modular function

In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details. Formally, a function j is called modular or a modular function iff it satisfies the following properties:

j is meromorphic in the upper half plane H.
For every matrix M in the modular group Γ, j(Mτ) = j(τ).
The Laurent series of j has the form

j(\tau) = \sum_{n=-m}^\infty a(n) e^{2i\pi n\tau}.

It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant J, and that every rational function of J is a modular function; furthermore, all modular functions are modular forms, although the converse does not hold. If a modular function j is not identically 0, then it can be shown that the number of zeroes of j is equal to the number of poles of j in the closure of the fundamental region R_Γ.

Other uses

There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.

<< Previous

Word Browser

Next >>

34 circe
monstrous moonshine
35 leukothea
tupiniquim
theodore roosevelt regional airport
marian harkin
36 atalante
uss plunger (ss 179)
big joe 1
37 fides
ernie bushmiller

outpost (computer game)
unsere besten
half period ratio
antonio sanchez de bustamante
list of cathedrals in the united kingdom
memory barrier
jose luis bustamente
doom gameplay
king mackerel
38 leda
adolf butenandt

bismarck municipal airport
39 laetitia
jean pierre blanchard
40 harmonia
benjamin franklin butler
j.b. salsberg
charles butler
uinta mountains
david butler
milo butler
bolvar province, ecuador

paul butler
william butler
caar
william morgan butler
william orlando butler
carchi
thomas & mack center
maryanne kusaka
izu islands
chimborazo
hechos