# modular functions

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##### 2: 23.16 Graphics
###### §23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions $\lambda$, $J$, and $\eta$. …
##### 3: 23.19 Interrelations
###### §23.19 Interrelations
23.19.1 $\lambda\left(\tau\right)=16\left(\frac{{\eta}^{2}\left(2\tau\right)\eta\left(% \tfrac{1}{2}\tau\right)}{{\eta}^{3}\left(\tau\right)}\right)^{8},$
23.19.2 $J\left(\tau\right)=\frac{4}{27}\frac{\left(1-\lambda\left(\tau\right)+{\lambda% }^{2}\left(\tau\right)\right)^{3}}{\left(\lambda\left(\tau\right)\left(1-% \lambda\left(\tau\right)\right)\right)^{2}},$
##### 4: 23.1 Special Notation
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 6: 23.21 Physical Applications
###### §23.21(iv) ModularFunctions
Physical applications of modular functions include: …
• String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

##### 8: 23.18 Modular Transformations
###### Elliptic ModularFunction
23.18.5 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
##### 10: 23.20 Mathematical Applications
For conformal mappings via modular functions see Apostol (1990, §2.7). …