Automatic Control Systems by Richard M. Phelan

By Richard M. Phelan

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The entire classical calculus (integer calculus) is based on it. • Cosmic Time: It corresponds to non-equably flowing time as shown in Fig. 20. The change of scale of the cosmic time is described using the homogeneous time scale as a reference scale. In other words, one can interprets the homogeneous time scale as an ideal notion, which is necessary for developing mathematical models describing inhomogeneous cosmic time and its change. 89) where t1 = kt and τ1 = kτ, k > 0. Let us now consider a three-dimensional space with co-ordinate (τ, g, f ) and plot the function gt (τ ) for 0 ≤ τ ≤ t on the plane (τ, g) as shown in Fig.

Dτ = n 0 0 (t − τ )n−1 f (τ )dτ. (n − 1)! 6) n This completes the proof. It is observed that: • the above generalized integration is valid only for the integer points; • it contains a factorial function; However, one very sound mathematical observation comes into picture that the above definition contains the factorial function. ) means factorial points” as shown in Fig. 1? Riemann-Liouville found that the generalized gamma function can provide a smooth curve connecting the factorial points. It is defined as ∞ e−t t z−1 dt, Γ (z) = 0 which converges in the right half of the complex plane for R(z) > 0.

In other words, one can interprets the homogeneous time scale as an ideal notion, which is necessary for developing mathematical models describing inhomogeneous cosmic time and its change. 89) where t1 = kt and τ1 = kτ, k > 0. Let us now consider a three-dimensional space with co-ordinate (τ, g, f ) and plot the function gt (τ ) for 0 ≤ τ ≤ t on the plane (τ, g) as shown in Fig. 21. Along the obtained curve, by varying the height f (τ ) one can build a fence. Finally fence can be projected onto two surfaces 28 1 Essence of Fractional Order Calculus, Physical Interpretation and Applications Fig.

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