# Dedekind eta function

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##### 1: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions $\lambda$, $J$, and $\eta$. …
##### 2: 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 $\eta\left(\tau\right)$. …
##### 4: 23.15 Definitions
###### Dedekind’s EtaFunction (or Dedekind Modular Function)
23.15.9 $\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/% 3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3\tau\right).$
##### 5: 27.14 Unrestricted Partitions
Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta\left(\tau\right)$, defined by
27.14.12 $\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\prod_{n=1}^{\infty}(1-e^{2\pi% \mathrm{i}n\tau}),$ $\Im\tau>0$.
27.14.14 $\eta\left(\frac{a\tau+b}{c\tau+d}\right)=\varepsilon(-\mathrm{i}(c\tau+d))^{% \frac{1}{2}}\eta\left(\tau\right),$
For further properties of the function $\eta\left(\tau\right)$ see §§23.1523.19. …
27.14.16 $\Delta\left(\tau\right)=(2\pi)^{12}(\eta\left(\tau\right))^{24},$ $\Im\tau>0$,
##### 8: 23.18 Modular Transformations
###### Dedekind’s EtaFunction
23.18.5 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
##### 9: Bibliography M
• I. G. Macdonald (1972) Affine root systems and Dedekind’s $\eta$-function. Invent. Math. 15 (2), pp. 91–143.
• ##### 10: 23.22 Methods of Computation
The modular functions $\lambda\left(\tau\right)$, $J\left(\tau\right)$, and $\eta\left(\tau\right)$ are also obtainable in a similar manner from their definitions in §23.15(ii). …