*7.5. Application to Represetenting Generalized Mittag-Leffler Function (2)*

The impulse response to fractional oscillators in Class I by using the generalized Mittag-Leffler function is in the form (Uchaikin ([38], Chapter 7))

$$h\_1(t) = t^{\kappa - 1} E\_{\mathfrak{a}, \mathfrak{a}} \left[ - (\omega\_n t)^a \right], 1 < \mathfrak{a} \le 2, t \ge 0. \tag{206}$$

In this section, we propose the representation of (206) by elementary functions.

**Corollary 16.** *The generalized Mittag-Leffler function in the form (206) can be expressed by the elementary functions in Theorem 13, for 1 < α* ≤ *2 and t* ≥ *0, in the form*

$$t^{a-1}E\_{\boldsymbol{\kappa},\boldsymbol{a}}\left[-\left(\omega\_{\boldsymbol{n}}t\right)^{a}\right] = \frac{e^{-\frac{\omega\_{\boldsymbol{n}}\sin\frac{a\pi}{2}}{2\left|\cos\frac{\boldsymbol{n}\cdot\boldsymbol{a}}{2}\right|}t}\sin\frac{\omega\_{\boldsymbol{n}}\sqrt{1-\frac{\omega^{2a}\sin^{2}\frac{\boldsymbol{n}\cdot\boldsymbol{a}}{2}}{4\omega\_{\boldsymbol{n}}^{2}\left|\cos\frac{\boldsymbol{n}\cdot\boldsymbol{a}}{2}\right|}}}{m\omega\_{\boldsymbol{n}}\sqrt{\omega^{a-2}\left|\cos\frac{\boldsymbol{n}\cdot\boldsymbol{a}}{2}\right|}\sqrt{1-\frac{\omega^{2a}\sin^{2}\frac{\boldsymbol{n}\cdot\boldsymbol{a}}{2}}{4\omega\_{\boldsymbol{n}}^{2}\left|\cos\frac{\boldsymbol{n}\cdot\boldsymbol{a}}{2}\right|}}}.\tag{207}$$

The proof is straightforward from Theorem 13 and (206).

### **8. Step Responses to Three Classes of Fractional Oscillators**

In this section, we shall put forward the unit step responses to three classes of fractional oscillators in the analytic closed forms with elementary functions. Besides, we shall sugges<sup>t</sup> a novel expression of a certain generalized Mittag-Leffler function by using elementary functions.
