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1 Complex Algebra

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2 Complex Number A complex number is an expression of the type x+iy, where both x and y are real numbers and the symbol real part imaginary part complex conjugate

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3 Argand Diagram Argand 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 imaginary axis. 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 of each are different.

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4 Modulus and Argument For each of the complex numbers represented by the lines OP and OQ, the lengths 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 the alternative 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 is usually written, in polar coordinates

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5 Principal Value When a complex number is expressed in polar coordinate, the principal of value 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 (0.645+2n )radians, and amp(4i-3) would equal to (2.495+2n )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 amplitude of a negative real number is conventionally taken as +.

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6 Algebraic 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 z 3 is the sum of z 1 represented by OP and z 2 represented by OQ, then Eqs(*) give the coordinates of z 3 on the Argand diagram as shown in Figure by the line OR. Using the same values as before, In the same way, the subtraction of two complex numbers can be expressed in the form of the addition z 1 to minus z 2 where

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7 Algebraic 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 two complex numbers are expressed in terms of polar coordinates. Thus To multiply two complex numbers, it is necessary to multiply the moduli and add the argument.

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8 Algebraic Operations on Argand Diagram 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 z 5 and z 6. Thus consider the complex numbers z 1 =(3i-4) and z2=(i-1) with arguments 1 =143 and 2 =135. z 1 z 2 =1-7i and 5 = 1 + 2 =278. The principal value of the argument lying between -180 and +180 is -82. To divide two complex numbers, it is necessary to divide the moduli and subtract the argument.

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9 Conjugate Numbers Two 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 the real axis. Usually, the conjugate of a complex number z is written as If 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 always real. Thus, The division of a true complex by its conjugate will not produce a real number.

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10 Conjugate Numbers

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11 De Moivres Theorem De Moivres theorem: For all rational values of n (positive or negative integer, or a real fraction Note: is not included!

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12 Trigonometric-Exponential Identities Hyperbolic Functions

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13 Derivatives of a Complex Variable Consider the complex variable to be a continuous function, and let and. Then the partial derivative of w w.r.t. x, is: or Similarly, the partial derivative of w w.r.t. y, is: or Cauchy-Riemann conditions They must be satisfied for the derivative of a complex number to have any meaning.

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14 Analytic Functions A 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 z 0 in the region satisfies 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 z 0 which satisfies the Cauchy- Riemann conditions Only analytic functions can be utilized in pure and applied mathematics.

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15 If w = z 3, show that the function satisfies the Cauchy-Riemann conditions and state the region wherein the function is analytic. Cauchy-Riemann conditions Satisfy! Also, for all finite values of z, w is finite. Hence the function w = z 3 is analytic in any region of finite size. (Note, w is not analytic when z =.) Example

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16 If w = z -1, show that the function satisfies the Cauchy-Riemann conditions and state the region wherein the function is analytic. Satisfy! Except from the origin For all finite values of z, except of 0, w is finite. Hence the function w = z -1 is analytic everywhere in the z plane with exception of the one point z = 0. ? Example Cauchy-Riemann conditions

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17 At the origin, y = 0 At the origin, x = 0 As x tends to zero through either positive or negative values, it tends to negative infinity. As 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. Example

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18 Singularities We have seen that the function w = z 3 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 in the 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 single-valued functions (c) Branch points multivalued functions

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19 Poles or Unessential Singularities A pole is a point in the complex plane at which the value of a function becomes infinite. For example, w = z -1 is infinite at z = 0, and we say that the function 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.

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20 If 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 for which g(z) is analytic at z = a. pole, (a=0) n 1 w pole Order = 1 0 a pole w 0 pole order 3 a pole order 4 Order of a Pole

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21 Essential 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 finite factor, they cannot be poles. This type of singularity is called an essential singularity and is portrayed by functions which can be expanded in a descending power series of the variable. Example: e 1/z has an essential singularity at z = 0.

