DedekindEta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

η(z) is an analytical function of z which is defined over the upper half of the complex z -plane.

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

Poles and essential singularities

On the boundary of analyticity (the real axis) the function η(z) has a dense set of poles.

Branch points

The function η(z) does not have branch points.

Branch cuts

The function λ(z) does not have branch cuts.

Natural boundary of analyticity

The real axis Im(z)==0 is the natural boundary of the region of analyticity.

Series representations

Exponential Fourier series

q-series

Expansions at z==0

Expansions at z==i ∞

Other series representations

Integral representations

On the real axis

Of the direct function

Product representations

Differential equations

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Functional identities

Differentiation

Low-order differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Operations

Limit operation

Representations through equivalent functions

With related functions

Involving Weierstrass functions

Involving theta functions

Involving other related functions

Theorems

Modular form of order 12

The 24th power of the Dedekind Eta function η(z) is a modular form of order 12. (This means that under all argument substitutions of the form z->z'=(a z + b)/(c z + d) where a, b, c, and d are integers with a==1 mod 12,  b==0 mod 12,  c==0 mod 12,  d==1 mod 12, and a d-b c==1 the identity ((η(z'))^24)=(((c z + d))^12)((η(z))^24) holds.)

Kronecker's first limit formula

(Underscript[(∑_(k,l=0)^∞), {k,l}≠{0,0}])((y^s)/(|k (x+i y)+l((|)^(2s))))=(π/(s-1))+2π(γ-log(2)-((log((Sqrt[y])|η(x+i y)|))^2)+O(s-1)/;Re(s)>1∧x,y∈ℝ

The number of representations of n as a sum of triangular numbers

The number (Subscript[ν, n]) of representations of n as a sum of triangular numbers is (Subscript[ν, n])==([(q^n)]  ((q^(-1/8))(((η(-i log(q)/π))^2)/(η(-i log(q)/(2π)))))).

One product-series representation

((η(τ))^((n^2)+2))==(((-1))^((({{n+1}, {2}})))) ((∏_(k=1)^(n-1))(1/(k!))) (∑_(Underscript[(Subscript[k, 1]),(Subscript[k, 2]),…,(Subscript[k, n])=-∞, (Subscript[k, i])mod n==l])^∞)(((((∑_(i=1)^n)(Subscript[k, i]))/n)-((n+1)/2)) (((-1))^((∑_(i=1)^n)(Subscript[k, i])))(∏_(1≤r<s≤n))((Subscript[k, s])-(Subscript[k, r]))(q^(1/(2n)(∑_(i=1)^n)(((v(i)-(n/2)))^2))))/;q=(e^(i 2 π τ))

One series representation

(η^24)(z)==(∑_(k=1)^∞)τ(k)(e^(k π i z/12))/;τ(n)==(1/288)(∑_(Underscript[(Underscript[(Subscript[k, 1]),(Subscript[k, 2]),(Subscript[k, 3]),(Subscript[k, 4]),(Subscript[k, 5])=-∞{(Subscript[k, 1]),(Subscript[k, 2]),(Subscript[k, 3]),(Subscript[k, 4]),(Subscript[k, 5])}mod 5=={1,2,3,4,5}, (Subscript[k, 1])+(Subscript[k, 2])+(Subscript[k, 3])+(Subscript[k, 4])+(Subscript[k, 5])==1]), (Subscript[k, 1]^2)+(Subscript[k, 2]^2)+(Subscript[k, 3]^2)+(Subscript[k, 4]^2)+(Subscript[k, 5]^2)==10n])^∞)((Subscript[k, 1])-(Subscript[k, 2]))((Subscript[k, 1])-(Subscript[k, 3]))((Subscript[k, 1])-(Subscript[k, 4]))((Subscript[k, 1])-(Subscript[k, 5]))((Subscript[k, 2])-(Subscript[k, 3]))((Subscript[k, 2])-(Subscript[k, 4]))((Subscript[k, 2])-(Subscript[k, 5]))((Subscript[k, 3])-(Subscript[k, 4]))((Subscript[k, 3])-(Subscript[k, 5]))((Subscript[k, 4])-(Subscript[k, 5]))

History

– R. Dedekind (1877) introduced the name "elliptic modular function"

Copyright