Real part of a complex

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

If the real part of a complex pole is zero, then p = coj. We have a purely sinusoidal behavior with frequency co. If the pole is zero, it is at the origin and corresponds to the integrator 1/s. In time domain, we d have a constant, or a step function. 

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... 

In physics it is often convenient to represent measurable quantities (which are real) by the real part of a complex function, m(t), as in f(t) = 3 In such cases the Fourier transform of the real function is... 

An acoustic wave is a traveling periodic pressure disturbance. This wave travels at a speed c dependent on the properties of the medium and the type of motion associated with the wave. The periodic nature of the acoustic wave is (for present purposes) taken to be a sinusoidal oscillation occurring at a frequency f. At any location x and instant in time t, the pressure associated with this traveling wave can be expressed as a cosine wave, or in a mathematically equivalent form as the real part of a complex exponential ... 

The set of real numbers The rt-dimensional vector space Real part of a complex number Sample-to-detector distance Guinier radius (i.e. radius of gyration)... 

Similarly, a cosinusoidal oscillation can be represented by the real part of a complex function ... 

What is the position and width of an investigated resonance In order to answer this fundamental question, one can use complex coordinates within computational methods that had been originally developed for bound states. The real part of a complex energy, e, that one obtains for the resonance, constitutes its position, i.e., the actual real energy. The imaginary part gives the width of the resonance, P = —2/m(e), so it determines the lifetime of the state, T = p. 

The real part of a complex number is denoted by x = 8fz and the imaginary part by y = Sz. The complex conjugate z (written as z in some books) is the number obtained by changing i to —i ... 

With the surface of section technique, it can be observed that a limit cycle can undergo, for example, a Hopf bifurcation. When the stability analysis of the Poincare section is carried out, the real part of a complex conjugate pair of eigenvalues is seen to pass from negative to positive a small perturbation added to the limit cycle will evolve away from the cycle in an oscillatory fashion. This type of bifurcation results, then, in the appearance of a second... 

Re Reynolds number or real part of a complex function R(q) Fluctuation function f Reduced drop radius... 

The Kramers-Kronig relations allow raie to calculate the real part of a complex function from the imaginary part and the imaginary part from the real part mily. They were initially applied to electrical impedances by Bode [558] and further discussed and applied to EIS [559-568]. To satisfy Kramers-Kronig relations, the complex function must satisfy four criteria, as follows [223, 559-561] ... 

Eq. (14.3) is the iterative equivalent of the TDSE. It does not contain the imaginary number i. If x, t) is just the real part of a complex wavefunction, then the equation will propagate forward, completely exactly, this real part of the complete wavefunction. The advantage of this is both one of storage and of computer time. Only half the computer storage is required for the real part of a wavefunction as compared for the complete complex wavefunction. The multiplication of two complex numbers requires four times as many operations as the multiplication of two real numbers. 

Note that the Cauchy-Riemann equations imply that, if the real part of a complex function is known, its imaginary part is determined to within a constant. This point is discussed later, from another viewpoint. 

The above results can be obtained by using complex notation and expressing the actual strain as the real part of a complex strain defined by... 

Let us consider a complex number plane. On the abscissa we shall put the real part of a complex number and on the ordinates its imaginary part (Figure 2.6). Then any complex number can be written as... 

---