## 2 Theory
### Analytic Continuation, Global Analytic Functions, and Riemann Surfaces
#### 2.2.2 Global Analytic Functions and Riemann Surfaces  
To handle a multivalued global analytic function \(\mathcal{G}\) as a function in the ordinary sense, it becomes mandatory to redefine a suitable domain \(\mathbb{S}(\mathcal{G})\) for \(\mathcal{G}\) and to operate in a way that it becomes an ordinary single-valued function from the new domain \(\mathbb{S}(\mathcal{G})\) to \(\mathbb{C}\). To this aim, instead of dealing with analytic function elements \((\Omega,f(\cdot))\), where \(\Omega\) is an arbitrary domain, we have to consider pairs of the form \((\Delta,f(\cdot))\) where \(\Delta\subseteq\Omega\) is a disk [35]. Between these elements, it is possible to introduce an equivalence relation [35, 37]. **Definition 2**. : _Two pairs \((\Delta^{\prime},f^{\prime}(\cdot))\) and \((\Delta^{\prime\prime},f^{\prime\prime}(\cdot))\), with disks \(\Delta^{\prime}\) and \(\Delta^{\prime\prime}\) centered in \(z^{\prime}\) and \(z^{\prime\prime}\), respectively, are equivalent if the following is the case:_  
\[\left(z^{\prime}=z^{\prime\prime}\right)\wedge\left(f^{\prime}(\cdot)=f^{ \prime\prime}(\cdot)\right) \tag{15}\]  
_in some neighborhood of the point \(z^{\prime}\)._  
From it, we derive the possibility to partition \(\mathcal{G}\) into particular subsets, called _germs_[37]. **Definition 3**. : _The set of all the equivalent pairs \((\Delta,f(\cdot))\in\mathcal{G}\) is called germ, centered at \(z\), and denoted by the symbol \([f]_{z}\). The value of a germ \([f]_{z}\) at its center \(z\), \([\bar{f}]_{z}\), is the value \(w=f(z)\) given by any of its elements at \(z\)._  
The germs can be collected together as sets [37]. **Definition 4**. : _The set of all germs in which all elements \((\Delta,f(\cdot))\) belonging to a global analytic function \(\mathcal{G}\) can be collected is denoted by \(\mathcal{G}^{*}\). The set of all germs in \(\mathcal{G}^{*}\) with the same center \(z\) is denoted by \(\mathcal{G}^{*}_{z}\)._  
Using the concept of a set of germs, we can introduce a theorem that formalizes what was first said about the number of values that a global analytic function \(\mathcal{G}\) could assume in correspondence to a given point \(z\) of its domain of existence \(\mathbb{M}\) (for its demonstration, see [32]). **Theorem 1** (Poincare-Volterra). : _Let \(\mathcal{G}\) be a global analytic function. For all \(z\in\mathbb{C}\) and \(\mathcal{G}_{z}^{*}\in\mathcal{G}^{*}\), the cardinality of \(\mathcal{G}_{z}^{*}\) is at most countable._  
Based on the Poincare-Volterra theorem, we are in the condition to introduce the concept of Riemann surface \(\mathbb{S}(\mathcal{G})\) for a global analytic function \(\mathcal{G}\). It consists of a number of _abstract sheets_ lying above \(\mathbb{C}\). These sheets (or layers) can be built by associating an array to each \(z\in\mathbb{C}\), for which its elements are germs belonging to set \(\mathcal{G}_{z}^{*}\) with the center at \(z\). The array can be empty (no germs at all) or, at most, have an infinite (countable) dimension [35; 37]. In a more formal language, we have the following. **Definition 5**. : _The set_  
\[\mathbb{S}(\mathcal{G})=((z,[f]_{z}):z\in\mathbb{C},[f]_{z}\in\mathcal{G}_{z} ^{*}) \tag{16}\]  
_is the Riemann surface related to \(\mathcal{G}\)._  
**Remark 2**. : _In the case of the global analytic logarithm \(\mathcal{L}\), its Riemann surface \(\mathbb{S}(\mathcal{L})\) is realized by all pairs \((z,[f]_{z})\) where \(z\in\mathbb{C}\) and \([f]_{z}=log(\cdot)+2p\pi i\) where \(p\in\mathbb{Z}\). The \(p\)-th sheet of \(\mathbb{S}(\mathcal{L})\), \(\mathbb{S}_{p}(\mathcal{L})\), consists of all couples of \((z,log(\cdot)+2p\pi i)\) with \(p\) fixed._  
In other words, \(\mathbb{S}(\mathcal{G})\) is a subset of the Cartesian product between \(\mathbb{C}\) and \(\mathcal{G}\). Using the Riemann surface \(\mathbb{S}(\mathcal{G})\) in lieu of \(\mathbb{M}\), it is possible to redefine \(\mathcal{G}\) in a way that it realizes a function from its Riemann surface \(\mathbb{S}(\mathcal{G})\) to \(\mathbb{C}\)[37]. **Definition 6**.