Weber modular functionIn mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber. Let where τ is an element of the upper half-plane. Then the Weber functions are These are also the definitions in Duke's paper "Continued Fractions and Modular Functions". The function is the Dedekind eta function and should be interpreted as . The descriptions as quotients immediately imply The transformation τ → –1/τ fixes f and exchanges f1 and f2.
Q-Pochhammer symbolIn mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with It is a q-analog of the Pochhammer symbol , in the sense that The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series.
Lemniscate constantIn mathematics, the lemniscate constant π is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of pi for the circle. Equivalently, the perimeter of the lemniscate is 2π. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. The symbol π is a cursive variant of π; see Pi § Variant pi. Gauss's constant, denoted by G, is equal to π /pi ≈ 0.8346268.
Elliptic functionIn the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass -function. Further development of this theory led to hyperelliptic functions and modular forms.
Dedekind eta functionIn mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. For any complex number τ with Im(τ) > 0, let q = e2πiτ; then the eta function is defined by, Raising the eta equation to the 24th power and multiplying by (2π)12 gives where Δ is the modular discriminant.
