQ">Jacobi theta functions</a>. The notations given in the Wikipedia article define the original function </p> ϑ 00 ( z ; τ ) = ∑ n = − ∞ ∞ exp ( π i n 2 τ + 2 π i n z ) {\displaystyle \vartheta _{00}(z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz)} <p>which is equivalent to </p> ϑ 00 ( w , q ) = ∑ n = − ∞ ∞ q n 2 w 2 n {\displaystyle \vartheta _{00}(w,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}w^{2n}} <p>where q = e π i τ {\displaystyle q=e^{\pi i\tau }} and w = e π i z {\displaystyle w=e^{\pi iz}} . </p><p>However, a similar notation is defined somewhat differently in <a href="/facts/Whittaker_and_Watson/RI1vXWBT">Whittaker and Watson</a>, p. 487: </p> ϑ 0 , 0 ( x ) = ∑ n = − ∞ ∞ q n 2 exp ( 2 π i n x / a ) {\displaystyle \vartheta _{0,0}(x)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2\pi inx/a)} <p>This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define </p> ϑ 1 , 1 ( x ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 exp ( π i ( 2 n + 1 ) x / a ) {\displaystyle \vartheta _{1,1}(x)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}\exp(\pi i(2n+1)x/a)} <p>This is a factor of <i>i</i> off from the definition of ϑ 11 {\displaystyle \vartheta _{11}} as defined in the Wikipedia article. These definitions can be made at least proportional by <i>x</i> = <i>za</i>, but other definitions cannot. Whittaker and Watson, Abramowitz and Stegun, and Gradshteyn and Ryzhik all follow Tannery and Molk, in which </p> ϑ 1 ( z ) = − i ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) {\displaystyle \vartheta _{1}(z)=-i\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}\exp((2n+1)iz)} ϑ 2 ( z ) = ∑ n = − ∞ ∞ q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) {\displaystyle \vartheta _{2}(z)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\exp((2n+1)iz)} ϑ 3 ( z ) = ∑ n = − ∞ ∞ q n 2 exp ( 2 n i z ) {\displaystyle \vartheta _{3}(z)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2niz)} ϑ 4 ( z ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2 exp ( 2 n i z ) {\displaystyle \vartheta _{4}(z)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\exp(2niz)} <p>Note that there is no factor of π in the argument as in the previous definitions. </p><p>Whittaker and Watson refer to still other definitions of ϑ j {\displaystyle \vartheta _{j}} . 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 ) {\displaystyle \vartheta (z)} should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of ϑ ( z ) {\displaystyle \vartheta (z)} is intended. </p> <ul><li><a href="/facts/Milton_Abramowitz/6yRyez7P">Abramowitz, Milton</a>; <a href="/facts/Irene_Stegun/FjSS1LDR">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. "Chapter 16.27ff.". <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61272-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/65012253">65-12253</a>.</li> <li><a href="/facts/Izrail_Solomonovich_Gradshteyn/SKShStsu">Gradshteyn, Izrail Solomonovich</a>; <a href="/facts/Iosif_Moiseevich_Ryzhik/SKShStsu">Ryzhik, Iosif Moiseevich</a>; <a href="/facts/Yuri_Veniaminovich_Geronimus/SKShStsu">Geronimus, Yuri Veniaminovich</a>; <a href="/facts/Michail_Yulyevich_Tseytlin/SKShStsu">Tseytlin, Michail Yulyevich</a> (1980). "8.18.". In Jeffrey, Alan (ed.). <a href="/facts/Gradshteyn_and_Ryzhik/SKShStsu"><i>Table of Integrals, Series, and Products</i></a>. Translated by Scripta Technica, Inc. (4th corrected and enlarged ed.). <a href="/facts/Academic_Press,_Inc./qxoKjzFy">Academic Press, Inc.</a> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-294760-6. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/79027143">79027143</a>.</li> <li><a href="/facts/E._T._Whittaker/b7wl3DMJ">E. T. Whittaker</a> and <a href="/facts/G._N._Watson/VLfFjgjh">G. N. Watson</a>, <i><a href="/facts/A_Course_in_Modern_Analysis/RI1vXWBT">A Course in Modern Analysis</a></i>, fourth edition, Cambridge University Press, 1927. <i>(See chapter XXI for the history of Jacobi's θ functions)</i></li></ul>