NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

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1 NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 29

2 Complex Analysis Module: 4: Complex Integration Lecture: 2: Contour Inegration and Cauchy s Theorem A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 2 / 29

3 Complex integral Contour Definition 1 A contour is a finite collection of smooth arcs joined end to end, or in other words, a contour is a piecewise continuous smooth arc that are joined end to end. 2 A contour is said to be simple, if it doesn t intersect itself,i.e no t 1, and t 2 such that z(t 1 ) = z(t 2 ), a t 1 t 2 b or z(t) is one-one function (univalent). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 3 / 29

4 Complex integral Contour y contour x A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 4 / 29

5 Complex integral Contour An arc is closed, if z(a) = z(b) for a t b (or) if there exist no initial (or) final point. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 29

6 Complex integral Contour An arc is closed, if z(a) = z(b) for a t b (or) if there exist no initial (or) final point. A simple closed contour is called Jordan curve. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 29

7 Complex integral Contour An arc is closed, if z(a) = z(b) for a t b (or) if there exist no initial (or) final point. A simple closed contour is called Jordan curve. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 29

8 Complex integral Contour An arc is closed, if z(a) = z(b) for a t b (or) if there exist no initial (or) final point. A simple closed contour is called Jordan curve. Jordan curve theorem Theorem Every Jordan curve divides C into two parts interior and exterior, of which one is bounded and another is unbounded. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 29

9 Complex integral Jordan curve Jordan curve is connected. The positive orientation is taken such that interior lies in the left side of the contour. An arc is closed, if it is simple, except for end points. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 6 / 29

10 Complex integral Jordan curve Jordan curve is connected. The positive orientation is taken such that interior lies in the left side of the contour. An arc is closed, if it is simple, except for end points. Example For z < 1, the positive orientation is anticlockwise and for z > 1, the positive orientation is clockwise. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 6 / 29

11 Complex integral Theorem on anti-derivative Theorem Let f be continuous on [a,b]. If there exist an analytic function F, such that F (z) = f (z) on [a,b], then b a f (z)dz = F(b) F(a). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 7 / 29

12 Complex integral Theorem on anti-derivative Example z2 dz Let f (z) = z 2, define F(z) = z3, F(z) is analytic in [3,4] and 3 F (z) = f (z) in [3,4]. = 4 3 z2 dz = [ z3 3 ]4 3 = = A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 8 / 29

13 Complex integral Theorem on anti-derivative Example 2. π 0 eit dt. f (z) = e it. define F(z) = e it then F in [0, π]and F (z) = ie it = if (z) in [0, π]. = π 0 eit = [ F(z) ] π F(π) F(0) 0 = = [1 + 1] = 2. i i A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 9 / 29

14 Complex integral Theorem on anti-derivative Example 3. C (z z 0) n dz where C = { z z 0 = r} is traversed once in the positive direction. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 29

15 Complex integral Theorem on anti-derivative Example 3. C (z z 0) n dz where C = { z z 0 = r} is traversed once in the positive direction. = z = z 0 + re it, 0 t 2π. (z z 0 ) n = r n e int, dz = ire it dt I = i 2π 0 r n+1 e i(n+1)t dt = ir n+1 2π 0 e i(n+1)t dt. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 29

16 Complex integral Theorem on anti-derivative Example 3. C (z z 0) n dz where C = { z z 0 = r} is traversed once in the positive direction. = z = z 0 + re it, 0 t 2π. (z z 0 ) n = r n e int, dz = ire it dt I = i 2π 0 r n+1 e i(n+1)t dt = ir n+1 2π 0 e i(n+1)t dt. Case(i): Let n 1 then F(z) = ei(n+1)t n + 1 Hence I=ir n+1 1 n + 1 [ei(n+1)2π e i(n+1)0 ] = 0 in [0, 2π]. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 29

17 Complex integral Theorem on anti-derivative Example 3. C (z z 0) n dz where C = { z z 0 = r} is traversed once in the positive direction. = z = z 0 + re it, 0 t 2π. (z z 0 ) n = r n e int, dz = ire it dt I = i 2π 0 r n+1 e i(n+1)t dt = ir n+1 2π 0 e i(n+1)t dt. Case(i): Let n 1 then F(z) = ei(n+1)t n + 1 Hence I=ir n+1 1 n + 1 [ei(n+1)2π e i(n+1)0 ] = 0 Case(ii): Let n = 1. then in [0, 2π]. 2π 2π I = ir n+1 e i(n+1)t dt = ir 1+1 e i( 1+1)t dt = i 0 0 2π 0 dt = 2πi. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 29

18 Complex integral Theorem on anti-derivative Example Hence C (z z 0 ) n dt = { 0 if n 1, 2πi if n = 1.. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 29

19 Complex integral Theorem on anti-derivative Example Hence C (z z 0 ) n dt = { 0 if n 1, 2πi if n = 1.. Important note. This example is important for an interesting concept at a later stage. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 29

20 Complex integration Cauchy theorems A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 12 / 29

21 Complex integration Green s theorem Theorem Let P and Q be functions defined on a region D R 2 such that P and Q are continuous with continuous first order partial derivatives in D R 2. Then for any piecewise differentiable curve C defined on the region D bounded by a smooth surface, C (Pdx + Qdy) = D ( Q x P x )dxdy. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 13 / 29

22 Complex integration Cauchy Fundamental theorem Theorem Let f be analytic and f be continuous in a simply connected domain D. Then for any closed contour C, lying entirely in D, f (z)dz = 0. C A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 14 / 29

