2Complex NumberA complex number is an expression of the type x+iy, where both x and y are real numbers and the symbolcomplex conjugatereal partimaginary part
3Argand DiagramArgand suggested that a complex number could be represented by a line in a plane in much the same way as a vector is represented. The value of the complex number could be expressed in terms of two axes of reference, and he suggested that one axis be called the real axis and the other axis arranged perpendicular to the first be called the imaginaryaxis. Then a complex number z=x+iy could be represented by a line in the plane having projections x on the real axis and y on the imaginary axis.The line OP represents the complex number 4+3i whose real part has a value of 4 and imaginary part 3. On the other hand, the line OQ represents the complex number 4i-3 with real part -3 and imaginary part 4. The lengths of OP and OQ are equal but the complex numbers they represent are unequal because the real parts and the imaginary parts ofeach are different.
4Modulus and ArgumentFor each of the complex numbers represented by the lines OP and OQ, thelengths of the lines are 5. That is,The value of the length of the line representing the complex number is called the ”modulus” or “absolute” of the number and is usually written in thealternative forms,The inclination of the line representing the complex number to the positive real axis is called the “amplitude”, “argument”, or “phase” of z, and isusually written,in polar coordinates
5Principal ValueWhen a complex number is expressed in polar coordinate, the principal ofvalue of is always implied unless otherwise stated. That is,If the angle is not restricted to its principal value amp(4+3i) would be equal to ( n)radians, and amp(4i-3) would equal to ( n)radians,where n could be zero or an integer. When the complex number is represented on the Argand diagram the principal value is the smaller of two angles between the positive real axis and the line. The sign of the angle depends upon the sense of rotation from the positive real axis, and this implies that the principal value lies in the range – to + . The principal value of the amplitudeof a negative real number is conventionally taken as + .
6Algebraic Operations on Argand Diagram If the Argand diagram describes all the properties of complex numbers it should be possible to carry out the above algebraic operations on the diagram. Thus consider Figure in which the complex numbers represented by the lines OP and OQ are redrawn. If z3 is the sum of z1represented by OP and z2 represented by OQ, thenEqs(*) give the coordinates of z3 on the Argand diagram as shown in Figure by the line OR. Usingthe same values as before,In the same way, the subtraction of two complex numbers can be expressed in the form of theaddition z1 to minus z2 where
7Algebraic Operations on Argand Diagram In order to illustrate multiplication and division on the Argand diagram, it is first necessary to show how the multiplication and division of twocomplex numbers are expressed in terms of polar coordinates. ThusTo multiply two complex numbers, it is necessary to multiply themoduli and add the argument.
8Algebraic Operations on Argand Diagram To divide two complex numbers, it is necessary to divide the moduli andsubtract the argument.In the multiplication and division operations 1 and 2 were the principal values of the arguments of the numbers. However (1+2) and (1-2) need not be the principal values of the arguments of z5 and z6. Thus consider the complex numbers z1=(3i-4) and z2=(i-1) with arguments 1=143 and 2=135. z1z2=1-7i and 5=1+2=278. Theprincipal value of the argument lying between -180 and +180 is -82.
9Conjugate NumbersTwo complex numbers such as x+iy and x-iy of which the real and imaginary parts are of equal magnitude, but in which the imaginary parts are of opposite sign are said to be conjugate numbers. On the Argand diagram they can be considered to be mirror images of each other in thereal axis. Usually, the conjugate of a complex number z is written asIf the imaginary part is zero, the real number is its own conjugate. The sum and the product of a complex number with its conjugate are alwaysreal. Thus,The division of a true complex by its conjugate will not produce a realnumber.
13Derivatives of a Complex Variable Consider the complex variable to be a continuous function,and let andThen the partial derivative of w w.r.t. x, is:orSimilarly, the partial derivative of w w.r.t. y, is:orCauchy-Riemann conditionsThey must be satisfied for the derivative of a complex number to have any meaning.
14Analytic FunctionsA function w=f(z) of the complex variable z=x+iy is called an analytic or regular function within a region R, if all points z0 in the regionsatisfies the following conditions:(1)It is single valued in the region R.(2)It has a unique finite value.(3)It has a unique finite derivative at z0 which satisfies the Cauchy-Riemann conditionsOnly analytic functions can be utilized in pure and applied mathematics.
