EEECCC111?7H?n?Tao Zhou(HHH?nnn?)111?1 / 32?Outlineznezn?sinz, cosz, V-sinhz, coshz,wAC3E?2XAC2?E?)5?ECAk?5Tao Zhou(HHH?nnn?)111?2 / 32?Outlineznezn?sinz, cosz, V-sinhz, coshz,wAC3E?2XAC2?E?)5?ECAk?5Tao Zhou(HHH?nnn?)111?2 / 32?Outlineznezn?sinz, cosz, V-sinhz, coshz,wAC3E?2XAC2?E?)5?ECAk?5Tao Zhou(HHH?nnn?)111?2 / 32?Outlineznezn?sinz, cosz, V-sinhz, coshz,wAC3E?2XAC2?E?)5?ECAk?5Tao Zhou(HHH?nnn?)111?2 / 32?znF?n = 0,1,2,zn3?)?n = 1,2,z = :?n = 1,2,3,znz = 0?)3z = )(zn)0= nzn1ddknP(z)Q(z)KKK00?Tao Zhou(HHH?nnn?)111?3 / 32?znF?n = 0,1,2,zn3?)?n = 1,2,z = :?n = 1,2,3,znz = 0?)3z = )(zn)0= nzn1ddknP(z)Q(z)KKK00?Tao Zhou(HHH?nnn?)111?3 / 32?znF?n = 0,1,2,zn3?)?n = 1,2,z = :?n = 1,2,3,znz = 0?)3z = )(zn)0= nzn1ddknP(z)Q(z)KKK00?Tao Zhou(HHH?nnn?)111?3 / 32?znF?n = 0,1,2,zn3?)?n = 1,2,z = :?n = 1,2,3,znz = 0?)3z = )(zn)0= nzn1ddknP(z)Q(z)KKK00?Tao Zhou(HHH?nnn?)111?3 / 32?znF?n = 0,1,2,zn3?)?n = 1,2,z = :?n = 1,2,3,znz = 0?)3z = )(zn)0= nzn1ddknP(z)Q(z)KKK00?Tao Zhou(HHH?nnn?)111?3 / 32?znF?n = 0,1,2,zn3?)?n = 1,2,z = :?n = 1,2,3,znz = 0?)3z = )(zn)0= nzn1ddknP(z)Q(z)KKK00?Tao Zhou(HHH?nnn?)111?3 / 32?ezDefinitionez= ex+iy ex eiy= ex(cosy + i siny)00.511.52x051015y-50500.511.5x00.511.52x051015y-50500.511.5xTao Zhou(HHH?nnn?)111?4 / 32?555/?u0$KuEE,ez1 ez2= ex1+iy1 ex2+iy2= ex1 ex2 eiy1 eiy2= ex1+x2 ei(y1+y2)= e(x1+x2)+i(y1+y2)= ez1+z2ez3?)(ez)0= ezTao Zhou(HHH?nnn?)111?5 / 32?555/?u0$KuEE,ez1 ez2= ex1+iy1 ex2+iy2= ex1 ex2 eiy1 eiy2= ex1+x2 ei(y1+y2)= e(x1+x2)+i(y1+y2)= ez1+z2ez3?)(ez)0= ezTao Zhou(HHH?nnn?)111?5 / 32?555ez3?:X?z?!KJu ez%C?z = ez?:E?Ak552 iez+2 i= ex+i(y+2)= excos(y + 2) + isin(y + 2)= excosy + isiny = ex+iy= ezXez= a(a 6= 0)kTao Zhou(HHH?nnn?)111?6 / 32?555ez3?:X?z?!KJu ez%C?z = ez?:E?Ak552 iez+2 i= ex+i(y+2)= excos(y + 2) + isin(y + 2)= excosy + isiny = ex+iy= ezXez= a(a 6= 0)kTao Zhou(HHH?nnn?)111?6 / 32?555ez3?:X?z?!KJu ez%C?z = ez?:E?Ak552 iez+2 i= ex+i(y+2)= excos(y + 2) + isin(y + 2)= excosy + isiny = ex+iy= ezXez= a(a 6= 0)kTao Zhou(HHH?nnn?)111?6 / 32?nnn?sinz, cosz,En?sinz, coszEeiz= cosz + i sinzeiz= cosz i sinzsinz =eiz eiz2i,cosz =eiz+ eiz2Tao Zhou(HHH?nnn?)111?7 / 32?nnn?sinz, cosz,En?sinz, coszEeiz= cosz + i sinzeiz= cosz i sinzLf?sinz =eiz eiz2icosz =eiz+ eiz2sinz =eiz eiz2i,cosz =eiz+ eiz2Tao Zhou(HHH?nnn?)111?7 / 32?nnn?sinz, cosz,En?sinz, coszEsinz =eiz eiz2i,cosz =eiz+ eiz2Tao Zhou(HHH?nnn?)111?7 / 32?nnn?sinz, cosz,En?sinz, coszEsinz =eiz eiz2i,cosz =eiz+ eiz2eiz= eixy= eycosx + ieysinxeiz= eix+y= eycosx ieysinxTao Zhou(HHH?nnn?)111?7 / 32?nnn?sinz, cosz,En?sinz, coszEsinz =eiz eiz2i,cosz =eiz+ eiz2eiz= eixy= eycosx + ieysinxeiz= eix+y= eycosx ieysinxcosz =ey+ ey2cosx ey ey2i sinx= coshy cosx i sinhy sinxTao Zhou(HHH?nnn?)111?7 / 32?nnn?sinz, cosz,cosz = coshy cosx i sinhy sinx051015x00.511.52y-202051015x051015x00.511.52y-202051015xTao Zhou(HHH?nnn?)111?8 / 32?nnn?555sinz, cosz3?)(sinz)0= cosz,(cosz)0= sinzz = ?:sinzcosz2sinzcosz?u1cosi =e1+ e12= 1.5430806isini =e1 e12= 1.1752011Tao Zhou(HHH?nnn?)111?9 / 32?nnn?555sinz, cosz3?)(sinz)0= cosz,(cosz)0= sinzz = ?:sinzcosz2sinzcosz?u1cosi =e1+ e12= 1.5430806isini =e1 e12= 1.1752011Tao Zhou(HHH?nnn?)111?9 / 32?nnn?555sinz, cosz3?)(sinz)0= cosz,(cosz)0= sinzz = ?:sinzcosz2sinzcosz?u1cosi =e1+ e12= 1.5430806isini =e1 e12= 1.1752011Tao Zhou(HHH?nnn?)111?9 / 32?nnn?555sinz, cosz3?)(sinz)0= cosz,(cosz)0= sinzz = ?:sinzcosz2sinzcosz?u1cosi =e1+ e12= 1.5430806isini =e1 e12= 1.1752011Tao Zhou(HHH?nnn?)111?9 / 32?nnn?555n?tanz,cotz,secz,csczsinzcosz /?tanz =sinzcosz,cotz =coszsinzsecz =1cosz,cscz =1sinzn?uEn?E,Xsin2z + cos2z = 1Tao Zhou(HHH?nnn?)111?10 / 32?nnn?555n?tanz,cotz,secz,csczsinzcosz /?tanz =sinzcosz,cotz =coszsinzsecz =1cosz,cscz =1sinzn?uEn?E,Xsin2z + cos2z = 1Tao Zhou(HHH?nnn?)111?10 / 32?VVV-sinhz,coshz,V-sinhz,coshzLE?sinhz =ez ez2coshz =ez+ ez2tanhz =sinhzcoshzcothz =coshzsinhzsechz =1coshzcschz =1sinhzV-n?pzd?KV-sinhz = isiniz,coshz = cosiz,tanhz = itanizTao Zhou(HHH?nnn?)111?11 / 32?VVV-sinhz,coshz,V-sinhz,coshzLE?sinhz =ez ez2coshz =ez+ ez2tanhz =sinhzcoshzcothz =coshzsinhzsechz =1coshzcschz =1sinhzV-n?pzd?KV-sinhz = isiniz,coshz = cosiz,tanhz = itanizTao Zhou(HHH?nnn?)111?11 / 32?VVV-?5555V-sinhz,coshz?2 i?(sinhz)0= coshz(coshz)0= sinhz(tanhz)0= sech2zTao Zhou(HHH?nnn?)111?12 / 32?VVV-?5555V-sinhz,coshz?2 i?(sinhz)0= coshz(coshz)0= sinhz(tanhz)0= sech2zTao Zhou(HHH?nnn?)111?12 / 32n?9A3)nk-/ SEC?J:Fz aDefinition (w =z)gCzv?w2= z?wz?z?z2= w=w = z2?5Ly3AugCzz?Tao Zhou(HHH?nnn?)111?13 / 32Fz aDefinition (w =z)gCzv?w2= z?wz?z?z2= w=w = z2?5Ly3AugCzz?Tao Zhou(HHH?nnn?)111?13 / 32Fz aDefinition (w =z)gCzv?w2= z?wz?z?z2= w=w = z2?5Ly3AugCzz?Tao Zhou(HHH?nnn?)111?13 / 32?/w?5y3cew =z a4ILw = ei,z a = rei =r, =2+ n,n = 0,1,2,u?zkwAw1(z) =rei/2?u?n = 0,2,w2(z) =rei(+/2)= rei/2?un = 1,3,Tao Zhou(HHH?nnn?)111?14 / 32w =z a4ILw = ei,z a = rei =r, =2+ n,n = 0,1,2,u?zkwAw1(z) =rei/2?u?n = 0,2,w2(z) =rei(+/2)= rei/2?un = 1,3,Tao Zhou(HHH?nnn?)111?14 / 32(w =z a?55?u?5O(5?umz a(?gCz)?55?LyKw?(rw =z a(L|w| =p|z a|,argw =12arg(z a)Tao Zhou(HHH?nnn?)111?15 / 32(w =z a?55?u?5O(5?umz a(?gCz)?55?LyKw?(rw =z a(L|w| =p|z a|,argw =12arg(z a)Tao Zhou(HHH?nnn?)111?15 / 32?:5z,:arg(z a)?z-YCzA?w?YCz?z4-(=g?4-)Cz?Uy(Ja:3w =z akA?/ ?z7a:=?z7a:=?A?Tao Zhou(HHH?nnn?)111?16 / 32?:a:3w =z akA?/ ?z7a:=?z7a:=?A?Tao Zhou(HHH?nnn?)111?16 / 32?:5z,:arg(z a)?z-YCzA?w?YCz?z4-(=g?4-)Cz?Uy(Ja:3w =z akA?/ ?z7a:=?z7a:=?A?Tao Zhou(HHH?nnn?)111?16 / 32?:a:w =z a?:z = w =z a?:XXX?:u5k(?mmm?uuu0?:,?LT:7?5?7L5?vvv?“?”(?SkT:U?:k:Tao Zhou(HHH?nnn?)111?17 / 32?:a:w =z a?:z = w =z a?:XXXnnn)777?:?”3v?Ekk?:XXX?:u5k(?mmm?uuu0?:,?LT:7?5?7L5?vvv?“?”(?SkT:U?:k:Tao Zhou(HHH?nnn?)111?17 / 32?:a:w =z a?:z = w =z a?:F ?S?X?z = aXXX?:u5k(?mmm?uuu0?:,?LT:7?5?7L5?vvv?“?”(?SkT:U?:k:Tao Zhou(HHH?nnn?)111?17 / 32?:a:w =z a?:z = w =z a?:XXX?:u5k(?mmm?uuu0?:,?LT:7?5?7L5?vvv?“?”