## 2 Theory
### Analytic Continuation, Global Analytic Functions, and Riemann Surfaces
#### 2.2.2 Global Analytic Functions and Riemann Surfaces
: _The function \(\hat{\mathcal{G}}[\cdot]\) is as follows:_  
\[\hat{\mathcal{G}}[(z,[f]_{z})]=w\quad(z,[f]_{z})\in\mathbb{S}(\mathcal{G}),w \in\mathbb{C}, \tag{17}\]  
_where \(w=\left[\widehat{f}\right]_{z}\) is called the lift of \(\mathcal{G}\) on its Riemann surface \(\mathbb{S}(\mathcal{G})\)._  
Another function of paramount importance is the following. **Definition 7**. : _The function \(\hat{\mathcal{P}}[\cdot]\)_  
\[\hat{\mathcal{P}}[(z,[f]_{z})]=z\quad(z,[f]_{z})\in\mathbb{S}(\mathcal{G}),z \in\mathbb{C} \tag{18}\]  
_is called the projection of \(\mathbb{S}(\mathcal{G})\) to \(\mathbb{C}\)._  
To clarify the idea behind the lift operation, let us consider set \(\hat{\Omega}\) of all pairs \((z,[f]_{z})\) such that \(z\in\hat{\Omega}\), and \([f]_{z}\) is germ-derived by \((\hat{\Omega},\tilde{f}(\cdot))\). This set, \(\hat{\Omega}\), is a copy of \(\Omega\) on \(\mathbb{S}(\mathcal{G})\) (we recall that \(\hat{\mathcal{P}}[(z,[f]_{z})]=z\in\hat{\Omega}\)). Let \((z,[f]_{z})\) be an element of \(\hat{\Omega}\); then, according to definition (6), we have the following. \[\hat{\mathcal{G}}[(z,[f]_{z})]=\left[\widehat{f}\right]_{z}=\tilde{f}(z) \tag{19}\]  
In otherwords, \(\hat{\mathcal{G}}[\cdot]\) assigns to each point in \((z,[f]_{z})\in\hat{\Omega}\subset\mathbb{S}(\mathcal{G})\) precisely the same value that \(\tilde{f}(\cdot)\) assigns to each point in \(z\in\hat{\Omega}\subset\mathbb{C}\). In sum, we have that the lift of \(\mathcal{G}\) on its Riemann surface, \(\hat{\mathcal{G}}[\cdot]\), and it is restricted to \(\hat{\Omega}\) and behaves exactly as \(\tilde{f}(\cdot)\) on \(\hat{\Omega}\)[37]. **Remark 3**. : _The lift of \(\mathcal{L}\) on \(\mathbb{S}(\mathcal{L})\) is given by the following._  
\[\hat{\mathcal{L}}[(z,[f]_{z})]=\left[\widehat{f}\right]_{z}=log(z)+2p\pi i \tag{20}\]_If \((\Omega,f(\cdot))\in\mathcal{L}\), the set \(\hat{\Omega}\) of all pairs_  
\[(z,log(\cdot)+2p\pi i),\quad z\in\Omega \tag{21}\]  
_is a copy of \(\Omega\) on \(\mathbb{S}(\mathcal{L})\). The integers \(p\) fulfilling Equation (21) define sheets \(\mathbb{S}_{p}(\mathcal{L})\) where \(\hat{\Omega}\) lies._  
Kramers-Kronig Relations and Phase Unwrapping Approach: A Unique Point of View within the Framework of the Global Analytic Functions on Riemann Surfaces  
As stated in Sections 2.1.1 and 2.1.2, Equation (1) is solved by evaluating \(h(\cdot)\) via the K-K relations or by the phase unwrapping approach, and both methods are equivalent to analytic continuation. It is quite straightforward to encompass these methods in the theoretical framework given in Section 2.2.2. Specifically, from Remark 1, the set of all the solutions of Equation (9) is a global analytic function, i.e, the global analytic logarithm \(\mathcal{L}[\cdot]\); from Remark 2, we have the explict structure of its Riemann surface \(\mathbb{S}(\mathcal{L})\); finally, from Remark 3, the values assumed by the lift of the global analytic logarithm \(\hat{\mathcal{L}}[(\cdot)]\) on sub-domain \(\mathbb{D}\) established by pairs \((z,[f]_{z})\) are _equal_ to the values taken by \(h(\cdot)\) on \(\mathbb{D}\); i.e., between the lift of global analytic logarithm, \(\hat{\mathcal{L}}[\cdot]\), and the right inverse \(h(\cdot)\), relation (20) must hold. \[\hat{\mathcal{L}}[(z,[f]_{z}])]=[\bar{f}]_{z}=h(z)\quad\forall z\in\mathbb{D} \tag{22}\]  
That is, we have the following. \[[\bar{f}]_{z}=log(z)+2p\pi i=h(z)\quad\forall z\in\mathbb{D} \tag{23}\]  
_Only in the context of the theory described above can the equalities (22) and (23) be derived and have meaning._ Since \(log(\cdot)=ln|\cdot|+i[arg_{\pi}(\cdot)]\), we can write the following. \[\mathsf{Im}[h(z)]=[arg_{\pi}(z)+2p\pi]=\quad\forall z\in\mathbb{D} \tag{24}\]  
Considering Equation (8), the above relation can be written as follows.