By David V. Widder

Vintage textual content leads from user-friendly calculus into extra theoretic difficulties. distinctive method with definitions, theorems, proofs, examples and routines. issues comprise partial differentiation, vectors, differential geometry, Stieltjes imperative, endless sequence, gamma functionality, Fourier sequence, Laplace remodel, even more. a variety of graded workouts with chosen solutions. 1961 edition.

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It then follows that: limx↓0 log(φ(x)) log(x) = limx↓0 log φ(x) ψ (x) ψH (x) H log(x) φ(x) ψH (x) ≤ C2 , = (100) = limx↓0 log(ψH (x)) log(x) + limx↓0 log φ(x) ψH (x) log(x) = dimH (F ). This means that the Hausdorff dimension does not depend on the representative of the equivalence class to which ψH belongs. (iii) Finally, if ψH (x) = xu for some u > 0, then 0 < dimH (F ) = sH = rH = u < ∞. Alternatively, suppose that ψH ∈ / P. s Since the scale F is a well-ordered set, then at least one of the sets {s > 0 : x ≺ ψH }, Local Fractional Derivatives 31 {r > 0 : ψH ≺ xr } is non-empty.

Of ψξ,c This behavior is easily illustrated with the following example. Example 102. Consider the function f (x) = |x|r , 0 < r < 1. From (161), we easily obtain for ψ ∈ Q: 0, if ψ ≺ xr ψ σ σ, if ψ = xr Dx f (0 ) = , for σ = ± ∞, if xr ≺ ψ x Also for the derivatives of order r, we have: (Dxr f ) (xσ ) = σ, if x = 0 , 0, if x = 0 for σ = ± The last equation reveals that, contrary to ordinary derivatives, local fractional derivatives do not obey Darboux’s theorem. Local Fractional Derivatives 47 Proposition 103.

Altogether, ψ ∈ D. Next, let φ(x) = x 2 ∈ D. For x > 0: 3 ψ(x) = x− 2 φ(x) Consider the sequence xn = ψ(xn ) φ(xn ) = π 2 (2n π 2 (2n x τ 1 + sin 0 −1 + 1) + 1) 3 2 1 τ dτ + 1 (70) . It follows that 2 π(2n+1) 0 τ 1 + sin 1 τ dτ + 1 = (71) = π 2 (2n + 1) 3 2 2 π(2n+1) n) If n is odd, then ψ(x φ(xn ) → 1, whereas φ are not comparable. [1 + (−1)n ] + O ψ(xn ) φ(xn ) 1 (2n+1)2 +1 → ∞, when n is even. We conclude that ψ and An important subset of D is the set of powers. Example 28. Let us consider the powers xr , with r > 0.