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22 Essential singularities can be distinguished from poles by the fact that they cannot be removed by multiplying by a factor of finite value. Example: infinite at the origin We try to remove the singularity of the function at the origin by multiplying z p It consists of a finite number of positive powers of z, followed by an infinite number of negative powers of z. All terms are positive It is impossible to find a finite value of p which will remove the singularity in e 1/z at the origin. The singularity is essential. Essential Singularities

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23 Branch Points The singularities described above arise from the non-analytic behaviour of single-valued functions. However, multi-valued functions frequently arise in the solution of engineering problems. For example: For any value of z represented by a point on the circumference of the circle in the z plane, there will be two corresponding values of w represented by points in the w plane.

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24 and Cauchy-Riemann conditions in polar coordinates when 0 2 The particular value of z at which the function becomes infinite or zero is called the branch point. The origin is the branch point here. Branch Points

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25 Branch Points A function is only multi-valued around closed contours which enclose the branch point. It is only necessary to eliminate such contours and the function will become single valued. -The simplest way of doing this is to erect a barrier from the branch point to infinity and not allow any curve to cross the barrier. -The function becomes single valued and analytic for all permitted curves.

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26 Barrier - Branch Cut The barrier must start from the branch point but it can go to infinity in any direction in the z plane, and may be either curved or straight. In most normal applications, the barrier is drawn along the 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 which the argument of z is within the range - < <.

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27 Integration of Functions of Complex Variables The integral of f(z) with respect to z is the sum of the product f M (z) z along the curve in the complex plane, where f M (z) is the mean value of f(z) in the length z of the curve. That is, where the suffix C under the integral sign specifies the curve in the z plane 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.

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28 Show that the value of z 2 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 ACDB Along AC, x = 0, z = iy Along CDB, r = 10, z = 10e i Independent of path Example

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29 Evaluate around a circle with its centre at the origin Let z = re i Although the function is not analytic at the origin, Evaluate around a circle with its centre at the origin Let z = re i Example

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30 Cauchys Theorem If 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 upon the 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 a surface integral using Stokes theorem: But for an analytic function, each integral on the right-hand side is zero according to the Cauchy-Riemann conditions.

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31 Cauchys Integral Formula A complex function f(z) is analytic upon and within the solid line contour C. Let a be a point within the closed contour such that f(z) is not 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, the integral around this modified contour is 0 (Cauchys theorem). The straight dotted lines joining the outside contour C and the inner circle are drawn very close together and their paths are synonymous.

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32 Cauchys 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 straight dotted lines will be zero: Let the value of f(z) on be ; where is a small quantity. 0, where is small Cauchys integral formula: It permits the evaluation of a function at any point within a closed contour when the value of the function on the contour is known.

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33 If a coordinate system with its origin at the singularity of f(z) and no other singularities 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 be expanded in a power series in z and f(z) will thus be: Laurent expansion of the complex function The infinite series of positive powers of z is analytic within and upon C and the integral of these terms will be zero by Cauchys theorem. the residue of the function at the pole If the pole is not at the origin but at z 0 The Theory of Residues

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34 Evaluate around a circle centred at the origin If |z| < |a|, the function is analytic within the contour Cauchys theorem If |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. Example

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35 Evaluation 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: If a simple pole exits at z = a, then g(z) = (z-a)G(z) Laurent expansion multiply both sides by (z-a) a z

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36 Evaluate the residues of Two poles at z = 3 and z = - 4 The residue at z = 3: B 1 = 3/(3+4) = 3/7 The residue at z = - 4: B 1 = - 4/(- 4 - 3) = 4/7 Evaluate the residues of Two poles at z = iw and z = - iw The residue at z = iw: B 1 = e iw /2iw The residue at z = - iw: B 1 = -e iw /2iw Example

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37 If the denominator cannot be factorized, the residue of f(z) at z = a is indeterminate LHôpitals rule Evaluate around a circle with centre at the origin and radius |z| < /n Example

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38 Evaluation 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) n The residue at z = a is the coefficient of (z-a) -1 The residue at a pole of order n situated at z = a is:

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39 Evaluate around a circle of radius |z| > |a|. has a pole of order 3 at z = a, and the residue is: Example

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