15ExampleIf w = z3, show that the function satisfies the Cauchy-Riemann conditions andstate the region wherein the function is analytic.Satisfy!Cauchy-Riemann conditionsAlso, for all finite values of z, w is finite. Hence the function w = z3 is analytic in any region of finite size.(Note, w is not analytic when z = .)
16ExampleIf w = z-1, show that the function satisfies the Cauchy-Riemann conditions andstate the region wherein the function is analytic.Satisfy!Except from the origin?Cauchy-Riemann conditionsFor all finite values of z, except of 0, w is finite.Hence the function w = z-1 is analytic everywhere in thez plane with exception of the one point z = 0.
17Example At the origin, y = 0 As x tends to zero through either positive or negative values, it tends to negative infinity.At the origin, x = 0As y tends to zero through either positive or negative values, it tends to positive infinity.Consider half of the Cauchy-Riemann condition , which is not satisfied at the origin.Although the other half of the condition is satisfied, i.e.
18SingularitiesWe have seen that the function w = z3 is analytic everywhere except at z = whilst the function w = z-1 is analytic everywhere except at z = 0.In fact, NO function except a constant is analytic throughout the complex plane, and every function of a complex variable has one or more points inthe z plane where it ceases to be analytic.These points are called “singularities”.Three types of singularities exist:(a) Poles or unessential singularities“single-valued” functions(b) Essential singularities(c) Branch points“multivalued” functions
19Poles or Unessential Singularities A pole is a point in the complex plane at which the value ofa function becomes infinite.For example, w = z-1 is infinite at z = 0, and we say that thefunction w = z-1 has a pole at the origin.A pole has an “order”:The pole in w = z-1 is first order.The pole in w = z-2 is second order.
20Order of a PoleIf w = f(z) becomes infinite at the point z = a, we define:where n is an integer.If it is possible to find a finite value of n which makes g(z) analytic at z = a,then, the pole of f(z) has been “removed” in forming g(z).The order of the pole is defined as the minimum integer value of n forwhich g(z) is analytic at z = a.什麼意思呢？比如： 在原點為 pole, (a=0)則Order = 1在 0 和 a 各有一個pole，則 w在 0 這個 pole 的 order 為 3在 a 這個 pole 的 order 為 4n 最小需大於 1，使得 w 在原點的 pole 消失。
21Essential Singularities Certain functions of complex variables have an infinite number of terms which all approach infinity as the complex variable approaches a specific value. These could be thought of as poles of infinite order, but as the singularity cannot be removed by multiplying the function by a finitefactor, they cannot be poles.This type of singularity is called an essential singularity and is portrayed by functions which can be expanded in adescending power series of the variable.Example: e1/z has an essential singularity at z = 0.
22Essential Singularities Essential singularities can be distinguished from poles by the fact thatthey cannot be removed by multiplying by a factor of finite value.Example:infinite at the originWe try to remove the singularity of the function at the origin by multiplying zpIt consists of a finite number of positive powers of z, followed by an infinitenumber of negative powers of z.All terms are positiveIt is impossible to find a finite value of p which will remove the singularity in e1/z at the origin.The singularity is “essential”.
23Branch PointsThe singularities described above arise from the non-analytic behaviourof single-valued functions.However, multi-valued functions frequently arise in the solution ofengineering problems.For example:For any value of z represented by a point onthe circumference of the circle in the z plane, there will be two corresponding values of wrepresented by points in the w plane.
24Branch Points and when 0 2 The particular value of z at which the function becomes infinite or zerois called the “branch point”.Cauchy-Riemann conditions in polar coordinatesThe origin is the branch point here.
25Branch Points A function is only multi-valued around closed contours which enclose the branch point.It is only necessary to eliminate such contours and thefunction will become single valued.-The simplest way of doing this is to erect a barrier from thebranch point to infinity and not allow any curve to crossthe barrier.-The function becomes single valued and analytic for allpermitted curves.
26Barrier - Branch CutThe barrier must start from the branch point but it can go to infinity in any direction in the z plane, and may beeither curved or straight.In most normal applications, the barrier is drawn alongthe negative real axis.-The branch is termed the “principle branch”.-The barrier is termed the “branch cut”.-For the example given in the previous slide, the region,the barrier confines the function to the region in whichthe argument of z is within the range - < <.