(?SkT:U?:k:Tao Zhou(HHH?nnn?)111?17 / 32?:a:w =z a?:z = w =z a?:XXX?:u5k(?mmm?uuu0?:,?LT:7?5?7L5?vvv?“?”(?SkT:U?:k:Tao Zhou(HHH?nnn?)111?17 / 32?)57Lz?w =z 150 arg(z 1) 2w(2), w(i), w(0), w(i)argw =12arg(z 1)d0 arg(z 1) 2arg(z 1)?z=2= 0,w(2) = 1arg(z 1)?z=i=34,w(i) =42e3 i/8arg(z 1)?z=0= ,w(0) = e i/2= iarg(z 1)?z=i=54,w(i) =42e5 i/8Tao Zhou(HHH?nnn?)111?18 / 32?)57Lz?555mmmz a?CCCzzz?mz a?3,Sw =z a?/(?w /(X50 arg(z a) 22 arg(z a) 4?w =z 150 arg(z 1) 2w(2), w(i), w(0), w(i)argw =12arg(z 1)d0 arg(z 1) 2arg(z 1)?z=2= 0,w(2) = 1arg(z 1)?z=i=34,w(i) =42e3 i/8arg(z 1)?z=0= ,w(0) = e i/2= iarg(z 1)?z=i=54,w(i) =42e5 i/8Tao Zhou(HHH?nnn?)111?18 / 32?)57Lz?w =z 150 arg(z 1) 2w(2), w(i), w(0), w(i)argw =12arg(z 1)d0 arg(z 1) 2arg(z 1)?z=2= 0,w(2) = 1arg(z 1)?z=i=34,w(i) =42e3 i/8arg(z 1)?z=0= ,w(0) = e i/2= iarg(z 1)?z=i=54,w(i) =42e5 i/8Tao Zhou(HHH?nnn?)111?18 / 32?)57Lz?w =z 150 arg(z 1) 2w(2), w(i), w(0), w(i)argw =12arg(z 1)d0 arg(z 1) 2arg(z 1)?z=2= 0,w(2) = 1arg(z 1)?z=i=34,w(i) =42e3 i/8arg(z 1)?z=0= ,w(0) = e i/2= iarg(z 1)?z=i=54,w(i) =42e5 i/8Tao Zhou(HHH?nnn?)111?18 / 32?3?50 arg(z a) 2ew?30 argw =?3w =z agCzm3A?X平面平面34 arg(z a) 6, 6 arg(z a) 8, 2 arg(z a) 0, 4 arg(z a) 2,?5e-Ey?(JTao Zhou(HHH?nnn?)111?19 / 32?XJ52 arg(z a) 4K argw 2w3e平面平面34 arg(z a) 6, 6 arg(z a) 8, 2 arg(z a) 0, 4 arg(z a) 2,?5e-Ey?(JTao Zhou(HHH?nnn?)111?19 / 32?34 arg(z a) 6, 6 arg(z a) 8, 2 arg(z a) 0, 4 arg(z a) 2,?5e-Ey?(JTao Zhou(HHH?nnn?)111?19 / 32?l?w?5m?Czz?Cz?z?ow =z akOw?e0 arg(z a) 2 .0 argw 2 arg(z a) 4 / argw 2m?Cz?5?X arg(z a) arg(z a) 33/2 arg(za) /2/2 arg(za) 5/2Tao Zhou(HHH?nnn?)111?20 / 32?l?w?5m?Czz?Cz?z?ow =z akOw?e0 arg(z a) 2 .0 argw 2 arg(z a) 4 / argw 2m?Cz?5?X arg(z a) arg(z a) 33/2 arg(za) /2/2 arg(za) 5/2Tao Zhou(HHH?nnn?)111?20 / 32?l?w?5m?Czz?Cz?z?ow =z akOw?e0 arg(z a) 2 .0 argw 2 arg(z a) 4 / argw 2m?Cz?5?X arg(z a) arg(z a) 33/2 arg(za) /2/2 arg(za) 5/2Tao Zhou(HHH?nnn?)111?20 / 32?w =z akOw?e0 arg(z a) 2 .0 argw 2 arg(z a) 4 / argw 2m?Cz?5?X arg(z a) arg(z a) 33/2 arg(za) /2/2 arg(za) 5/2Tao Zhou(HHH?nnn?)111?20 / 32?yeZ($)z?Cz#Nz7:=?X3?fz?7z = a:=?u3z1ulz = a:mY?:XJ53Warg(z a) = 0.XJ53Warg(z a) = 2/5?w=?w?z ?Czdu(?:z = adz2U?7:=?(d7:=?#N?)Tao Zhou(HHH?nnn?)111?21 / 32?u3z1ulz = a:mY?:XJ53Warg(z a) = 0.XJ53Warg(z a) = 2/5?w=?w?z ?Czdu(?:z = adz2U?7:=?(d7:=?#N?)Tao Zhou(HHH?nnn?)111?21 / 32?u3z1ulz = a:mY?:XJ53Warg(z a) = 0.XJ53Warg(z a) = 2/5?w=?w?z ?Czdu(?:z = adz2U?7:=?(d7:=?#N?)Tao Zhou(HHH?nnn?)111?21 / 32?Q,?=?z?Cz?yUB7:=?d?K($E,?-xxx?KKK7Lu:u:z:?kxBuK?x?z?Cz(U?)?3x?E:?m?3?EzA?Tao Zhou(HHH?nnn?)111?22 / 32?Q,?=?z?Cz?yUB7:=?d?K($E,?-xxx?KKK7Lu:u:z:?kxBuK?x?z?Cz(U?)?3x?E:?m?3?EzA?Tao Zhou(HHH?nnn?)111?22 / 32?Example1 ?w =p(z a)(z b)?:!x?Cz)U?:(z a)(z b)?u0?:=z = a!z = bz = ,?n:?:N?z = az = b:?z = :I5?m?)?z?=z az bTao Zhou(HHH?nnn?)111?23 / 32?Example1 ?w =p(z a)(z b)?:!x?Cz)U?:(z a)(z b)?u0?:=z = a!z = bz = ,?n:?:N?z = az = b:?z = :I5?m?)?z?=z az bTao Zhou(HHH?nnn?)111?23 / 32?Example1 ?w =p(z a)(z b)?:!x?Cz)U?:(z a)(z b)?u0?:=z = a!z = bz = ,?n:?:N?z = az = b:?z = :I5?m?)?z?=z az bTao Zhou(HHH?nnn?)111?23 / 32?(1)(2)(3)(4)Tao Zhou(HHH?nnn?)111?24 / 32?x?:r?nnn?”?)5uk?z a?0=12z a?:Au=3:NC3Au?Tao Zhou(HHH?nnn?)111?25 / 32?x?:r?nnn?”?)5uk?z a?0=12z a?:Au=3:NC3Au?Tao Zhou(HHH?nnn?)111?25 / 32?y?:k?5?m?CzU5?E,?K?:,?(gCAX?555w333,:z0?(z?YYYCCCzzz?z-YYYCCCzzzwYYYCCCzzzTao Zhou(HHH?nnn?)111?26 / 32?y?:k?5?m?CzU5?E,?K?:,?(gCAX?555w333,:z0?(z?YYYCCCzzz?z-YYYCCCzzzwYYYCCCzzzTao Zhou(HHH?nnn?)111?26 / 32CCCzzz?2 Ew =z 15w(2) = 1?zC1C2YCz?:wC1C2z = 1?%!1?e?)?zC2?z = 0arg(z 1) = argw = 2w(0) = ei/2= iTao Zhou(HHH?nnn?)111?27 / 32CCCzzz?)w,?zC1?z = 0arg(z 1) = argw =12arg(z 1) =2w(0) = ei/2= i?zC2?z = 0arg(z 1) = argw = 2w(0) = ei/2= iTao Zhou(HHH?nnn?)111?27 / 32CCCzzz?)?zC2?z = 0arg(z 1) = argw = 2w(0) = ei/2= iTao Zhou(HHH?nnn?)111?27 / 32iii-z?Cz?l$?,?um?z?51?eW(arg(z 1) = 2)1?W(arg(z 1) = 2);1?W(arg(z 1) = 0)1?eW(arg(z 1) = 4)?“Riemann-Tao Zhou(HHH?nnn?)111?28 / 32iii-z?Cz?l$?,?um?z?51?eW(arg(z 1) = 2)1?W(arg(z 1) = 2);1?W(arg(z 1) = 0)1?eW(arg(z 1) = 4)?“Riemann-Tao Zhou(HHH?nnn?)111?28 / 32iii-?“Riemann-arg10z arg12z Tao Zhou(HHH?nnn?)111?28 / 32iii-?“Riemann-Tao Zhou(HHH?nnn?)111?28 / 32iii-?“Riemann-Ur?5U?)5Tao Zhou(HHH?nnn?)111?28 / 32Definitionw = lnz?gCz?vew= z?kww = lnz?w = ez?-w = u + ivz = rei?eu eiv= reiu = lnr = ln|z|,v = + 2n(n = 0,1,2,)rw = lnz(Lw = lnz = ln|z| + i( + 2n) = ln|z| + i argzTao Zhou(HHH?nnn?)111?29 / 32Definitionw = lnz?gCz?vew= z?kww = lnz?w = ez?-w = u + ivz = rei?eu eiv= reiu = lnr = ln|z|,v = + 2n(n = 0,1,2,)rw = lnz(Lw = lnz = ln|z| + i( + 2n) = ln|z| + i argzTao Zhou(HHH?nnn?)111?29 / 32Definitionw = lnz?gCz?vew= z?kww = lnz?w = ez?-w = u + ivz = rei?eu eiv= reiu = lnr = ln|z|,v = + 2n(n = 0,1,2,)rw = lnz(Lw = lnz = ln|z| + i( + 2n) = ln|z| + i argzTao Zhou(HHH?nnn?)111?29 / 32w = lnz?5?5?mz?55?LyKw?JAzzkw?J?2?w = lnz?:z = 0?05?argz=?w = lnz?w = lnzkzSkddz(lnz) =1z?Tao Zhou(HHH?nnn?)111?30 / 32w = lnz?5?5?mz?55?LyKw?JAzzkw?J?2?w = lnz?:z = 0?05?argz=?w = lnz?w = lnzkzSkddz(lnz) =1z?Tao Zhou(HHH?nnn?)111?30 / 32w = lnz?5?5?mz?55?LyKw?JAzzkw?J?2?w = lnz?:z = 0?05?argz=?w = lnz?w = lnzkzSkddz(lnz) =1z?Tao Zhou(HHH?nnn?)111?30 / 32w = lnz?5?5?mz?55?LyKw?JAzzkw?J?2?w = lnz?:z = 0?05?argz=?w = lnz?w = lnzkzSkddz(lnz) =1z?Tao Zhou(HHH?nnn?)111?30 / 32w = lnzkRiemann“?平面平面lnwz=?i-Tao Zhou(HHH?nnn?)111?31 / 32?Ny?5ULLX?z= elnz(?E)Xn?covkarcsinz =1iln?iz+p1z2?arccosz =1iln?z+pz21?arctanz =12iln1 + iz1 izTao Zhou(HHH?nnn?)111?32 / 32?Ny?5ULLX?z= elnz(?E)Xn?covkarcsinz =1iln?iz+p1z2?arccosz =1iln?z+pz21?arctanz =12iln1 + iz1 izTao Zhou(HHH?nnn?)111?32 / 32?Ny?5ULLX?z= elnz(?E)Xn?covkarcsinz =1iln?iz+p1z2?arccosz =1iln?z+pz21?arctanz =12iln1 + iz1 izTao Zhou(HHH?nnn?)111?32 / 32?Ny?5ULLX?z= elnz(?E)Xn?covkarcsinz =1iln?iz+p1z2?arccosz =1iln?z+pz21?arctanz =12iln1 + iz1 izTao Zhou(HHH?nnn?)111?32 / 32