# modular functions

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## 11—20 of 43 matching pages

##### 11: 20.9 Relations to Other Functions
###### §20.9(ii) Elliptic Functions and ModularFunctions
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). As a function of $\tau$, $k^{2}$ is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6). …
##### 12: Ranjan Roy
He also authored another two advanced mathematics books: Sources in the development of mathematics (Roy, 2011), Elliptic and modular functions from Gauss to Dedekind to Hecke (Roy, 2017). …
##### 13: 27.14 Unrestricted Partitions
###### §27.14(iv) Relation to ModularFunctions
Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta\left(\tau\right)$, defined by
27.14.12 $\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\prod_{n=1}^{\infty}(1-e^{2\pi% \mathrm{i}n\tau}),$ $\Im\tau>0$.
For further properties of the function $\eta\left(\tau\right)$ see §§23.1523.19. …
27.14.16 $\Delta\left(\tau\right)=(2\pi)^{12}(\eta\left(\tau\right))^{24},$ $\Im\tau>0$,
##### 14: 21.5 Modular Transformations
###### §21.5(i) Riemann Theta Functions
Equation (21.5.4) is the modular transformation property for Riemann theta functions. The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: …
###### §21.5(ii) Riemann Theta Functions with Characteristics
For explicit results in the case $g=1$, see §20.7(viii).
##### 16: 23.2 Definitions and Periodic Properties
23.2.4 ${}\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right),$
23.2.5 ${}\zeta\left(z\right)=\frac{1}{z}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),$
23.2.6 ${}\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-% \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$
##### 17: 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). …
##### 18: 20.3 Graphics Figure 20.3.2: θ 1 ⁡ ( π ⁢ x , q ) , 0 ≤ x ≤ 2 , q = 0. …Here q Dedekind = e - π ⁢ y 0 = 0.19 approximately, where y = y 0 corresponds to the maximum value of Dedekind’s eta function η ⁡ ( i ⁢ y ) as depicted in Figure 23.16.1. Magnify
##### 19: Bibliography R
• R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.
• ##### 20: 20.7 Identities
20.7.33 $(-i\tau)^{1/2}\theta_{4}\left(z\middle|\tau\right)=\exp\left(i\tau^{\prime}z^{% 2}/\pi\right)\theta_{2}\left(z\tau^{\prime}\middle|\tau^{\prime}\right).$