The Exponential Function

To investigate the behaviour of this function it is useful to draw up a table of values, as below  

First of all notice that the exponential function is never negative the exponential of a negative number is always a fraction however. For positive values of x, the exponential function increases rapidly. This is what is meant by the commonly used term exponential growth , although strictly speaking this should only be used when the exact exponential function is used. 

If you have verified any of the values in the above table, you may have noticed that to generate exponentials involved a sequence such as SHIFT followed by In . There is a reason for this, which we will return to in Chapter 22. 

The graph in Fig. 21.2 shows the behaviour of the function /(x) = e, which is known as exponential decay . Note that the value of the exponential becomes ever closer to zero but never actually reaches it. 

Kinetics is the study of the speeds with which reactions take place, and a study of it provides us with valuable information on the mechanisms of reactions. As we saw in Chapter 11, the rate of a reaction is quoted in units of mol dm s and shows how the concentration of a reactant varies with time. It is the mechanism of a reaction which determines its order n that appears in the general equation 

---

This expression corresponds to the Arrhenius equation with an exponential dependence on the tlireshold energy and the temperature T. The factor in front of the exponential function contains the collision cross section and implicitly also the mean velocity of the electrons. 

Using splitting schemes of the exponential function allows for a generation of numerical integrators. For example [24, 22] ... 

The exponential function with base b can also be defined as the inverse of the logarithmic function. The most common exponential function in applications corresponds to choosing Z the transcendental number e. 

This is the Wilson-Frenkel rate. With that rate an individual kink moves along a step by adsorbing more atoms from the vapour phase than desorbing. The growth rate of the step is then simply obtained as a multiple of Zd vF and the kink density. For small A/i the exponential function can be hnearized so that the step on a crystal surface follows a linear growth law... 

Next consider the exponential function, which is important in Idnetics. Let F(t) =... 

In this equation, a is a constant determining the size (radial extent) of the function. The exponential function is multiplied by powers (possibly 0) of x, y, and z, and a constant for normalization so that the integral of over all space is 1 (note that therefore c must also be a funrtion of a). 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

The precision of the rate constants as a function of temperature determines the standard deviations of the activation parameters. The absolute error, not the percentage error in the activation parameters, represents the agreement to the model, because of the exponential functions. If, for example, one wished to examine the values of AS for two reactions that were reported as -4 3 and 26 3 J mol 1K 1, then it should be concluded that the two are known to the same accuracy. Since AS and A// are correlated parameters, the uncertainty in AS will be about 1/Tav times that in A//. At ambient temperature this amounts to an approximate factor of three (that is, 1000/T, converting from joules for AS to kilojoules for A// ). Thus, the uncertainty in A//, 0 of 2.50 kJ mol 1 is consistent with the uncertainty in ASn of 7.21 J mol1 K-1 at Tav - 350 K. 

In addition to the chemical inferences that can be drawn from the values of AS and AH, considered in Section 7.6, the activation parameters provide a reliable means of storing and retrieving the kinetic data. With them one can easily interpolate a rate constant at any intermediate temperature. And, with some risk, rate constants outside the experimental range can be calculated as well, although the assumption of temperature-independent activation parameters must be kept in mind. For archival purposes, values of AS and AH should be given to more places than might seem warranted so as to avoid roundoff error when the exponential functions are used to reconstruct the rate constants. 

Equation (38) is solved and gives the amplitude tf in Eq. (40a). Disregarding the time dependence of the other parts of the amplitude in comparison with that of the exponential function, the actual form is given as follows,... 

What does this equation tell us This wavefunction also falls exponentially toward zero as r increases. Notice, though, that there is a factor r that multiplies the exponential function, so vjr is zero at the nucleus (at r = 0) as well as far away from it. We discuss the angular dependence shortly. 

In this specific case, the predictive power of the polynomial (see Fig. 3.9) and the exponential function are about equal in the x-interval of interest. The peak height corresponding to an unknown sample amount would be... 

B. THE EXPONENTIAL FUNCTION WITH AN EQUIDISTANT GRID For the example... 

D. THE EXPONENTIAL FUNCTION WITH A LOGARITHMICALLY EQUIDISTANT GRID... 

The optimum interval length goes as l/v nand the error as exp (—7T /n). This is certainly a much faster convergence than for the choice of an equidistant grid for the exponential function as studied in appendix B. 

This relation can be easily verified by employing the Euler formula c = cos 6 i sin 0 for the pure imaginary part of the exponential function and by observing the definite integrals ... 

In order to introduce a time-dependence of the Hamiltonian in Eq. (92), use is made of the exponential functions and of Eq. (93) such that we obtain ... 

This property can be derived by means of a series expansion of the exponential function in eq. (39.43) and by neglecting higher order terms in 0. 

To conclude this section on two-compartment models we note that the hybrid constants a and p in the exponential function are eigenvalues of the matrix of coefficients of the system of linear differential equations ... 

The exponential functions, the weighted sums of which determine the time courses in the various departments, can thus be regarded as eigenfunctions of the phar-... 

In the general case there will be n roots which are the eigenvalues of the transfer matrix K. Each of the eigenvalues defines a particular phase of the time course of the contents in the n compartments of the model. The eigenvalues are the hybrid transfer constants which appear in the exponents of the exponential function. For example, for the ith compartment we obtain the general solution ... 

These data can be integrated numerically between times 0 and 120 minutes. The remaining part between 120 minutes and infinity must be extrapolated from the downslope of the curve (P-phase) which can be modelled by means of the exponential function ... 

For thin absorbers with t exponential function in the transmission integral can be developed in a series, the first two terms of which can be solved yielding the following expression for the count rate in the detector ... 

In addition, the time dependence of the solution, meaning the exponential function, arises from the left hand side of Eq. (2-2), the linear differential operator. In fact, we may recall that the left hand side of (2-2) gives rise to the so-called characteristic equation (or characteristic polynomial). 

The exponential function is still based on the root s = -a, but the actual time dependence will decay slower because of the (oc2t +. ..) terms. 

The real part of a complex pole in (3-19) is -Zjx, meaning that the exponential function forcing the oscillation to decay to zero is e- x as in Eq. (3-23). If we draw an analogy to a first order transfer function, the time constant of an underdamped second order function is x/t,. Thus to settle within 5% of the final value, we can choose the settling time as 1... 

Many classical control techniques are developed to work only with polynomials in s, and we need some way to tackle the exponential function. 

To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called Pade approximation, which puts the function as a ratio of two polynomials. The simplest is the first order (1/1) Pade approximation ... 

We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in Eq. (3-30) and match the terms to a Taylor s expansion of the exponential function. If we do, we ll find that the (1/1) Pade approximation is equivalent to a third order Taylor... 

There are several things that we want to take note of. First, the exponential function is dependent only on x, or in other words, the pole at -1/x. Second, with Eq. (3-49), the actual time response depends on whether x < xz, x > xz, or xz < 0 (Fig. 3.5). Third, when x = xz, the time response is just a horizontal line aty = 1, corresponding to the input x = u(t). This is also obvious from (3-47) which becomes just Y = X. When a zero equals to a pole, we have what is called a... 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

One idea (not that we really do that) is to apply the Taylor series expansion on the exponential function of A, and evaluate the state transition matrix with... 

Instead of an infinite series, we can derive a closed form expression for the exponential function. For an n x n matrix A, we have... 

We again take that we can expand the exponential function as in Eq. (9-5). Thus we have... 

---