27Integration of Functions of Complex Variables The integral of f(z) with respect to z is the sum of the product fM(z)zalong the curve in the complex plane, where fM(z) is the mean value off(z) in the length z of the curve. That is,where the suffix C under the integral sign specifies the curve in the zplane along which the integration is performed.When w and z are both real (i.e. v = y = 0):This is the form that we have learnt about integration; actually,this is only a special case of a contour integration along the real axis.
28ExampleShow that the value of z2 dz between z = 0 and z = 8 + 6i is the same whether the integration is carried out along the path AB or around the path ACDB.The path of AB is given by the equation:Consider the integration along the curve ACDBIndependent of pathAlong AC, x = 0, z = iyAlong CDB, r = 10, z = 10ei
29Example Let z = rei Let z = rei Evaluate around a circle with its centre at the originLet z = reiAlthough the function is not analytic at the origin,Evaluate around a circle with its centre at the originLet z = rei
30Cauchy’s TheoremIf any function is analytic within and upon a closed contour, the integral taken around the contour is zero.If KLMN represents a closed curve and there are no singularities of f(z) within or uponthe contour, the value of the integral of f(z) around the contour is:Since the curve is closed, each integral on the right-hand side can be restated as asurface integral using Stokes’ theorem:But for an analytic function, each integral on the right-hand side iszero according to the Cauchy-Riemann conditions.
31Cauchy’s Integral Formula A complex function f(z) is analytic upon and within the solid linecontour C. Let a be a point within the closed contour such that f(z) isnot zero and define a new function g(z):g(z) is analytic within the contour C except at the point a (simple pole).If the pole is isolated by drawing a circle around a and joining to C, theintegral around this modified contour is 0 (Cauchy’s theorem).The straight dotted lines joining the outside contour C and the inner circle are drawn very close together and their paths are synonymous.
32Cauchy’s Integral Formula Since integration along them will be in opposite directions and g(z) is analytic in the region containing them, the net value of the integral along the straightdotted lines will be zero:Let the value of f(z) on be ; where is a small quantity.0, where is smallCauchy’s integral formula: It permits the evaluation of a function at any point within a closed contour when thevalue of the function on the contour is known.
33The Theory of ResiduesIf a coordinate system with its origin at the singularity of f(z) and no othersingularities of f(z). If the singularity at the origin is a pole of order N, then:will be analytic at all points within the contour C. g(z) can then beexpanded in a power series in z and f(z) will thus be:Laurent expansion of the complex functionThe infinite series of positive powers of z is analytic within and upon C and the integral of these terms will be zero by Cauchy’s theorem.the residue of the function at the poleIf the pole is not at the origin but at z0
34Example Evaluate around a circle centred at the origin If |z| < |a|, the function is analytic within the contourCauchy’s theoremIf |z| > |a|, there is a pole of order 3 at z = a within the contour.Therefore transfer the origin to z = a by putting = z - a.
35Evaluation of residues without Laurent Expansion The complex function f(z) can be expressed in terms of a numerator and a denominator if it has any singularities:Laurent expansionmultiply both sides by (z-a)azIf a simple pole exits at z = a, then g(z) = (z-a)G(z)
36Example Evaluate the residues of Evaluate the residues of Two poles at z = 3 and z = - 4The residue at z = 3:B1= 3/(3+4) = 3/7The residue at z = - 4:B1= - 4/( ) = 4/7Evaluate the residues ofTwo poles at z = iw and z = - iwThe residue at z = iw:B1= eiw/2iwThe residue at z = - iw:B1= -eiw/2iw
37ExampleIf the denominator cannot be factorized, the residue of f(z) at z = a isindeterminateL’Hôpital’s ruleEvaluate around a circle with centre at the origin and radius |z| < /n
38Evaluation of Residues at Multiple Poles If f(z) has a pole of order n at z = a and no other singularity, f(z) is:where n is a finite integer, and F(z) is analytic at z = a.F(z) can be expanded by the Taylor series:Dividing throughout by (z-a)nThe residue at z = a is thecoefficient of (z-a)-1The residue at a pole of order n situated at z = a is:
39Example Evaluate around a circle of radius |z| > |a|. has a pole of order 3 at z = a, and the residue is: