Post Graduate Programme M.Sc. Mathematics (Previous) Distance Educations Self Instructional Material Block:1 Unit-1 Complex Integration Unit-2 Calculus of Residue 1 | P a g e Paper IV Complex Analysis Block -1 Complex Analysis Introduction : The fundamental idea of extending the real number system by the introduction of complex numbers was first necessitated by the solution of algebraical equations L. Euler (1707-1783) was the first mathematician who introduced the symbol I =√-1 C.F. Gaurs (1777-1855) studied that the algebraic equation with real coefficients has complex roots of the from a+ ib where a and b are real numbers . W.R. Hamilton (1805-1865) also made considerable contribution to the development of the theory of complex number. The BLOCK contains two units. In unit 1 we will discuss complex integration. Under this we will studies the proofs at various theorems based on complex integration like Cauchy’s integral formula. Morera’s theorem, Schwarz lemma etc. in unit 2 we will study about residues and evaluation of integrals using it. Complex Analysis: Unit-1 Complex integration Cauchy –Gousat theorem Cauchy integral formula. Higher order derivatives. Morera’s Theorem Cauchy’s inequality and Liouville’s theorem. The Fundamental theorem of algebra. Taylor’s theorem. Maximum modulus principle. Schwarz lemma. Laurent’s series. Lsoluted singularities Menomarphlc Punctions. The argument principle . Rouche’s theorem. Inverse function theorem. Unit–II Residues. Cauchy’s residue theorem. Evaluation of integrals. Branches of many valued functions with special n reference to argz. logzandz . 2 | P a g e Unit -1 Complex Integration Objectives: After reading this unit, your should to able to, understand the concept of complex integration prove many results based on it understand singularities STRUCTURE: 1.1 Complex integration 1.2 Basic Definitions 1.2.1 Partition 1.2.2 Jordanarc 1.2.3 Rectifiable Arcs 1.2.4 Contours 1.2.5 Simple and Multiply connected Regional. 1.2.6 Analytic functions 1.3 Cauchy’s –Goursat. Theorem 1.4 Cauchy’s Integral Derivalives 1.5 Higher order Inequality 1.6 Cauchy’s Inequality 1.7 Liouville’s theorem 1.8 Taylor’s theorem 1.9 Zero’s of Analysis functions 1.91 Singularities 1.92 Types of Singularity 1.10 Meromorphic functions 1.11 Argument Principle 1.12 Rouche’s Theorem 1.13 Fundamental theorem of algebra 1.14 Maximum modules principle 1.15 Schwarz lemma 1.16 summary 1.17 Self Assessment Test 1.18 Further Readings 3 | P a g e 1.1 Complex integration: In the domain of real variable there are two points of view from which integration may be considered. In the domain of complex variable use start with the definition of an integral as the limit of a sum and latter on establish the connection between differentiation and integration. 1.2 Basic Definitions: 1.2.1 Partitions: Let [a,b] be a closed interval where a,b are real number. Subdivide the interval [a,b] into n Sub- intervals [t0,t1], [t1,t2], _ _ _ _ _ _ [tn,tn], by inserting n-1 intermediate. points t1, t2, _ _ _ tn-1, Satisfying the inequality a= t0 < t1 < t2 _ _ _ _ < tn = b Then the set P={ t0, t1 _ _ _ _ tn } is called partition of the interval [a,b]. 