Thompson - Computing for Scientists and Engineers, страница 9

Derive them yourself.Exercise 2.12Use Euler’s theorem to show that(2.44)which gives the values for successive rotations in the complex plane by r/2.Compare these results with Figure 2.1. nThe exponential formis generally much more symmetric and therefore iseasier to handle analytically than are the cosine and sine functions, with their awkward function changes and sign changes upon differentiation compared with thesimplicity of differentiating the complex exponential. This simplicity is very powerful when used in discussing the solution of differential equations in Chapter 8, andalso in deriving the fast Fourier transform (FFT) algorithm in Chapter 9.3.An interesting application of the complex-exponential function that anticipates itsuse in the FFT algorithm is made in the following exercise.Exercise 2.13Consider the distributive and recurrence properties of the complex-exponentialfunction defined by(2.45)(a) Prove the following properties of powers of E(N):34COMPLEX VARIABLES(2.46)(2.47)for any a, b, and for any p 0.(b) Using these results, show that if N = 2V, where v is a positive integer,then, no matter how many integer powers of E(N) are required, only one evaluation of this complex-exponential function is required.

nAs a further topic in our review of functions of complex variables, let us consider the hyperbolic and circular functions.Hyperbolic functions and their circular analogsExponential functions with complex arguments are required when studying the solutions of differential equations in Chapters 7 and 8. A frequently occurring combination is made from exponentially damped and exponentially increasing functions.This leads to the definition of hyperbolic functions, as follows.The hyperbolic cosine, called “cosh,” is defined by(2.48)while the hyperbolic sine, pronounced “sinsh,” is defined by(2.49)If u is real, then the hyperbolic functions are real-valued.

The name “hyperbolic”comes from noting the identity(2.50)in which, if u describes the x and y coordinates parametrically by(2.5 1)then an x - y plot is a rectangular hyperbola with lines at /4 to the x and y axesas asymptotes.Exercise 2.14(a) Noting the theorem of Pythagoras,(2.52)2.3 FUNCTIONS OF COMPLEX VARIABLES35for any (complex) u, as proved in Exercise 2.11, explain why the cosine andsine functions are called “circular” functions.(b) Derive the following relations between hyperbolic and circular functions(2.53)a n d(2.54)valid for any complex-valued u. nThese two equations may be used to provide a general rule relating signs in identitiesfor hyperbolic functions to identities for circular functions:An algebraic identity for hyperbolic functions is the same as that for circularfunctions, except that in the former the product (or implied product) of two sinhfunctions has the opposite sign to that for two sin functions.For example, given the identity for the circular functions(2.55)we immediately have the identity for the hyperbolic functions(2.56)Exercise 2.15Provide a brief general proof of the hyperbolic-circular rule stated above.

nNote that derivatives, and therefore integrals, of hyperbolic and circular functions donot satisfy the above general rule. The derivatives of the hyperbolic functions aregiven by(2.57)and by(2.58)in both of which the real argument, u, is in radians. There is no sign change on differentiating the hyperbolic cosine, unlike the analogous result for the circular cosine.36COMPLEX VARIABLESThe differential equations satisfied by the circular and hyperbolic functions alsodiffer by signs, since the cosine and sine are solutions of(2.59)which has oscillatory solutions, whereas the cosh and sinh are solutions of(2.60)which has solutions exponentially increasing or exponentially decreasing.Exercise 2.16(a) Prove the two derivative relations (2.57) and (2.58) by starting with the defining equations for the cosh and sinh.(b) Use the relations between hyperbolic and circular functions, (2.53) and(2.54), to compute the derivatives of the hyperbolic functions in terms of thosefor the circular functions.(c) Verify the appropriateness of the circular and hyperbolic functions as solutions of the differential equations (2.59) and (2.60), respectively.

nTo complete the analogy with the circular functions, one also defines the hyperbolic tangent, called “tansh,” by(2.6 1)which is analogous to the circular function, the tangent, defined by(2.62)Among these six hyperbolic and circular functions, for real arguments there arethree that are bounded by ±1 (sin, cos, tanh) and three that are unbounded (sinh,cosh, tan). Therefore we show them in a pair of figures, Figures 2.3 and 2.4, withappropriate scales.By displaying the bounded hyperbolic tangent on the same scale as the sine function in Figure 2.3, we notice an interesting fact- these two functions are equal towithin 10% for | x | < 2, so may often be used nearly interchangeably.

