An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function

Abstract

A significant advance in characterizing the nature of the zeros and organizing the Mittag-Leffler functions into phases according to their behavior is presented. Regions have been identified in the domain of and where the Mittag-Leffler functions , ( ) have not only the same type of zeros but also exhibit similar functional behavior, and this permits the establishment of an -phase diagram., ( ) ≈ −

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

“…However, this monotonicity is not necessarily preserved when (α, β) ∈ Ω, since either the function or its derivative could have roots. On the other hand, it was established in [20] that E α,β has at most a finite number of roots when (α, β) ∈ Ω. Thus, E α,β (−t) is monotone for sufficiently large t. Furthermore, it was shown in [20] that there exists only one zero when α is sufficiently close to 1, while the number of zeros increases as α increases toward 2.…”

“…In Figure 1, the curve ϕ is the boundary given in Table 1 in [20] such that E α,β has a finite number of real roots when (α, β) is below it and none above it. However, the function in the region below ϕ(α) differs with respect to the number of roots.…”

“…The existing rational approximants are effective when the MLF is completely monotone, which is the case when α ∈ (0, 1) and β ≥ α. However, when α > 1, as discussed in [19][20][21], for some α-β combinations, E α,β is often oscillatory with multiple real zeros. In these situations, the rational approximants might fail to trace the oscillation profile and match the large zeros.…”

“…The data for the boundary ϕ ([20], Table1) and the boundary ψ. The function E α,β possesses real roots below the curve ϕ(α) and has no real roots above it.…”