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9998. 45) where a ≤ 1 and b = 1 − a 2 . 24), and the starting points are slightly different from each other. 1650). 4. 25 but different starting points. 1650). 1 Demonstration Different orbits for Hénon’s model can be plotted if different starting points are randomly chosen. 24 case, with random initial conditions. 5. 25 but random starting points. 9 Generation of Special Functions from Their Recursion Relations* In this section, we go back to more classical mathematics. We consider the case of the special functions of mathematical physics.

In the terminology of the circuit engineer, the voltage source VS is called the input to the circuit, and the current I and the voltage V are called the circuit outputs. Thus, this is an example of a system with one input and two outputs. As you may have studied in high school physics courses, all of circuit analysis with resistors as elements can be accomplished using Kirchhoff’s current law, Kirchoff’s voltage law, and Ohm’s law. • Kirchoff’s voltage law: The sum of all voltage drops around a closed loop is balanced by the sum of all voltage sources around the same loop.

Usually, the infinite sum is reduced to a finite sum because the inputs with negative indexes are usually assumed to be zeros. 29) where, of course, n is the order of the difference equation. 4. However, the most powerful technique to directly solve the linear difference equation with constant coefficients is, as pointed out earlier, the z-transform technique. Each of the above formulations of the input-output problem has distinct advantages in different circumstances. The direct difference equation formulation is the most amenable to numerical computations because of lower computer memory requirements, while the convolution-summation technique has the advantage of being suitable for developing mathematical proofs and finding general features for the difference equation.