1.3 Vectors and Polar Forms 01 人91-b-- Figure 1.14 Crank-and-piston linkage For engineering analysis it is important to be able to relate the crankshaft's angular coordinates-position,velocity,and acceleration ordinates for the piston.Although this calculation -to the corresponding linear co- can be carried oul using vector analysis,the following complex variable technique is more"automatic." Let the crankshaft pivot lie at the origin of the coordinate system,and let z be the complex number giving the location of the base of the piston rod,as depicted in Fig.1.14, z=1+id, where/gives the pision's(linear)excursion and d is a fixed offset.The crankarm is described by A- (cos+isin)and the connecting arm by B=b(cos+ isin ).(0 is negativein Fig.1.14).Exploit the obvious identity A+B==1id to derive the expression relating the piston position to the crankshaft angle: 1=acas4+bcossin(-am） b 29.Suppose the mechanism in Prob.28 has the dimensions a=0.1m,b=0.2m,d=0.1m and the crankshaft rotates at a uniform velocity of 2 rad/s.Compute the position and velocity of the piston when=. 30.For the linkage illustrated in Fig.1.15,use complex variables to outline a scheme for expressing the angular position,velocity.and acceleration of arm c in terms of those of arin a.(You needn't work out the equations.). b Figure 1.15 Linkage in Prob.30
Complex Numbers 26 1.4 The Complex Exponential The familiar exponential function f()=e has a natural and extremely useful ex- tension to thempepne.Indeed efuctionprovidabasic propagation,and iime-invariant physical systems in general. To find a suitable definition for e when =+iy,we want to preserve the basic identities satisfed by the real,first of all,we postulate that the multiplicative property should persist: e24e2=e21t22 (1) This simplifies matters considerably,since Eq.(1)enables the decomposition ei=extiy =etely (2) and wesee that to definee we need ony specify e(in other words we will be able to exponentiate complex numbers once we discover how to exponentiate pure iinaginary ones). Next we propose that the differentiation law dez (3) be preserved.Differentiation with respect to a complex variable=iy is a very profound and.at this stage,ambiguo operation;indeed Chapter 2 is devoted to a painstaking study of this onp(dheof th book is sequences).But thanks to the factorization displayed in Eq.(2)we need only consider(for the moment)a special case of Eq.(3)-namely, deiy di列se or,equivalently(by the chain rule), deiy sfe4 (4) The consequences of postulating Eq.(4)become more apparent if we differentiate again d2eiy dyd lier d =P2ely =-ey:
1.4 The Complex Exponentlal 27 in other words,the function g(y):=ey satisfies the differential equation 导 (5) Now observe that any function of the form A cosy+B siny (A.B constants) satisfiesEq.(5).In fact,from the theory of differential equations it is known that every solution of Eq.(5)must have this form.Hence we can write 8(y)=A cosy+B sin y. (6) To evaluate A and B we use the conditions that 8(0)=ei0=e=1=A cos0+B sin0 and 票0=g商-1=-Asn0+Bm0 Thus A=I and B=i,leading us to the identification ely cosy+i siny. Equation (7)is known as Euler's equation.Combining Eqs.(T)and(2)we formulate the following. Definition 5.If=x+y,then e is defined to be the complex number e :=e*(cosy +isin y). (⑧) It is not difficult to verify directly that e,as defined above satisfies the usual al gebraicpropertiesofthe,exponcntialfunction in particular,the multiplicative iden tity (I)and the associated division rule ee2 (⑨ (see Prob.15a).In Sec.2.5 we will obtain further confirmation that we have made the right choice"by showing that Definition 5 produces a function that has the extremely desirable property of analyricity.Another confirmation is exhibited in the following example. Leonhard Euler (1707-1783)
Complex Numbers 28 Example 1 Show that Euler's equation is formally consistent with the usual Taylor series expan sions 。=1+x+元+引+项++… cosx=1-+年 nx=x-3+ Solntion. We shall study series representations of complex functionsin full de- tail in Chapter 5.For now we ignore questions of convergence,etc.,and simply sub- stitutex=iy into the exponential series: =+w+婴+警+++… 2! 3! 2 y3 =cosy+isiny.■ Euler's equation (7)enables us to write the polar form (Sec.1.3)of a complex z=rciso=r(cos0+i sine)=rel0 Thus we can (and do)drop the awkward"cis"artifice and use,as the standard polar representation, 2=reio lzle arBz. (10) In paricular.notice the following identities: foaieai edmi =emt=1. el=/2)=i, et-12)i=i,eri =-1. (Students of mathematics,including Euler himself,have often marveled at the last identity.The constant e comes from calculus,comes from geometry,and i comes from algebraand the combination e gives-1,the basic unit for generating the arithmetic system from the c unting numbers,or cardinals!) Observe also thatand that Euler'squation leads to the following representations of the customary trigonometric functions: cos0-Re efo efe1 2 (1) sina=Im afo foa-0 (12)
1.4 The Complex Exponentlal 29 The rules derived in Sec.1.3 for multiplying and dividing complex numbers in polar form now find very natural expressions: z12=（e）(2e）=0zea+ (13) 器-（月） (14) and complex conjugation ofz=re is accomplished by changing the sign ofi in the exponent: 乏=re-i9 (15) Example 2 Compute(a)(1+i)/(3-i)and (b)(1+i)24 Solution. (a)This quotient was evaluated using the cis operator in Example 1.11 of Sec.1.3:using the exponential the calculations take the form 1+i=2cis(r/4)=3-i=2cis(-n/6)=2e-im/6 and,therefore, 1i eiMi12 5-7=20-5=2 (b)The exponential forms become (1+i)24=(W2em424=(W224e2*/4=-22e6a=2.☐ y把脚2P品Aryiel men more formula involving trigonometric functions.which we describe in the next example. Example3 Prove De Moivre's forula: (cos0+i sin0)"cosn0+isinne,n 1,2,3.... (16 Solution.By the multiplicative property,Eq.(1), (e0)=ge…e0=e9+i94t0=e0. (n times) Now applying Euler's formula(7)to the first and last members of this equation string, we deduce(I6).■ De Moivre's formula can be a convenient tool for deducing multiple-angle trigono- metric identities,as is illustrated by the following example.(See also Probs.12 and 20.) Published by Abraham De Moivre n 1707