1.2.2 Jordan Arc: Suppose a point Z lies on an ace L is defined by Z=Z(t)= x (t) +iy (t) Where t runs through the interval a ≤ t ≤ b and x (t), y(t) are continuous function of t Then L is called simple or Jordan are if Z(t1)= Z(t2) only when t1 = t2 1.2.3 Rectifiable Arcs: Suppose L denotes a Jordan arc defined by Z=Z(t)= x (t) + iy (t), a ≤ t ≤ b Z0 Zn-1 Z1 Z3 Z2 Zn 4 | P a g e Let P={a=t0, t1 , t2 , _ _ _ _ _ tn = b } be any partition of [a,b] let Z0 ,Z1, …...Zn be the points of the are wr responding to their value. Then the length of this polygonal are is; = ∑ = ∑ | | If this sum tends to a unique value then it is called rectifiable. 1.2.4 Contour: A contour is a continuous chain of a finite number of regular arc (rectifiable arc) 1.2.5 Simply and Multiply Connected Regions: A region in the argand plane in which every closed curve can be shrunk to a point without passing out of the region is called a simply connected region otherwise it is multiply connected. 1.2.6 Analytic Functions: A function of Z which is one valued and differentiable at every point of a domain D. Save possibly for a finite number of exceptional points is called analytic in the domain D. Those exceptional points are called Singularities. 1.3 Cauchy-Goursat Theorem Statement: If a function f(Z) is analytic and single valued inside and on a simple closed contour C, then ∫ ( ) Proof: For the proof of the theorem use will use two lemmas which are as follows; Lemma1: if C is a closed contour, then the two integrals ∫ ∫ Lemma 2 : Let f(z) be analytic within and on a closed contour C. Then for any given >0, it is always possible to divide the region inside C into number of meshes either 5 | P a g e complete square Cn or partial Squarer Dn Such that within each mesh there exists a point Z0 such that | ( ) ( ) ( ) ( )| | | For every Z within or on the mesh. Now we divide the interior of C into complete squares C1, C2, _ _ _ _ Cm and partial Squares D1, D2, ____ _ _ _ Dn. Now ∫ ( ) ∑ ∫ ( ) ∑ ∫ ( ) y A D B C x 0 For the proof of the theorem. We will consider two cases. Case- I : Consider the mesh Cm Then ( ) ( ) ( ) ( ) ( ) ( ) ( ) Where | ( )| ̇ ∫ ( ) ∫ ,( ) ( ) ( )- ∫ ( ) ( ) From lemma (1) we get ∫ ( ) ( ) as | ( )| |∫ ( ) | √ Where denotes the length of diagonal of Cm 6 | P a g e Case II – Consider the irregular sub regions Dn. let Sn denotes the length of the curved part of Dn Then |∫ ( ) ( ) | √ ( ) Now (1) becomes |∫ ( ) | ∑ |∫ ( ) | ∑ ∫ ( ) = ∑ |∫ ( ) ( ) | ∑ |∫ ( ) ( ) | ̇ |∫ ( ) | √ ∑( ) √ ∑ all the terms in the right hand side are constant ̇ |∫ ( ) | this is only true when ∫ ( ) 1.4 Cauchy’s Integral Formula: Statement : let ( ) be analytic within and on a closed contour C and let any point within C . Then ; ( ) ( ) ∫ Proof : We define a circle by the equation | | = where and is the distance from to the neares point of C. 7 | P a g e Consider a function ( ) ( ) ( ) ( ) ∫ ∫ where C and are both traversed in the positive sense. As ( ) is continuous at so by defintion for given | ( ) ( )| whenever | | -----------(1) Now ( ) ∫ ( ) ( ) ( ) ∫ ∫ --------------(2) Put = We have ( ) ∫ ( ) ∫ ( ) ∫ ( ) Hence (2) becomes ( ) ( ) ( ) ∫ ∫ ( ) or ( ) ( ) ( ) ∫ ( ) ∫ ---------(3) Since we may choose as small as wee please, we take . Thus the inequality (1) is satisfied for all points on . 