By L. Mirsky

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99) that the H-function can be simplified to the G-function by setting c0 = 0. 1 summarizes the advanced functions studied in previous sections along with their defining series and Laplace Transforms. 10 Preliminaries to the Fractional Trigonometry Development The previous sections of this chapter have considered the R-function as it relates to the other important functions of the fractional calculus and to the traditional elementary functions. Furthermore, various identities have been shown. Some specific tools and properties, however, are still needed.

5, the voltage transfer function from the input terminals to the supercapacitor terminals is found to be √ R + √???? + sC1 RCs + ????C s + 1 Vo (s) s . 33) = = √ Vi (s) sL + R + √???? + 1 LCs2 + RCs + ????C s + 1 s For this example, we let RC = 1, sC ????C = 1, and LC = 1. Then, √ s+ s+1 Vo (s) . 5. Clearly, there are two poles in the right half of the w-plane, but to the left of the stability boundary. These pole locations correspond to complex stable poles in the s-plane and imply a damped oscillatory impulse response.

Let the series converge uniformly to s(x). 16 Term-by-Term Operations is a convergent series and ∞ b∑ ∫a uk (x)dx = k=1 ∞ ∑ k=1 b ∫a uk (x)dx. 159) For term-by-term differentiation of an infinite series, Miller [94], p. 160) be an infinite series of functions. Let each ui (x), i = 1, 2, … be defined and differentiable on an interval [a,b]. 160) converge to the sum s(x) and let the series of derivatives u′1 (x) + u′2 (x) + · · · + u′n (x) + · · · converge uniformly to w(x). ” In other words, ∑ d d∑ uk (x) = u (x).

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