# 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
For small \(|z|<<1\) and large values of \(|z|>>1\), this function has the following behavior  
\[f(z)=\left( \begin{array}{c}\left|\frac{a_{0}}{b_{0}}\right|+\left(\left| \frac{a_{1}}{b_{0}}\right|-\left|\frac{b_{1}}{b_{0}}\right|\right)|z|+\left( \frac{a_{2}}{b_{0}}-\frac{a_{0}b_{1}}{b_{0}^{2}}\right)|z|^{2}+\ldots\;\;at|z| <<1,\\ \left(\frac{a_{n-1}}{b_{n}}\right)\frac{1}{|z|}+\left(\frac{a_{n-2}}{b_{n}}- \frac{b_{n-1}}{b_{n}}\right)\frac{1}{|z|^{2}}+\ldots at|z|>>1.\end{array} \right. \tag{17}\]  
These decompositions are similar to the previous decomposition (3). We should note here that expression (17) is valid for cases (a) and (b) considered above. Therefore, if the passive RCL elements are similar to each other and satisfy the scaling property \(R_{n}=R_{0}b^{n}\), then one might expect the appearance of a CPE with power-law behavior accompanied always by complex-conjugate supplements. To complete the consideration of examples (1) and (17), we can also recall the famous product first proposed by [PERSON] with his pupils in the well-known paper [9] that also leads to the CPE from the product  
\[s^{\alpha}\cong\prod_{n=-N}^{N}\frac{s+\omega_{1,n}}{s+\omega_{n}} \tag{18}\]  
Instead of expression (18), we consider the more general product  
\[P(z)=\prod_{n=-N}^{N}f(z\xi^{n}),\;\;f(z\xi^{n})=\frac{A_{0}+a_{1}z}{1+b_{1}z}. \tag{19}\]  
Here \(z\) is an arbitrary complex variable \(z=i\omega\), \(a_{1}\), \(b_{1}\) > 0 are some positive constants, the constant \(A_{0}=|A_{0}|e^{i\varphi_{0}}\) can be complex and accept any value. It is easy to note that the product (19) satisfies the following relationship  
\[P(z\xi)=\frac{f\Big{(}z\xi^{N+1}\Big{)}}{f\Big{(}z\xi^{-N}\Big{)}}P(z) \tag{20}\]  
Further analysis of the product (20) at \(N>>1\) at different values of \(\xi\), leads to the following expressions  
\[P(z\xi)\cong\lambda_{1}P(z) \tag{21}\]  
law and more exponents. Obviously, expression (9) can be generalized. Let us _assume_ that from the SSP one can write the following relationship  
\[F\Big{(}z\xi^{L}\Big{)}=\sum\limits_{q=0}^{L-1}w_{q}F(z\xi^{q}),\quad\sum\limits_ {q=0}^{L-1}w_{q}=1 \tag{26}\]  
where the previous measurements form a specific self-similar memory. Here \(\xi\), defines the mean scaling parameter. We also assume that the set of weights \(w_{q}\) is _normalized_. In fact, relationship (25) itself represents the functional equation with solutions similar to (13). The complete solution of (25) can be found in [4]:  
\[ \begin{array}{l}F(z)=\sum\limits_{q=1}^{L-1}z^{\
u_{q}}\cdot\Pr_{q}(\ln(z)) \,+\Pr(\ln(z)),\,\,\
u_{k}=\frac{\ln(\lambda_{k})}{\ln(L)},\\ \mbox{where }(r_{k})^{\ln(z)/\ln\,(\xi)}=z^{\ln\,(\kappa)/\ln\,(\xi)}\equiv z^{ \
u_{k}}\\ \Pr_{q}(\ln(z)=A_{0}^{(q)}+\\ +\sum\limits_{k=1}^{K>>1}\Big{[}Ac_{k}^{(q)}\cos\Big{(}\frac{2\pi k}{\ln(z)} \ln(z)\Big{)}+As_{k}^{(q)}\sin\Big{(}\frac{2\pi k}{\ln(z)}\ln(z)\Big{)}\Big{]} \end{array}  \tag{27}\]  
Here the roots \(r_{q}\) satisfy Equation (27) and they are assumed to be positive. \[P(r)=r^{L}-\sum\limits_{q=0}^{L-1}w_{q}r^{q}=0 \tag{28}\]  
The log-periodic function \(\Pr(\ln(z))\) in the first row in (26) should be separated because of the normalization condition (25). It corresponds to the solution for the root \(r=1\) that is verified easily by substituting solution (26) in (27). For a negative root from (27), solution (26) should be rewritten as  
\[ \begin{array}{l}F(z)=|\lambda_{a}|^{\ln(z)/\ln\,\xi}P_{a}(\ln z)\equiv z^{ \
u_{a}}P_{a}(\ln z),\,\,\
u_{a}=\frac{|\ln(\lambda_{a})|}{\ln\,\xi}\\ P_{a}(\ln z)=\sum\limits_{k=1}^{K>1}\Big{(}Ac_{k}\cos\Big{(}(2k-1)\pi\frac{\ln (z)}{\ln(\xi)}\Big{)}+As_{k}\sin\Big{(}(2k-1)\pi\frac{\ln(z)}{\ln(\xi)}\Big{)} \Big{)},\\ P_{a}(\ln z\pm\ln\xi)=-P_{a}(\ln z)\end{array}