## 4 Taylor and Pade Expansions of Mie Coefficients
In this section, we briefly summarize references [63; 72; 85] and the results found therein. As a first example, the \([3/3]\) Pade expansion for \(a_{1}\) reads  
\[a_{1}^{p}\approx-i\frac{2}{3}\frac{\varepsilon_{1}-1}{\varepsilon_{1}+2}\frac {x^{3}}{\left(1-\frac{3}{5}\varepsilon_{1}+2}x^{2}-i\frac{2}{3}\frac{ \varepsilon_{1}-1}{\varepsilon_{1}+2}x^{3}\right) \tag{20}\]  
We note that Equation (20) can be decomposed into three distinctive parts. First, the \(\frac{\varepsilon_{1}-1}{\varepsilon_{1}+2}\) is the normalized static polarizability term, similarly with Equation (15). This term contributes a system pole at \(\varepsilon_{1}=-2\) only for the static case [88]. The second part is \(\frac{\varepsilon_{1}-2}{\varepsilon_{1}+2}x^{2}\) that introduces the size effects over the coefficient and to the overall scattering procedure, and thus it is recognized as the _dynamic depolarization_ term [89]. Lastly, the imaginary term of order \(x^{3}\) in the denominator is recognized as the _radiative damping_ part. For the lossless case (\(\varepsilon^{\prime\prime}=0\)), this part prevents the coefficient from infinite growth and represents the intrinsic radiative damping/reaction mechanism of the scattering process [46; 49; 90]. Ignoring the dynamic depolarization term in Equation (20) and utilizing the expression (19) we observe that  
\[a_{1}^{[3/3]}=\frac{a_{1}^{T}}{1+a_{1}^{T}}=\frac{-i\frac{2}{9}\alpha x^{3}}{ 1-i\frac{2}{9}\alpha x^{3}} \tag{21}\]  
This is an equivalent expression of the polarizability model that includes the induced radiative reaction on the single subwalength scatterer, as has been repeatedly exposed by many workers [6; 46; 89; 90; 91; 92; 93; 94]. Expression (20) exhibits a pole (resonance) at \(\varepsilon_{1}=-2\frac{15i+9 ix^{2}-5x^{3}}{15i-9 ix^{2}+10x^{3}}\), which gives, for small \(x\)  
\[\varepsilon_{1}=-2-\frac{12}{5}x^{2}(1+\frac{3}{5}x^{2})-2 ix^{3}(1+\frac{7}{5 }x^{2})+... \tag{22}\]As before, the resonant condition contains a static term (\(-2\)), a dynamic depolarization term (\(x^{2}\)), and a radiative damping imaginary term (\(x^{3}\)). Note that, for an open resonator, the resonant frequencies should be generally complex, known as _natural frequencies_[16]. Again, the real terms of Equation (22) represent the shift that the position of the resonance exhibits with respect to the size. However, the imaginary term dictates also both the width of the resonance and the amount of losses required in order to maximize the absorption. The latter can be seen as a form of a conjugate matching between the incident wave and the scatterer [20; 63; 95; 96; 97],  
Let us now apply the same Pade stratagem for the magnetic coefficient \(b_{n}\). The \([5/5]\) Pade expansion of the first magnetic dipole coefficient \(b_{1}\) reads  
\[b_{1}^{T}\approx-i\frac{2}{3}\frac{\mu_{1}-1}{\mu_{1}+2}x^{3}-i\frac{1}{5} \frac{\mu_{1}^{2}(1+\varepsilon_{1})-6\mu_{1}+4}{(\mu_{1}+2)^{2}}x^{5}+O[x]^{6} \tag{16}\]0  
with the truncated term being  
\[b_{1}^{T}\approx-i\frac{2}{3}\frac{\mu_{1}-1}{\mu_{1}+2}x^{3}-i\frac{1}{5} \frac{\mu_{1}^{2}(1+\varepsilon_{1})-6\mu_{1}+4}{(\mu_{1}+2)^{2}}x^{5}+O[x]^{6} \tag{16}\]1  
Interestingly, expression (23) gives a resonant condition that yields to the value  
\[b_{1}^{T}\approx-i\frac{2}{3}\frac{\mu_{1}-1}{\mu_{1}+2}x^{3}-i\frac{1}{5} \frac{\mu_{1}^{2}(1+\varepsilon_{1})-6\mu_{1}+4}{(\mu_{1}+2)^{2}}x^{5}+O[x]^{6} \tag{16}\]2  
where the accuracy was kept up to the second decimal. The condition (25) predicts, in contrast to the Taylor-Mie example, a pole for the magnetic dipole coefficients, suggesting a clear intuitive picture regarding the nature of the magnetic resonances enabled on a dielectric sphere. First, the pole condition in Equation (25) suggests an inverse square dependence, showing that the magnetic resonance for small spheres can be reached only for large contrast materials, making the observation of this kind of resonances for very small spheres practically impossible.