23 Complex integration Proof of Cauchy Fundamental theorem Since f is analytic f = u + iv satisfies C R equations = u x = v y and v x = u y. Further f is continuous implies u x, u y, v x, v y are continuous. Now C f (z)dz = = C C = I 1 + I 2. (u + iv)(dx + idy) (udx vdy) + i (vdx + udy) f analytic implies, u and v are continuous. Hence the hypothesis of Green s Theorem is satisfied. C A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 15 / 29

24 Complex integration Proof of Cauchy Fundamental theorem This gives I 1 = I 2 = C C (udx vdy) = (vdx + udy) = D D ( v x = D u y )dxdy 0dxdy = 0 ( u x v )dxdy = 0. y A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 16 / 29

25 Complex integration Cauchy Fundamental theorem Example Since the function z n, exp(z) are analytic in C, f (z)dz = 0 C for any simple closed curve C in C and any f (z) given by a polynomial or exp(z). A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 17 / 29

26 Complex integration A question Can we generalize Cauchy fundamental theorem by removing the condition f (z) is continuous in D? A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 18 / 29

27 Complex integration A question Can we generalize Cauchy fundamental theorem by removing the condition f (z) is continuous in D? YES! A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 18 / 29

28 Complex integration Cauchy-Goursat Theorem If a function f is analytic at all points inside and on a simple closed contour C, then f (z)dz = 0 C A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 19 / 29

29 Complex integration Cauchy-Goursat Theorem If a function f is analytic at all points inside and on a simple closed contour C, then f (z)dz = 0 Proof Beyond the scope of this course. C A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 19 / 29

30 Cauchy theorem Simply connected domain Theorem Let f be analytic in a simply connected domain D, then for any closed contour C, lying entirely on D, f (z)dz = 0. C A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 20 / 29

31 Cauchy theorem Simply connected domain Theorem Let f be analytic in a simply connected domain D, then for any closed contour C, lying entirely on D, f (z)dz = 0. Note C This is alternative statement of Cauchy-Goursat theorem. The result is true, if the simple closed contour is replaced by arbitrary closed contour. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 20 / 29

32 Complex integration Cauchy Theorem for multiply connected domains Theorem Let C, C 1, C 2,..., C m be m + 1 simple closed contours with positive orientation such that each C k, k = 1, 2,..., m lies inside C and the set of points interior to each C i has no points in common to the interior of C j, i, j = 1, 2,..., m. Let f (z) be analytic in a domain D that contains all the points between C and C 1, C 2,..., C m. Then C f (z)dz = m k=1 C k f (z)dz A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 21 / 29

33 Cauchy theorem for multiply connected domains y C 1 z 1 C 2 z 2 z m C m C O x A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 22 / 29

34 Cauchy Theorem for multiply connected domains Proof Proof? A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 23 / 29

35 Cauchy Theorem for multiply connected domains Proof Without loss of generality, let us assume D as bounded domain. We recall that a domain D is multiply connected or m-connected, if D is connected and the complement of D has m components. A domain D is m-connected, if its boundary has m Jordan curves of which the interior of m 1 Jordan curves have no points in common and that points in these interiors cannot be connected to the point at infinity. For example the annular region R 1 < z < R 2 is a finite doubly connected domain. Let us assume that the result of Cauchy-Goursat theorem for simply connected domain be true. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 24 / 29

36 Cauchy Theorem for multiply connected domains Proof Define C 1 C2 as the boundary of the domain D such that C 1 := {z : z = R 1 } and C 2 := {z : z = R 2 }. Define Γ 1 to be arc from any point z 11 of C 1 to a point z 21 in C 2. Define Γ 2 to be arc from any point z 22 of C 2 to a point z 12 in C 1. Assume that z 11 z 12 and z 21 z 22 are arbitrarily very small. Then Γ 1 and Γ 2 are of opposite orientation and Γ 1 f (z)dz = Γ 2 f (z)dz. Define C = C 1 C2 Γ1 Γ2 so that C is the Jordan curve and interior of C and its complement are connected. Now apply Cauchy Goursat theorem for C. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 25 / 29

37 Cauchy Theorem for multiply connected domains Proof Now C f (z)dz = (C 1 C 2 Γ 1 Γ 2 ) f (z)dz = f (z)dz + f (z)dz + f (z)dz + f (z)dz C 1 C 2 Γ 1 Γ 2 = f (z)dz + f (z)dz + f (z)dz f (z)dz C 1 C 2 Γ 1 Γ 1 = f (z)dz + f (z)dz. C 1 C 2 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 26 / 29

38 Complex Integration Independence of path Theorem Let f be continuous in D and has antiderivative F throughout D, i.e. d dz F = f in D. then for any contour Γ in D, with z I as initial point and z T as terminal point f (z)dz = F (z T ) F(z I ). Γ A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 27 / 29

39 Complex Integration Independence of path Theorem Let f be continuous in D and has antiderivative F throughout D, i.e. d dz F = f in D. then for any contour Γ in D, with z I as initial point and z T as terminal point f (z)dz = F (z T ) F(z I ). Γ This means - the parametric representation is not required. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 27 / 29

40 Independence of path Proof Let the contour Γ be partitioned into z I, z 1, z 2,..., z n, z T such that Γ 0 is from z I to z 1, Γ 1 is from z 1 to z 2,..., Γ n is from z n to z T. Then Γ f (z)dz = n j=1 Γ j f (z)dz. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 28 / 29

41 Independence of path Proof Choose n so large such that Γ k from z k to z k+1 leads to a small segment parallel to real or imaginary axis. Hence this Γ k = [a, b] with both a and b either real or purely imaginary. Hence b a f (z)dz = = = b a b a b a f (z(t))z (t)dt F (z(t))dz(t) d d(z(t)) F(z(t))d(z(t)) = F(z(t)) b = f (z(b)) F(z(a)). a a t b A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 29 / 29