exponential function

Thus, the Mittag-Leffler functions can be considered as the analogy of exponential functions for fractional differential operators, since they are the sum of all basis elements.

The Taylor Series for [e.sup.u] exponential function is as shown in Eq.

We determine [rho] from the condition that the scheme is exact when the solution of the problem (1.1)-(1.2) with constant coefficients is an exponential function. In this case, [rho] is the largest (by the absolute value) root of the quadratic equation

When [mu][lambda] < 0, the exponential function solution of system (44) can be expressed as

The exponential function can still be available as the prediction model for the normalized settlement.

The solution to system (1) is Y(f) = exp[(f - [t.sub.0])A] [Y.sub.0], and exp(x) is the tensor exponential function.

If [tau] = 0 and T = R, then the exponential function can be written as [e.sub.[alpha]](t,0) = [e.sup.at] for all t [member of] T.

As shown in Figure 2(c) the analysis of the semi-major axis values of TRAPPIST-1's planets b-h revealed an exponential function (or a quasi linear one when logarithmized values were used; Figure 2(d)).

The exponential function is employed to represent the relation of the arithmetic average surface roughness, the cutting force ratio, and the cutting parameters.

The temporal/spatial variation of the echo intensity is also simulated by the exponential function, and the formula is as follows:

where [E.sup.t.sub.q] = [[summation].sup.[infinity].sub.n=0][q.sup.n(n-1)/2]([t.sup.n]/[[n].sub.q]) is a q-analogue of the classical exponential function, [[u].sub.q] = (1 - [q.sup.u])/(1 - q), and [[n].sub.q!] = [[n].sub.q] [[n - 1].sub.q] ...