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11: 20.9 Relations to Other Functions
§20.9(ii) Elliptic Functions and Modular Functions
The relations (20.9.1) and (20.9.2) between k and τ (or q ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). As a function of τ , 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 Modular Functions
Dedekind sums occur in the transformation theory of the Dedekind modular function η ( τ ) , defined by
27.14.12 η ( τ ) = e π i τ / 12 n = 1 ( 1 - e 2 π i n τ ) , τ > 0 .
For further properties of the function η ( τ ) see §§23.1523.19. …
27.14.16 Δ ( τ ) = ( 2 π ) 12 ( η ( τ ) ) 24 , τ > 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 ξ ( Γ ) is determinate: …
§21.5(ii) Riemann Theta Functions with Characteristics
For explicit results in the case g = 1 , see §20.7(viii).
15: William P. Reinhardt
16: 23.2 Definitions and Periodic Properties
23.2.4 ( z ) = 1 z 2 + w 𝕃 { 0 } ( 1 ( z - w ) 2 - 1 w 2 ) ,
23.2.5 ζ ( z ) = 1 z + w 𝕃 { 0 } ( 1 z - w + 1 w + z w 2 ) ,
23.2.6 σ ( z ) = z w 𝕃 { 0 } ( ( 1 - z w ) exp ( z w + z 2 2 w 2 ) ) .
17: 23.22 Methods of Computation
The modular functions λ ( τ ) , J ( τ ) , and η ( τ ) are also obtainable in a similar manner from their definitions in §23.15(ii). …
18: 20.3 Graphics
See accompanying text
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 τ ) 1 / 2 θ 4 ( z | τ ) = exp ( i τ z 2 / π ) θ 2 ( z τ | τ ) .