# Can Self-Similarity Processes Be Reflected by the Power-Law Dependencies?
## 2 Confirmation of the SSP for the Weierstrass-Mandelbrot Function and in Other Models  
As is well-known [6; 7], the Weierstrass-Mandelbrot (W-M) function is defined by the following expression  
\[W(t)=\sum_{n=-N}^{N}\frac{1-\exp(ib^{n}t)e^{i\varphi_{n}t}}{b^{(2-D)n}}. \tag{1}\]  
For further purposes, it is convenient to present this function in a more general form  
\[S(z)=\sum_{n=-N}^{N}\left(be^{i\varphi}\right)^{n}\!f(z\xi^{n}). \tag{2}\]  
In Equation (2), the variable \(z\) can coincide with time \(t\), the complex frequency \(i\omega\), the Laplace complex parameter \(p\), etc. The phase \(\varphi\) in (2) takes into account the fact that parameter \(b\) can accept a negative value (\(\varphi=\pi\)) or can be complex (\(\varphi\
eq 0\)). For further manipulations with expression (2), we simply replace \((be^{i\varphi})\to b\) implying that number \(b\) can accept any arbitrary value (positive, negative or even complex). The complex function \(f(z)\) has the following decompositions  
\[f(z)=\left( \begin{array}{cc}c_{1}|z|+c_{2}|z|^{2}+\ldots&at|z|<<1,\\ \frac{A_{1}}{|z|}+\frac{A_{2}}{|z|^{2}}+\ldots at|z|>>1.\end{array} \right. \tag{3}\]  
At the conditions imposed above, one can obtain the following relationship  
\[S(z\xi)=\frac{1}{b}S(z)+b^{N}f\Big{(}z\xi^{N+1}\Big{)}-b^{-N-1}f\Big{(}z\xi^{- N}\Big{)}. \tag{4}\]  
This function was analyzed in paper [8]. Four cases are possible:  
Case (a) when \(b\), \(\xi>1\). The contribution of two terms in (4) becomes negligible when \(\xi>b\). Case (b) when \(b\), \(\xi<1\). The contribution of two terms in (4) becomes negligible when \(\xi<b\). Case (c) when \(b>1\), \(\xi<1\). For this case, it is necessary that \(b\xi<1\). Case (d) when \(b<1\), \(\xi>1\). For this case, it is necessary that \(b\xi>1\). For \(b=1\), the contribution of the last two terms is valid also, if the function \(f(z)\) keeps its limiting values in accordance with decompositions (3). The numerical verification of the real part of the function \(S(z)\) was realized in paper [8]. Therefore, one can write approximately  
\[S(z\xi)\cong\frac{1}{b}S(z),\;\;for\;\;\;\;\left|\frac{A_{2}}{A_{1}}\right|<< \left|z\right|<<\left|\frac{c_{1}}{c_{2}}\right|\!. \tag{5}\]Let us assume that the initial function itself represents a linear combination of three generalized W-M functions of type (2). \[ \begin{array}{l}F_{0}(z)=\sum\limits_{n=-N}^{N}b_{1}^{n}f_{1}(zk^{n})+\sum \limits_{n=-N}^{N}b_{2}^{n}f_{2}(zk^{n})+\\ \quad+\sum\limits_{n=-N}^{N}b_{3}^{n}f_{3}(zk^{n})\equiv\sum\limits_{k=1}^{3}S_ {k}(z).\end{array}  \tag{6}\]  
We assume that the \"microscopic\" functions \(f_{k}(z)\) fulfil comparable decompositions and that the evaluations of each sum \(S_{k}(z)\) given above are still accurate. Therefore, using the approximate relationship (5) one can write  
\[\begin{array}{l}F_{0}(z)=S_{1}(z)+S_{2}(z)+S_{3}(z),\\ F_{1}(z)=r_{1}S_{1  
Eliminating \(S_{1,2}(z)\) from two rows of (14) and substituting these expressions in the last row, one can obtain the following functional equation  
\[ \begin{array}{l}F\Big{(}z\xi^{2}\Big{)}=w_{1}F(z\xi)+w_{0}F(z),\\ w_{1}=r_{1}+r_{2},\;\;w_{0}=-r_{1}r_{2},\;\;r^{2}=w_{1}r+w_{0}.\end{array}  \tag{15}\]  
The solution of this equation can be written in complete analogy with expressions (12) and (13). Normalization to the unit value similar to (10) is possible if one puts \(r_{2}=1\). It follows from the calculations made above that the functional Equations (9) and (15) are derived from the approximate functional Equation (5) that is valid for variables located in the mesoscale. It is interesting to propagate this result for the frequency variable \(z=i\omega\) and consider the so-called constant-phase (CPE) or reind/recup elements. To this end, we represent the complex impedance for a combination of RCL passive elements in the form  
\[f(z)=\frac{P_{n-1}(z)}{P_{n}(z)}=\frac{a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_ {1}z+a_{0}}{b_{n}z^{n}+b_{n-1}z^{n-1}+\ldots+b_{1}z+b_{0}} \tag{16}\]  
where \(z\) is the complex/Laplace parameter.