Hidden Symmetries of Complex Analysis

We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.

connected Hausdorff space with a conformal structure defined by a family Φ of local homeomorphisms (parameters). Each ϕ ∈ Φ is a topological mapping of an open set V S ⊂ onto a relatively open set of the upper (closed) half plane is covered by open sets V S ′ ⊂ such that |V ϕ ′′ ∈ Φ , then ϕ ∈ Φ and finally the domains of all ϕ cover the whole space S. The conformal structure of Φ means that for every 1 1 V ϕ onto ( ) 2 2 V ϕ whose complex conjugate is conformal. The involution k generates the two-element group of transformations of S formed with k and the identity I. We have k I I k k = =   and k k I =  . This group is denoted by k and the respective symmetric Riemann surface is denoted by ( ) , S k . This is a symmetry in the sense of Klein (see [2]) and in this paper we will be talking only about this kind of symmetries. It is known that the conformal structure of ( ) , S k induces a dianalytic structure Ψ on the factor space X S k = , i.e. local parameters ψ ∈ Ψ are defined on X by ( ) ( )

Simple Examples of Hidden Symmetries
The simplest examples of symmetric Riemann surfaces are the complex plane and the Riemann sphere endowed with the fixed point free antianalytic involu- Local parameters 1 ϕ and 2 ϕ can be defined on where a and b are uniquely determined such that We have proved in [3], Theorem 3 that conformal images of symmetric domains are symmetric. Since Möbius transformations ( ) Proof: Indeed, it can be easily checked that and therefore where A ac bd = + ,

Blaschke Product Symmetries
In the infinite case it is customary to assign the values n n a a to the constants e n iθ , as long as 0 n a ≠ . The points 1 n a are poles of ( ) B z . It is known that in the infinite case a necessary and sufficient condition for the uniform convergence of ( ) Thus, ( ) B z has a meaning only if E is a subset of the unit circle. For finite Blaschke products E = ∅ . Since every Blaschke factor maps the unit circle onto itself, the same is true for any Blaschke product, with the specification that in the infinite case E should be removed. Also, the unit disc is mapped onto itself and the exterior of the unit disc is mapped onto itself. In general, these mappings are not bijective, yet there is a way to partition the complex plane into sets whose interiors are conformally mapped onto the whole complex plane with some slits. These are the fundamental domains of the Blaschke product.
For the simple case of a single zero of order 2 m ≥ , not only the fundamental domains are obvious, but so are the involved symmetries. Let We can find the fundamental domains of then it continues with a symmetric arc with respect to the unit circle such that the pole 1 n a is reached at the limit as w → +∞ . Hence the pre-image of the positive real half axis is formed with arcs connecting zeros and poles of ( ) B z . Continuing over the negative half axis we might reach the branch points of ( ) B z and this will allow us to find fundamental domains of ( ) B z . This is the case of Blaschke products which have symmetric zeros with respect to the origin.   . If we denote by Ω any one of these quadrants, and s ∈ Ω is such that ( ) is not defined at s. Then instead of Ω we should deal with ( ) . This is a hidden symmetry of ∆ . Figure 3 below illustrates this situation when ( ) We colored blue the pre-image of the positive real half axis, red that of the negative real half axis, black that of the circle 1 w = , green that of the circle 0.5 w = and brown that of the circle 1.5 w = .
Obviously, similar situations can be created when taking In the case where the zeros of ( ) B z are randomly located inside the unit disc, we need to find a different approach.   . It is obvious that they are symmetric with respect to the unit circle. Each one of them contains also the hidden symmetry This situation is illustrated in Figure 4.   Figure 5 illustrates the fundamental domain of cos z .

The Modular Function ( ) λ τ
The modular function ( ) λ τ effects a one-to-one conformal mapping (see [5], is mapped onto the whole complex plane with a slit alongside the real axis from −∞ to 1. This is a fundamental domain of ( ) λ τ . If instead ′ Ω we take ′′ Ω the symmetric of Ω  with respect to the circle 1 2 1 2 τ − = and we do the corresponding union we obtain another fundamental domain and the process can be continued indefinitely. The symmetric of these domains with respect to the real axis are also fundamental domains of ( ) λ τ . These domains accumulate to every point of the real axis in the sense that any neighborhood of such a point contains infinitely many fundamental domains. There are obvious symmetries of these domains, yet each one of them, say ∆ contains also a hidden symmetry, namely

The Hidden Symmetries of the Euler Gamma Function
There are a lot of ways to introduce this famous function. For our purpose the Weierstrass definition is the most useful, namely: