Ch.1:Complex Numbers Ch.1:Complex Numbers 1.1 The Algebra of Complex Numbers Chapter 1:Complex Numbers 1.2 Point Representation of Complex Numbers 1.3 Vectors and Polar Forms Li,Yongzhao 1.4 The Complex Exponential State Key Laboratory of Integrated Services Networks,Xidian University September 28,2010 1.5 Powers and Roots 1.6 Planar Sets Ch.1:Complex Numbers Ch.1:Complex Numbers L11 The Algetra of Complex Numbers 1.1 The Algebra of Complex Numbers Review of Real Numbers Review of Real Numbers(Cont.) Initially.we learned the positive integers 1,2,3,... Zero 0 is an interesting number 0.5 e=2.71828. Sometimes we need to calculate the equation 2-8,so we introduced the solution-6 which is a negative integer An apple is cut into two pieces,each is half(0.5) 2=1.414 Integers and fractions constitutes the rational number system (a/b) We can compare the magnitudes of any two real numbers One solution to the equation 2=2 is v2 which is an (larger,equal or smaller) irrational number One dimensional(represented by a straight line) Rational and irrational numbers form the real number Are real numbers enough? system
Ch.1: Complex Numbers Chapter 1: Complex Numbers Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University September 28, 2010 Ch.1: Complex Numbers Outline 1.1 The Algebra of Complex Numbers 1.2 Point Representation of Complex Numbers 1.3 Vectors and Polar Forms 1.4 The Complex Exponential 1.5 Powers and Roots 1.6 Planar Sets Ch.1: Complex Numbers 1.1 The Algebra of Complex Numbers Review of Real Numbers Initially, we learned the positive integers 1, 2, 3,... Zero 0 is an interesting number Sometimes we need to calculate the equation 2 − 8, so we introduced the solution −6 which is a negative integer An apple is cut into two pieces, each is half (0.5) Integers and fractions constitutes the rational number system (a/b) One solution to the equation x2 = 2 is √2 which is an irrational number Rational and irrational numbers form the real number system Ch.1: Complex Numbers 1.1 The Algebra of Complex Numbers Review of Real Numbers (Cont.) We can compare the magnitudes of any two real numbers (larger, equal or smaller) One dimensional (represented by a straight line) Are real numbers enough?
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.1 The Algebra of Complex Numbers L11 The Algebra of Complex Numbers Extend Real Numbers to Complex Numbers Basic Operations of Complex Numbers The problem of solving the equation 22=-1 One solution is -I (not a real number) Addition (or subtraction) Use a symbol i (or j)to designate v-1 (a+bi)士(c+d):=(a±c)+(b±d)i We get:2=-1 Multiplication With the aid of i,we get the definition of a Complex (a+bi)(c+di):=(ac-bd)+(bc+ad)i Number Division z:=a+bi (a+bi)ac+bd bc-ad = where real numbers a:=Rez =Rz and b:=Imz Sz are (c+di) 2+2+2+2 called the Real Part and Imaginary Part of z Ch.1:Complex Numbers Ch.1:Complex Numbers L11 The Algetra of Complex Numbers 12 Point Representation of Complex Numbers Comments to Complex Numbers Representing Complex Numbers in z-plane 1m母4 The set of all complex numbers is denoted as C(R for reals) \-23) No nature ordering for the elements of C -2+3/◆ 2+2i -2+3· 22 The real part and imaginary part are independent of each other 0 0 Retl A complex number can be represented as a point in a two-dimensional plane Or it can be viewed as a vector with two entries (a b) All reals are complex (a line in the two-dimensional plane) Cartesian Coordinate System z-plane Argand Diagram
Ch.1: Complex Numbers 1.1 The Algebra of Complex Numbers Extend Real Numbers to Complex Numbers The problem of solving the equation x2 = −1 One solution is √−1 (not a real number) Use a symbol i (or j) to designate √−1 We get: i2 = −1 With the aid of i, we get the definition of a Complex Number z := a + bi where real numbers a := Rez = z and b := Imz = z are called the Real Part and Imaginary Part of z Ch.1: Complex Numbers 1.1 The Algebra of Complex Numbers Basic Operations of Complex Numbers Addition (or subtraction) (a + bi) ± (c + di) := (a ± c)+(b ± d)i Multiplication (a + bi)(c + di) := (ac − bd)+(bc + ad)i Division (a + bi) (c + di) := ac + bd c2 + d2 + bc − ad c2 + d2 i Ch.1: Complex Numbers 1.1 The Algebra of Complex Numbers Comments to Complex Numbers The set of all complex numbers is denoted as C (R for reals) No nature ordering for the elements of C The real part and imaginary part are independent of each other A complex number can be represented as a point in a two-dimensional plane Or it can be viewed as a vector with two entries a b All reals are complex (a line in the two-dimensional plane) Ch.1: Complex Numbers 1.2 Point Representation of Complex Numbers Representing Complex Numbers in z-plane Argand Diagram
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.2 Point Representation of Complex Numbers L12 Point Representation of Complex Numbers Absolute Value of a Complex Number Complex Conjugate of a Complex Number The distance between two points z1=a1+bii,z2 =a2+b2i in the z-plane is v(a1-a2)2+(61-b2)2 Complex Conjugate or 2*) When 22=0 is the origin of the z-plane,we get the absolute z=a+bi:=a-bi value (or modulus)of z which is denoted by The function of complex conjugate is to change the sign of |a=Va12+b12 the imaginary part of a complex number Hence,the distance between z1 and z2 can be written as Features 21-22 ·Rez=a=(2+到/2mz=b=(z-到/2i(何=2， 21士22=2五士2,2122=2五·2互(21/22)=五/2 Equation z-zo=r (where zo is a fixed complex number ·=,2泛=l2 and r is a fixed non-negative real number)describes a circle of 、1/z=八22 radius r centered at zo Ch.1:Complex Numbers Ch.1:Complex Numbers L13 Vectors and Polar Forms 1.3 Vectors and Polar Forms Vectors in the Complex-plane Vector Addition and Subtraction Each point z in the complex plane 1十z corresponds to a directed line segment from the origin to the point z 11 21 The vector is determined by its 0 length and direction The length equals to the modulus of z,namely z Vector Addition Vector Subtraction The diagonal of the parallelogram
Ch.1: Complex Numbers 1.2 Point Representation of Complex Numbers Absolute Value of a Complex Number The distance between two points z1 = a1 + b1i, z2 = a2 + b2i in the z-plane is (a1 − a2)2 + (b1 − b2)2 When z2 = 0 is the origin of the z-plane, we get the absolute value (or modulus) of z1 which is denoted by |z1| := a12 + b12 Hence, the distance between z1 and z2 can be written as |z1 − z2| Equation |z − z0| = r (where z0 is a fixed complex number and r is a fixed non-negative real number) describes a circle of radius r centered at z0 Ch.1: Complex Numbers 1.2 Point Representation of Complex Numbers Complex Conjugate of a Complex Number Complex Conjugate (z or z∗ ) z = a + bi := a − bi The function of complex conjugate is to change the sign of the imaginary part of a complex number Features Rez = a = (z + z)/2 Imz = b = (z − z)/2i (z) = z, z1 ± z2 = z1 ± z2, z1z2 = z1 · z2,(z1/z2) = z1/z2 |z| = |z|,zz = |z|2 1/z = z/|z|2 Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Vectors in the Complex-plane Each point z in the complex plane corresponds to a directed line segment from the origin to the point z The vector is determined by its length and direction The length equals to the modulus of z, namely |z| Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Vector Addition and Subtraction
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.3 Vectors and Polar Forms L1.3 Vectors and Polar Forms Vector Addition and Subtraction (cont.) Polar Forms of Complex Numbers (x,)=→(r,) =rcoso,y=rsin Parallelogram Law for addition of two vectors (or complex z=r(cos0+isin)=rcis0 numbers) z=x+iy r=la=Vr2+y2 Triangle Inequality:|z1+z0; of the lengths of the other two sides arctan(y/x)±元，x<0 I to IV quadrant Corollary:z2l≤lal+l2-zl=→lz2l-lz1l≤lz2-zl argz=Argz+2kπ The difference of the lengths of any two sides of a triangle is 0 (k=0,±1，士2，.) no greater than the length of the third side Argz is the principal value of arg z Polar Coordinates Q:How to represent a product of two complex numbers in a 2-D plane? 4日10。+之+1生+意QG Ch.1:Complex Numbers Ch.1:Complex Numbers L13 Vectors and Polar Forms 14 The Complex Exponential Another Two Examples Euler's Equation The real exponential function f(x)=e where x is a real number 2122个 By replacing with z =z+iy,we get the complex Z2 exponential function f(z)=e= First,we postulate that the multiplication property should 0 persist:exti=eeiv,where e is still a real number and the second part eu needs to be defined According to Taylor'series expansion,we get the following 0 equation Geometric interpretation of the product Conjugate and the reciprocal ++++苦+… e=1+y+ (1)
Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Vector Addition and Subtraction (cont.) Parallelogram Law for addition of two vectors (or complex numbers) Triangle Inequality: |z1 + z2|≤|z1| + |z2| The length of any side of a triangle is no greater than the sum of the lengths of the other two sides Corollary:|z2|≤|z1| + |z2 − z1| =⇒ |z2|−|z1|≤|z2 − z1| The difference of the lengths of any two sides of a triangle is no greater than the length of the third side Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Polar Forms of Complex Numbers (x, y) =⇒ (r, θ) x = r cos θ, y = r sin θ z = r(cos θ + isin θ) = rcisθ r = |z| = x2 + y2 θ = arctan(y/x), x> 0; arctan(y/x) ± π, x < 0 I to IV quadrant arg z = Argz + 2kπ (k = 0, ±1, ±2,...) Argz is the principal value of arg z Q: How to represent a product of two complex numbers in a 2-D plane? Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Another Two Examples Ch.1: Complex Numbers 1.4 The Complex Exponential Euler’s Equation The real exponential function f(x) = ex where x is a real number By replacing x with z = x + iy, we get the complex exponential function f(z) = ez First, we postulate that the multiplication property should persist: ex+iy = exeiy, where ex is still a real number and the second part eiy needs to be defined According to Taylor’ series expansion, we get the following equation ey =1+ y + y22! + y33! + y44! + y55! + ... (1)
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.4 The Complex Exponential L14 The Complex Exponential Euler's Equation (Cont'd) Euler's Equation(Cont'd) By replacing y with iy in Eq.(1),we get the Taylor's By separating the real part and imaginary part of e,Eq.(2) expansion of ety as follows can be rewritten as =1++g+g+g+罗 + (2) --苦+若-+(-苦+黄-… We know the identities Note that the real part and the imaginary part of the above =i,2=-1,3=-i,4=1 are just the Taylor's expansions of cosy and siny,respectively 泸=i,=-1,7=-i,8=1, (3) Hence,We arrive at the famous Euler's equation as follows eiv cosy+isiny Hence,we deduce that i=im+4 is periodical function with By using the Euler's equation,we have the definition of a period 4 complex exponential function:e*:=e*(cosy+isiny) Ch.1:Complex Numbers Ch.1:Complex Numbers L14 The Complex Exponential 14 The Complex Exponential Comments to Euler's Equation Application of Complex Exponential Since leiu=cosy+isinyl Vcos2y+sin2y=1,ei is a 2—；sin0=3e0-e0-e-0 ·os9=Re0=e0+e0 2i vector which locates on the circle of radius 1 about origin Multiplication of two complex numbers: y is the angle of inclination of the vector eiy,measured 2a2=(r1ei01)(r2e）=(r12)e0+) positively in a counterclockwise sense from the positive real Division of two complex numbers: axis Recall that any complex number z can be written as the polar 号-受- T2 form:z=r(cos0+isin) Complex Conjugate:=re-i0 Euler's equation enables us to write it in another form: De Moivre's Formula:(cos0+isin0)n=cosne +isin no 2=rei leleiargs Q:Does this formula hold for arbitrary integers n(positive or negative)?
