Math 656 Complex Variables I - Information Services and ...

1 Math 656 Complex Variables IProf Cummings February 2, 2010 Contents1 Introduction to the Complex numbers .. Exponential representation of Complex numbers .. Algebra of Complex numbers .. Complex conjugates and inequalities .. Solving simple Complex equations ..92 Subsets of the Complex Introduction: Simple subsets of the Complex plane .. Elementary definitions .. Stereographic projection .. Paths in the Complex plane .. Paths and connectedness ..203 Analytic functions: The Complex sequences, series, and functions .. Limits and continuity of functions .. Differentiability of Complex functions .. The Cauchy-Riemann theorem .. Harmonic functions .. Multivalued functions .. General non-integer powers ..41 Department of Mathematical Sciences, Logarithms .. Contours and multivaluedness .. Branch cuts to ensure single-valuedness .. Composite multifunctions.

2 Example: Product of square-roots .. Example: Logarithm of a product .. Summary and concluding remarks on multifunctions ..484 Complex Basic methods of Complex integration .. General results for Complex integration .. Harmonic functions to construct analytic functions ..595 Cauchy s Cauchy s theorem I .. Cauchy s theorem II ..626 Beyond Cauchy s (first) Morera and Liouville theorems ..837 Taylor series and related Taylor s theorem .. Taylor expansion about infinity .. Zeros of analytic functions .. Identity theorem for analytic functions .. The Open Mapping theorem, and Maximum principles revisited1018 Laurent Laurent s theorem .. Singular points ..1129 Cauchy s Residue Preliminaries: Motivation .. The residue theorem .. applications of the theorem to simple integrals .. Jordan s inequality and Jordan s lemma .. More complicated integration contours.

3 Avoiding a singularity on the integration contour byindentation .. Integrals around sector contours .. Integrals of functions with branch-points .. Integrals around rectangular contours .. Principle of the argument, and Rouch e s theorem ..14310 Asymptotic expansions and Stokes Asymptotic sequences and asymptotic expansions .. Asymptotic expansions of Complex functions .. The Stokes phenomenon ..155 Ablowitz & Fokas. Complex Variables : Introduction andApplications (2nd edition).Cambridge University Press(2003).1 Introduction to the courseYou are familiar with the theory and calculus of functions of one (or more)real variable ,f(x),f(x, y),f(x, y, t) etc. This course is concerned with thetheory of Complex -valued functions of a Complex variable :f(z)=u(x, y)+iv(x, y),wherez=x+iyandi2= begin by introducing Complex numbers and their algebraic properties,together with some useful geometrical Complex numbersThe set of all Complex numbers is denoted byC, and is in many ways analo-gous to the set of all ordered pairs of real numbers,R2.

4 A Complex number isspecified by a pair of real numbers (x, y): we writez=x+iy, wherei2= say thatxis thereal partofz, andyis theimaginary partofz, usingthe equivalent notationx= (z)=Re(z),y= (z)=Im(z).Note that the relationi2= 1 leads to the identitiesi2m=( 1)m,i2m+1=( 1)mi,form Z.(1)3 Two Complex numbers are equal if and only if both their real and imaginaryparts are equal. Theabsolute valueormagnitudeof the Complex numberzis defined by the length of the vector (x, y) associated withz:|z|= x2+y2,(2)always a positive quantity (except whenz=0).Example Complex numberz1=3+4ihas magnitude|z|= 32+42=5, as has the Complex numberz2=4+ set of all real numbersRis a natural subset ofC, being the set ofall those numbers inCwith zero imaginary part. The standard geometryofR2provides a convenient and useful representation ofC(this geometricalstructure is known as thecomplex plane: the real numbers lie along thex-axis, known as the real axis, and the pure imaginary numbers lie along they-axis, known as the imaginary axis).

5 It is also often helpful to use the polar representation (r, ) of points in2D space, wherer2=x2+y2and tan =y/x(alternatively,x=rcos ory=rsin ). In this representation,z=x+iy=r(cos +isin ).Here againris the absolute value ofz,r=|z|, and the angle is known astheargumentofz, written =arg(z). Note that in this representation ofzis not single-valued: ifz0=r0(cos 0+isin 0) is one representation, thenreplacing 0by 0+2n also gives the same Complex numberz. Any suchvalue of is an argument of the Complex number, and arg(z) is therefore reallyan infinitesetof -values. It is often useful to define a uniqueprincipalargument, which is usually taken to be the argument lying in the range0 <2 (but sometimes the range < ; convenience usuallydictates the choice in a particular application). The principal argument issometimes written = Arg(z) to distinguish it from the set arg(z). Wewill see further consequences of this nonuniqueness later on, when we discussmulti-valued the polar form of the Complex numbers (i)z1=i, (ii)z2=1 i, (iii)z3= 3 :(i) Forz1we have|z1|= 1 and =cos 1(0) = sin 1(1) = /2+2n , for any integern.

