relations to theta functions

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1: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

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§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. ►The relations (20.9.1) and (20.9.2) between

k

and

\tau

(or

q

) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). … ►

§20.9(iii) Riemann Zeta Function

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2: 23.15 Definitions

§23.15 Definitions

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3: 22.2 Definitions

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22.2.9

\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.

ⓘ

Defines:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

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4: 20.3 Graphics

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See accompanying text — Figure 20.3.2: $\theta_{1}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0. … Magnify

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5: 21.2 Definitions

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§21.2(iii) Relation to Classical Theta Functions

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6: 19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …