长沙理工大学：《复变函数与积分变换》课程教学课件（英文讲稿）Chapter 1 Complex numbers and functions of a complex variable

Chapter1:ComplexNumbersandFunctions of a ComplexVariableFCV&ITSeptember2,20193/40Mineha Uni.of Sci&Tech)

Chapter 1: Complex Numbers and Functions of a Complex Variable Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 3 / 40

g1.1 Complex numbers and its four fundamentaloperationsaUni.ofSci&Tech)FCV&ITSeptember2,20194/40

§1.1 Complex numbers and its four fundamental operations Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 4 / 40

Introduction toxnumbersIntroductiontocomplexnumbersQuestion(GirolamoCardano,1500s)Solvethequadratic equaitons:2=1,2+2+2=0Definitioni=V-i2=V-1×V-1=-1Answer2=1has solutions±i;r2+2+2=0hassolutions-1±i.FCV&ITgsha Uni. of Sci & Tech)September2,20195/40MingLi(Chang

Introduction to complex numbers Introduction to complex numbers Question(Girolamo Cardano, 1500s) Solve the quadratic equaitons: x 2 = 1, x2 + 2x + 2 = 0 Definition i = √ −1 i 2 = √ −1 × √ −1 = −1 Answer x 2 = 1 has solutions ±i; x 2 + 2x + 2 = 0 has solutions −1 ± i. Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 5 / 40

Introduction tenumberDefinition ofcomplexnumberDefinitionItiscustomarytodenoteacomplexnumber(1)z=r+iywhere ,y are real numbers. ,y are known as the real and imaginarypartsof zrespectively,and wewrite(2)Rez=r,Imz=yRemark:Two complexnumberss areequal whenevertheyhavethesamereal partsand the same imaginaryparts,i.e.z1=z2if and only if 1 =2and y1 = y2.FCV&ITSeptember 2, 20196/40MineLilChsaUni.ofSci&Tech)

Introduction to complex numbers Definition of complex number Definition It is customary to denote a complex number: z = x + iy (1) where x, y are real numbers. x, y are known as the real and imaginary parts of z respectively, and we write Re z = x, Im z = y (2) Remark: Two complex numberss are equal whenever they have the same real parts and the same imaginary parts, i.e. z1 = z2 if and only if x1 = x2 and y1 = y2. Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 6 / 40

FourfuroperationsFour fundamental operationsadditionz1± z2 = (r1 + y1i) ±(r2 + y2i) =(r1± a2) +(y1 ±y2)imultiplicationziz2=(r1+y1i)× (2+y2i)=(r1r2-y1y2)+(riy2+2y1)iquotient(assume22=2+iy20)-i+iyi2+iy222(r1 + iy1)(r2 - iy2)(2+iy2)(2-i2)(i22+1y2)+i(291—T1y2)+yFCV&ITaUni.ofSci&Tech)September2,20197/40MineLlCh

Four fundamental operations Four fundamental operations addition z1 ± z2 = (x1 + y1i) ± (x2 + y2i) = (x1 ± x2) + (y1 ± y2)i multiplication z1 · z2 = (x1 + y1i) × (x2 + y2i) = (x1x2 − y1y2) + (x1y2 + x2y1)i quotient (assume z2 = x2 + iy2 6= 0) z1 z2 = x1 + iy1 x2 + iy2 = (x1 + iy1)(x2 − iy2) (x2 + iy2)(x2 − iy2) = (x1x2 + y1y2) + i(x2y1 − x1y2) x 2 2 + y 2 2 Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 7 / 40

FourfundannentaloperationsAdditive rulesz+w=w+zz+(w+s)=(z+w)+sz+0=z+ (-2) = 0Multiplication ruleszw=wz(zw)s = z(ws)1z=zz(z-1)=1 for z0Distributive rulez(w+s)=zw+zssha Uni. of Sci&Tech)FCV&ITSeptember2,20198/40MineLilChal

Four fundamental operations Additive rules z + w = w + z z + (w + s) = (z + w) + s z + 0 = z z + (−z) = 0 Multiplication rules zw = wz (zw)s = z(ws) 1z = z z(z −1 ) = 1 for z 6= 0 Distributive rule z(w + s) = zw + zs Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 8 / 40

FourfundaentaloperationsCompute2+3i13.(2)(1)2-3isha Uni. of Sci & Tech)FCV&ITSeptember2,20199/40MineLlCha

Four fundamental operations Compute (1) i 3 , (2) 2 + 3i 2 − 3i Answer: (1) − i, (2) −5 + 12i 13 Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 9 / 40

FourfundantaloperationsCompute2+3i3(1) (2)2-3iAnswer:-5 + 12i(2)(1)-i13tha Uni.ofSci&Tech)FCV&ITSeptember2,20199/40MineLilCha

Four fundamental operations Compute (1) i 3 , (2) 2 + 3i 2 − 3i Answer: (1) − i, (2) −5 + 12i 13 Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 9 / 40

FourfunLoneations1.2GeometricrepresentationofcomplexnumbersaUni.ofSci&Tech)FCV&ITSeptember2.201910/40MineLilCh

Four fundamental operations §1.2 Geometric representation of complex numbers Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 10 / 40

ceometric reoresentation of comolexnumbersGeometricrepresentationof complexnumbersImaginaryaxis(yaxis),z=a+bi0Real axis (r axis)vectorrepresentation:z=a+ibFCV&ITsha Uni.of Sci &Tech)September2.201911/40MineLlchs

Geometric representation of complex numbers Geometric representation of complex numbers O Real axis (x axis) Imaginary axis (y axis) z = a + bi θ r r cos θ r sin θ vector representation: z = a + ib triangle functions representation: z = r(cos θ + isin θ) polar coordinate representation: z = re iθ Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 11 / 40