6 The principal argument would be = /2. Inany case it is clear thatz1=cos( /2) +isin( /2).(ii) Forz2=1 iwe have|z2|= 2, and thus 1 = 2cos , 1= 2 sin .Thus = /4, modulo 2 , andz2= 2(cos( /4) +isin( /4)).(iii)|z3|= 2, and 3=2cos , 1 = 2 sin . Thus = /6 (modulo 2 ),andz3=2(cos( /6) +isin( /6)). Exponential representation of Complex numbersA more concise representation ofzis obtained by introducing the complexexponential function:ei =cos +isin , R,(3)where is the polar angle introduced above. This definition is a special caseof the more general Complex exponential function that we will introduce now we note that the definition makes sense in terms of the standardTaylor series representation of these functions:ei = n=0(i )nn!= m=0(i )2m(2m)!+ m=0(i )2m+1(2m+1)!= m=0( 1)m 2m(2m)!+i m=0( 1)m 2m+1(2m+1)!=cos +isin ,using the identities (1). The absolute value of the Complex numberw=ei is clearly|w|= 1, and as the argument varies, the point representingei in the Complex plane moves around the unit circle centered on the origin.

7 Ingeneral then, we have the equivalent representationsz=x+iy=r(cos +isin )=rei .(4)This last representation provides an easy way to take the power of a complexnumber:zn=rnein =rn(cos(n )+isin(n )).Note that this gives a simple derivation of the identity (cos +isin )n=cos(n )+isin(n ), a result known as De Moivre s the numbers in example we have (i)z1=ei /2, (ii)z2= 2e i /4, and (iii)z3=2e i Algebra of Complex numbersRecalling the identities (1) it is straightforward to extend the algebra of thereal numbers to Complex numbers. Ifz=x+iyandw=u+iv, thenz+w=x+u+i(y+v),zw=(x+iy)(u+iv)=xu yv+i(xv+yu).From these laws and the properties already noted, it is easy to show thatthe Complex numbersCform afield. The inverse of the Complex numberz,given by 1/z, satisfiesz 1=1z=1x+iy=x iy(x+iy)(x iy)=x iyx2+y2.(5)This identity may be alternatively obtained by the Complex polar represen-tation (4), which givesz 1=e i r=re i |z|2.

8 (6)The polar representation also gives the neatest interpretation of complexmultiplication (and division which, with the notion of an inverse given aboveis really just multiplication also), since ifz=rei andw=Rei thenzw=rRei( + ).Example product ofz1andz2in example isz1z2= 2ei /4=1+i; the product ofz1andz3isz1z3=2ei /3=1+i 3; and the product ofz2andz3isz2z3=2 2e 5i Complex conjugates and inequalitiesEvery Complex numberz=x+iyhas acomplex conjugatez, defined byz=x iy=r(cos isin )=re i .(7)The first equality above shows that geometrically, the Complex conjugate of apointz Cis obtained byreflectionofzin the real (x) axis. The definition6(7) also shows that the inverse ofz, given by (5) or (6) above, is related toits Complex conjugate byz 1=z|z| is evident thatz=z, z+z=2 (z),z z=2i (z),|z|=|z|,zz=|z| , for two Complex numberszandw,z+w=z+w,zw= above properties show that|zw|=|z||w|, but in general|z+w| =|z|+|w|.

9 Rather, the following inequalities hold, analogous to thetriangleinequalitiesof vector algebra:||z| |w|| |z+w| |z|+|w|.(8)The second inequality may be proved using elementary properties of complexnumbers. First note that|z+w|2=(z+w)(z+w)=|z|2+|w|2+2 (zw),and then that, for any Complex number, the absolute value of its real partmust be less than or equal to the absolute value of the Complex number itself,which gives (zw) | (zw)| |zw|=|z||w|.Thus,|z+w|2 |z|2+|w|2+2|z||w|=(|z|+|w|) that|z+w| 0 and|z|+|w| 0, the second inequality of (8) the first inequality, we use the result just proved, and setZ=z+w,W= w(arbitrary Complex numbers, sincezandware arbitrary). Thenz=Z+W,w= W, and|Z| |Z+W|+|W| |Z| |W| |Z+W|.7If|Z| |W|then the result follows; if not then repeat the above argumentwithW=z+wandZ= wto obtain|W| |Z| |Z+W|,proving the result for|W| |Z| right-hand inequality in (8) generalizes easily to an arbitrary sum ofcomplex numbers: N n=1zn N n=1|zn|.

10 When the Complex numbersz,ware plotted in the Complex plane|z|is justthe length of the straight line joiningzto the origin. The Complex numberz+wis found from the usual parallelogram law for vector addition. Therefore(8) has a geometrical interpretation in terms of lengths of sides of a :Determine the conditions under which equality holds in (8).Example the Complex numbersz=3+0iandw=0+4i,|z|=3,|w|=4, and|z+w|=|3+4i|=5. Clearly,||z| |w||=1 |z+w|=5 |z|+|w| the Complex numbersz=4,w=3,|z|+|w|=7=|z+w|.Similarly for the Complex numbersz=4i,w= ,w= 3,||z| |w||=1=|z+w|. Similarly forz=4i,w= :1. Prove that, for 0 =z C,|z| | (z)|+| (z)| 2|z|.Show, by examples, that either (but not both!) of these inequalities may bean Prove that, forz, w C,|1 zw|2 |z w|2=(1 |z|2)(1 |w|2).Deduce that, if|z|<1 and|w|<1, z w1 zw < that inequalities pertain only to theabsolute valueof com-plex numbers. Unlike the real numbers, there isno orderingof the field ofcomplex numbers, so no notion of one Complex number being greater thanor less than Solving simple Complex equationsAlgebraic Complex equations may be solved numerically in the same way asreal equations in two Variables : by varyingx= (z) andy= (z) until azero is found.