Ch.1: Complex Numbers 1.4 The Complex Exponential Euler’s Equation (Cont’d) By replacing y with iy in Eq. (1), we get the Taylor’s expansion of eiy as follows eiy =1+ iy + (iy)2 2! + (iy)3 3! + (iy)4 4! + (iy)5 5! + ... (2) We know the identities i1 = i, i2 = −1, i3 = −i, i4 = 1 i5 = i, i6 = −1, i7 = −i, i8 = 1,... (3) Hence, we deduce that in = in+4 is periodical function with period 4 Ch.1: Complex Numbers 1.4 The Complex Exponential Euler’s Equation (Cont’d) By separating the real part and imaginary part of eiy, Eq. (2) can be rewritten as eiy = 1 − y22! + y44! − ... + i y − y33! + y55! − ... (4) Note that the real part and the imaginary part of the above are just the Taylor’s expansions of cos y and sin y, respectively Hence, We arrive at the famous Euler’s equation as follows eiy = cos y + isin y By using the Euler’s equation, we have the definition of a complex exponential function: ez := ex(cos y + isin y) Ch.1: Complex Numbers 1.4 The Complex Exponential Comments to Euler’s Equation Since |eiy| = | cos y + isin y| = cos2 y + sin2 y = 1, eiy is a vector which locates on the circle of radius 1 about origin y is the angle of inclination of the vector eiy, measured positively in a counterclockwise sense from the positive real axis Recall that any complex number z can be written as the polar form: z = r(cos θ + isin θ) Euler’s equation enables us to write it in another form: z = reiθ = |z|ei arg z Ch.1: Complex Numbers 1.4 The Complex Exponential Application of Complex Exponential cos θ = eiθ = eiθ + e−iθ 2 ; sin θ = eiθ = eiθ − e−iθ 2i Multiplication of two complex numbers: z1z2 = r1eiθ1 r2eiθ2 = (r1r2)ei(θ1+θ2) Division of two complex numbers: z1 z2 = r1eiθ1 r2eiθ2 = r1 r2 ei(θ1−θ2) Complex Conjugate: z = re−iθ De Moivre’s Formula: (cos θ + isin θ)n = cos nθ + isin nθ Q: Does this formula hold for arbitrary integers n (positive or negative)?
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.5 Powers and Roots L15 Powers and Roots Powers of a Complex Number Roots of a Complex Number We can represent the complex number z in its polar form: The computation of the roots is more complicated than z=rei =r(cos0+isin) powers The n-th power of z is calculated by two steps: Let w=p(cos p+isin)be the m-th roots of Step 1:The n-th power of the modulus:r" z=r(cos0 +isin),so wm =z means Step 2:The n-fold of the angle of inclination:ne p"(cosm+isin mp)=r(cos0+isin) Finally,we get the n-th power of z,namely.2n=r"ein (5) The above rule is valid for both positive and negative integers Eq.(5)means The question arises whether the formula will work for pm =r.cosm cos0,sin mp=sin n=1/m 、Hence,.p=r/m,mp=0+2kr=→p= 0+2k元 n Ch.1:Complex Numbers Ch.1:Complex Numbers L15 Powers and Roots 1.5 Powers and Roots Roots of a Complex Number(Con't) An Example of Finding the Roots Find the Four fourth roots of v2+iv2 When =0,1,2,...,m-1,we get the m distinct roots for 2+iW2=2e Eq-(5)as wk =rl/m cos 8+2k元 +isin 0+2kx） h信 m When k m,m +1,m +2,...,2m-1,the same roots repeat again,… a.你习 Hence,there are only m distinct roots for 21/m 信 迈 迈。信)
Ch.1: Complex Numbers 1.5 Powers and Roots Powers of a Complex Number We can represent the complex number z in its polar form: z = reiθ = r(cos θ + isin θ) The n-th power of z is calculated by two steps: Step 1: The n-th power of the modulus: rn Step 2: The n-fold of the angle of inclination: nθ Finally, we get the n-th power of z, namely, zn = rneinθ The above rule is valid for both positive and negative integers The question arises whether the formula will work for n = 1/m Ch.1: Complex Numbers 1.5 Powers and Roots Roots of a Complex Number The computation of the roots is more complicated than powers Let w = ρ(cos ϕ + isin ϕ) be the m-th roots of z = r(cos θ + isin θ), so wm = z means ρm(cos mϕ + isin mϕ) = r(cos θ + isin θ) (5) Eq.(5) means ρm = r, cos mϕ = cos θ, sin mϕ = sin θ Hence, ρ = r1/m, mϕ = θ + 2kπ =⇒ ϕ = θ + 2kπ m Ch.1: Complex Numbers 1.5 Powers and Roots Roots of a Complex Number (Con’t) When k = 0, 1, 2,...,m − 1, we get the m distinct roots for Eq.(5) as wk = r1/m cos θ + 2kπ m + isin θ + 2kπ m When k = m, m + 1, m + 2,..., 2m − 1, the same roots repeat again,... Hence, there are only m distinct roots for z1/m Ch.1: Complex Numbers 1.5 Powers and Roots An Example of Finding the Roots Find the Four fourth roots of √2 + i√2
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.6 Planar Sets L16 Pbrar Sets Planar Sets Open Disk(Neighborhood)and Open Set In the calculus of functions of a real variable,the main The set of all points that satisfy the inequality theorems are typically stated for functions defined on an z-z0<P interval,such as(0,1),(0,1,[0,1),0,1] where p is a positive number,is called an open disk or The interval can be interpreted as a segment in the x-axis in circular neighborhood of z0 z-plane A point zo which lies in a set S is called an interior point of A complex number is two-dimensional,hence for the functions S if there is some circular neighborhood of zo that is of a complex variable,the basic results are formulated for completely contained in S functions defined on sets that are 2-dimensional "domains"or If every point of a set S is an interior point of S.we say that "closed regions" S is an open set 白·0+之。·急，是2风C Ch.1:Complex Numbers Ch.1:Complex Numbers L16 Planar Scts 1.6 Ptanar Sets Domain Domain (Cont'd) An open set S is said to be connected if every pair of points The extension result to functions of two real variables: z1.z2 in S can be joined by a polygonal path that lies entirely Suppose u(z,y)is a real-valued function defined in a domain in S.Roughly speaking,this means that S consists of a D.If the first partial derivative of u satisfy "Single Piece" Ou du =0 An open connected set is called a domain 8x y For real variables,the derivative of the function equals zero at all points of D.then u =constant in D implies that this function is identically constant on the defined If D is merely assumed to be an open set(not connected) interval the theorem is no longer true
Ch.1: Complex Numbers 1.6 Planar Sets Planar Sets In the calculus of functions of a real variable, the main theorems are typically stated for functions defined on an interval, such as (0, 1), (0, 1], [0, 1), [0, 1] The interval can be interpreted as a segment in the x-axis in z-plane A complex number is two-dimensional, hence for the functions of a complex variable, the basic results are formulated for functions defined on sets that are 2-dimensional ”domains” or ”closed regions” Ch.1: Complex Numbers 1.6 Planar Sets Open Disk (Neighborhood) and Open Set The set of all points that satisfy the inequality |z − z0| < ρ where ρ is a positive number, is called an open disk or circular neighborhood of z0 A point z0 which lies in a set S is called an interior point of S if there is some circular neighborhood of z0 that is completely contained in S If every point of a set S is an interior point of S, we say that S is an open set Ch.1: Complex Numbers 1.6 Planar Sets Domain An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S. Roughly speaking, this means that S consists of a ”Single Piece” An open connected set is called a domain For real variables, the derivative of the function equals zero implies that this function is identically constant on the defined interval Ch.1: Complex Numbers 1.6 Planar Sets Domain (Cont’d) The extension result to functions of two real variables: Suppose u(x, y) is a real-valued function defined in a domain D. If the first partial derivative of u satisfy ∂u ∂x = ∂u ∂y = 0 at all points of D, then u ≡constant in D If D is merely assumed to be an open set (not connected), the theorem is no longer true
Ch.1:Complex Numbers Ch.1:Complex Numbers L1.6 Planar Sets L16 Pbnar Sets Boundary Bounded and Region A point zo is said to be a boundary point of a set S if every neighborhood of zo contains at least one point not in S The set of all boundary points of S is called the boundary or A set of points S is said to be bounded if there exists a frontier of S positive real number R such that z0)is a closed set,for it contains its boundary z-zol =p.We call this set a closed disk 4日18。t+2+意0c
Ch.1: Complex Numbers 1.6 Planar Sets Boundary A point z0 is said to be a boundary point of a set S if every neighborhood of z0 contains at least one point not in S The set of all boundary points of S is called the boundary or frontier of S Since each point of a domain D is an interior point of D, it follows that a domain cannot contain any of its boundary points A set S is said to be closed if it contains all of its boundary points. The set of points z that satisfy the inequality |z − z0| ≤ ρ (ρ > 0) is a closed set, for it contains its boundary |z − z0| = ρ. We call this set a closed disk Ch.1: Complex Numbers 1.6 Planar Sets Bounded and Region A set of points S is said to be bounded if there exists a positive real number R such that |z| < R for every z in S A set is both closed and bounded is said to be compact A region is a domain together with some, none, or all of its boundary points. In particular, every domain is region
