Answers to Final Exam试卷答案 2017年1月4日22：17 Prid.Sime Cosze+ismio=eiv=(eio) (5) and (eie)’=(n0+9.o)3 (5) -(c'0-30203in20)+i(30-0smo-sin'o) Copuing the imginnry port it folimes thu cnlasin 2202er Solition. St人(se)=X),hr人(x+)=s2Xw-2. ad(62+9)X)-2=3-. (3) Hteme 2 X6)=9年1+5-07) 2 二1 。一6。+ s-3i (4) =品-而。，+古 Therefore =3(X)=25mt-6mt+6et 3) The aniwer io not unighe. Key prins. a2+5 1)f(3)should havt thi form ao C2+d (3) 2)9(-i)=0 (3) 分区教学的第1页
(10%) Using Euler’s equation together with the binomial formula to derive the identity: , where is real. 1. 2. (10%) Use the Laplace transform to solve the IVP 3. (10%) Find a Möbius transformation mapping the unit disc onto the right half-plane and taking to the origin. Answers to Final Exam 试卷答案 2017年1月4日 22:17 分区 教学 的第 1 页
2)P(-i)=0 (3) 3)寸1z1=I,thin Refia)=o. (2') 4)Re fo) (2) (Find he Len ens h Solution （2) (2) 2a-（u+2)e-的(+4+2-） 2+ (21 I+hold5m0<l2-41<4},2 5.(1)Find and cf the isolated singularity ofinthemn Solutim.o ond o are islitid lrities n驶-合云(s注-3 (3') -三高”) (3 -层a frr au lelo 0 cmovable3 arit西. (2) amd肉为mwu分nln巧：(2) 6.(1%)Evaluatesdz Sim时e克n专t-2iRe(em.ou) 1= 分区散学的第2页
4. (10%) Find the Laurent series for at the punctured neighborhood of z=4, and determine the annulus of convergence 5. (10%) Find and classify the isolated singularity of in the extended complex plane. 6. (10%) Evaluate 分区 教学 的第 2 页
and 2庭5w2=(1+之*22t…）(位-实2+…）》 (4') Hence I=2ht.I=2πd (2) .0t2wt比 (3) Ifw>0,t认n ') a: 4元 =Te如竖w. (2) i.t.()=-ine 。4+1 tt-e 分区教学的第3页
7. (10%) Find the Fourier transform of . 分区 教学 的第 3 页
。4小1 地七4+1 -Te-foow i〔-me0客”， 工以，（-°-meg 8.(1%)Find an analytie function satisfying Re f(z)=x+y-xy and f()=0 Solton.Set utx.y)=x+y-xy.un Vy=u×=1-. (3) It implie∞ ,-∫(-)山=y-兰+c) W C'∞=x=-4y=x-1,(3') it impli∽ cx)=空-×+c Hence fa)=u*iv=x4yy)+y-茎+登-x+c) =z+(登-2)+i0. (2 Snef)-0,tf%wn)=2+-2).(2) e6ee列年加e出仁置 (1(1)Using the chain rule fomally,show tha p叶“路=器+的器 aw2的 iu 2 分区散学的第4页
8. (10%) Find an analytic function f(z) satisfying and . 9. Set the function by introducing the change of variables . (1) (10%) Using the chain rule formally, show that . (2) (10%) Show that =0, if "is analytic function, i.e. “ " is analytic function” implies “f is independent of ”. 分区 教学 的第 4 页
z-4zy→z =之改-测 2 器兴斧+贺是 3 之兴+主别 2 法-w…)=止祭+的)+器》 4 =兴器)+（黄+器）-0 2-3(u-o）=-。 21 分区学的第5页
分区 教学 的第 5 页