The explanation for their agreement is given in Section 3.2, where their Maclaurin series arepresented. Figure 2.4 shows a similar near-coincidence of the cosh and sinh functions for x > 1.5, where they agree to better than 10% and the agreement improvesas x increases because they both tend to the exponential function.2.3FUNCTIONS OF COMPLEX VARIABLES37FIGURE 2.3 Bounded circular and hyperbolic functions, sine, cosine, and hyperbolic tangent.FIGURE 2.4 The unbounded circular and hyperbolic functions, tangent, hyperbolic cosine, andhyperbolic sine.

For x greater than about 1.5, the latter two functions are indistinguishable on thescale of this figure.The tangent function is undefined in the limit that the cosine function in the denominator of its definition (2.62) tends to zero. For example, in Figure 2.4 valuesof the argument of the tangent function within about 0.1 of x = ± /2 have beenomitted.38COMPLEX VARIABLESTrajectories in the complex planeAnother interesting concept and visualization method for complex quantities is thatof the trajectory in the complex plane. It is best introduced by analogy with particletrajectories in two space dimensions, as we now summarize.When studying motion in a real plane one often displays the path of the motion,called the trajectory, by plotting the coordinates x (t) and y (t), with time t beingthe parameter labeling points on the trajectory. For example, suppose thatx (t) = A cos ( t) and y (t) = B sin ( ), with A and B positive, then the trajectory is an ellipse with axes A and B, and it is symmetric about the origin of the X- ycoordinates.

As t increases from zero, x initially decreases and y initially increases. One may indicate this by labeling the trajectory to indicate the direction of increasing t. The intricate Lissajous figures in mechanics, obtained by superpositionof harmonic motions, provide a more-involved example of trajectories.Analogously to kinematic trajectories, in the complex plane real and imaginaryparts of a complex-valued function of a parameter may be displayed. For example,in Section 8.1 we discuss the motion of damped harmonic oscillators in terms of areal dimensionless damping parameter Expressed in polar-coordinate form, theamplitude of oscillation is(2.63)where the complex “frequency” (if x represents time) is given by(2.64)The trajectory of v depends on the range ofsquare root in (2.64).Exercise 2.17(a) Show that ifand on the sign associated with the1, which gives rise to damped oscillatory motion, then(2.65)and that the trajectory of v+ is a counterclockwise semicircle in the upper halfplane, while the trajectory of v- is a clockwise semicircle in the lower halfplane.

In both trajectories is given as increasing from -1 to +l.(b) Suppose that >1, which produces exponential decay called overdampedmotion. Show that v± is then purely real and negative, so the trajectory liesalong the real axis. Show that v- increases from -1 toward the origin as increases, while v+ decreases toward asincreases. nThe complex-plane trajectory, withas displayed in Figure 2.5.as parameter, expressed by (2.64) is therefore2.3 FUNCTIONS OF COMPLEX VARIABLES39FIGURE 2.5 Frequency trajectory in the complex plane according to (2.64) as a function of thedamping parameterAs a final note on this example, there is no acceptable solution for v- if < - 1and if x > 0 is considered in (2.63), since y (x) is then divergent.Another interesting example of a trajectory in the complex plane arises in theproblem of forced oscillations (Section 10.2) in the approximation that the energydependence is given by the Lorentzian(2.66)where c is a proportionality constant and the complex Lorentzian amplitudes L± aregiven by(2.67)where andare dimensionless frequency and damping parameters.