8 | P a g e Hence ( ) ( ) ( ) ( ) | ∫ | | ∫ )| ∫ | ( ) ( )| ∫ Thus ( ) ( ) | ∫ | −-------------------(4) Then it follows from (3) and (4) that ( ) | ∫ ( )| We now observe that is arbitrary and the left hand Side is independent of it. Thus implies that. ( ) ∫ ( ) ( ) ∫ ( ) or ( ) ∫ 1.5 Higher order derivatives: Statement: Let ( ) be analytic within and on the boundary C* of a simply connected region and let be any point within C. Then derivatives of all orders are analytic and given by 9 | P a g e ( ) ( ) ∫ ( ) Proof: We first show that ( ) is analytic inside C. To prove this it is enough to show that ( ) has a differential coefficient at every point inside C. We have by Cauchy’s formula for ( ) and ( ) ( ) ( ) ( ) ∫ [ ] ( ) ( ) ( ) ( ) ∫ ( ) ( ) ( ) follows that ( ) ( ) ( ) ∫ ( ) ( ) ( ) ∫ ( ) [ ] ( ) ( ) ( ) ( ) ∫ ( ) [ ] ( ) ( ) 0 ( ) 1 ∫ ( ) ( ) ( ) Where is the circle | | lying entirely within C. Hence ( ) ( ) ( ) | ∫ | ( ) 10 | P a g e | | | | | | ∫ | ( )|| | ( ) ( ) | | | | ( ⁄ ) by means of arguments parallel to those used in the proof to Cauchy’s formula for ( ) | | . | |/ Where M is the upper bound of ( ) in D. Hence when | | the right hand side of (1) will tends to Zero. We then have ( ) ( ) ( ) ∫ ( ) or ( ) ( ) ∫ ( ) Similarly ( ) ( ) ( ) ∫ ( ) ̇ ( ) ( ) ( ) ( ) [∫ ∫ ] ( ) ( ) ∫ 6 ( ) ( )( ) 7 ( ) As we have ( ) ( ) ( ) ( ) ( ) ( ) ∫ ( ) 11 | P a g e or ( ) ( ) ( ) ( ) ∫ ( ) Replacing by ( ) ( ) ∫ ( ) 1.6 Cauchy’s Inequality: Statement : let ( ) be analytic within and on a circle C ( ) defined by | | If | ( )| on C, then | ( )| Proof: By Cauchy’s formula for the nth derivative of an analytic function at a point we have ( ) ( ) ∫ ( ) ̇ | ( )| | | ( ) | ( )| ∫ | | ∫ | | | | ---------------(1) Now we may write the equation of the circle | | as So that Thus | | Hence (1) will give ( ) | ( )| ∫ Thus | ( )( )| Remark : If the function ( ) is analytic in every finite region of the Z plane then ( ) is called entire function. 1.7 Liouvilles’s Theorem: Statement: A bounded entire function is constant. Proof:- Let a be any arbitrary point of the Z plane. Now ( ) is by hypothesis analytic for | | 12 | P a g e however large R may be Moreover, | ( )| Satisfies the inequality. | ( )| On any circle | | where M denotes the upper bound at | ( )| for z lying in all finite regions of the z plane. Hence by Cauchy’s inequality we have | ( )| As we get ( ) But is an arbitrary point in the Z plane. Thus derivative of ( ) every where hence ( ) is constant. 1.8 Taylor’s Theorem: Statement: Let ( ) be analytic at all points within a circle C0 with center Z0 and radius R and Let Z be any point inside C. Then ( ) ( ) ( ) ∑ ( ) Proof: Let Z be any point inside the circle C0 with centre Z0 and radius R. Let | | and Let C be the Circle with centre Z0 and radius Such that then Z lies inside C. By Cauchy integral formula we have ( ) ( ) ∫ ( ) ∫ ( ) ( ) ( ) ∫ [ ] ( ) ( ) ∫ 6 ( ) ( ) ( ) 7 13 | P a g e ( ) ( ) ∫ ∫ ( ) ( ) ( ) ( ) ∫ ( )( ) Now we will apply Cauchy’s formula and we get ( ) ( ) ( ) ( ) Where ( ) ( ) ∫ ( )( ) Now for the proof of the theorem it is sufficient to show that as So ( ) ( ) | | | ∫ | ( )( ) | | | ( )| ∫ | | | | | || | ∫ | | ( ) ( ) ( ) as ( ) ( ) ( ) ( ) ∫ 14 | P a g e Remark: when Z0=0 the series is called Maclaurin series Example: Prove that ( ) ( ) ( ) + -----+ Solution : Let ( ) Then by Tuylor’s theorem ( ) ( ) ( ) ∫( ) ( ) ( ) ( ) ( ) ( ) ( ) ----- (A) Now replace Z by 1 in (a) ( ) ( ) ( ) ‘ ‘ ‘ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) 1.9 Zero’s of Analytic functions: A Zero of an analytic function ( )is any value of Z for which ( )vanishes.