Pure imaginary number

If the roots are pure imaginary numbers, the form of the response is purely oscillatory, and the magnitude will neither increase nor decay. The response, thus, remains in the neighbourhood of the steady-state solution and forms stable oscillations or limit cycles. 

The s = jco substitution in the integrating function leads to a pure imaginary number ... 

We know the system is stable if all the roots of the characteristic equation are in the LHP and unstable if any of the roots are in the RHP. Therefore the imaginary axis represents the stability boundary. On the imaginary axis s is equal to some pure imaginary number s = ioi. 

With the phase choices made for the basis functions in the present work, we obtain the real, positive line strength in equation (21) in the form Iwl, where s is real, that is, as the module square of a purely imaginary number. 

G[ico) is a pure imaginary number (its real part is zero), lying on the imaginary axis. It starts at minus infinity when o) is zero and goes to the origin as w — The Nyquist... 

C+ contour. On the contour the variable s is a pure imaginary number. Thus, s = io) as (o goes from 0 to +oc. Substituting io) for in the total openloop system transfer function gives... 

If ao < 0, at least one of the roots will be positive, and the solution will diverge for t 00. If 3i = 0 and ao > 0, the roots will be purely imaginary numbers, with oscillatory solutions for xandj. Thus, the necessary and sufficient conditions for stability (i.e., xa and T return to the steady state after removal of the perturbation or vand/ 0 as oo) are equations (10.4.2-8) and (10.4.2-9). In terms of the physical variables those equations can be written as follows ... 

From the definitions it is easy to notice that Cf XfH) and d/(t//2) are diagonal matrices of pure imaginary numbers since Xf= t on the contrary Cj(T,/2) and d,(T,/2) have a real part that is always positive since Xi=—t — iji, and therefore the integrand always vanishes in the limit of integration. 

A complex number is a number composed of a real part x and an imaginary part iy, where i = and normally is represented by the equation z — x + iy. A real number, then, is one in which y = 0, while a pure imaginary number is one in which x = 0. Thus, in a sense, all numbers can be thought of as complex numbers. 

---