Modular Function

A function is said to be modular (or ``elliptic modular'') if it satisfies:

1. is Meromorphic in the upper half-Plane ,

2. for every Matrix A in the modular group ,

3. The Laurent Series of has the form

An important property of modular functions is that if is modular and not identically 0, then the number of zeros of isequal to the number of poles of in the Closure of the fundamental region (Apostol 1997, p. 34).

References

Modular Functions

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976.

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, 1977.

Schoeneberg, B. Elliptic Modular Functions: An Introduction. Berlin: New York: Springer-Verlag, 1974.

• Jacobi Function of the First Kind
• Jacobi Function of the Second Kind
• Jacobi-Gauss Quadrature
• Jacobi Identities
• Jacobi Matrix
• Jacobi Method
• Jacobi Polynomial
• Jacobi Quadrature
• Jacobi Rotation Matrix
• Jacobi's Curvature Theorem
• Jacobi's Determinant Identity
• Jacobi's Imaginary Transformation
• Jacobi's Theorem
• Jacobi Symbol
• Jacobi Tensor
• Jacobi Theta Function
• Jacobi Transformation
• Jacobi Triple Product
• Jacobi Zeta Function
• Jacobsthal-Lucas Number
• Jacobsthal-Lucas Polynomial
• Jacobsthal Number
• Jacobsthal Polynomial
• Jaco-Shalen-Johannson Torus Decomposition
• Janko Groups