MSO202: Introduction To Complex Analysis

Transcription

1 1 MSO0: Introduction To Complex Analysis Lecture z a Geometrical Interpretation of Ha { z:im( ) 0}, b 0 b z We first give the geometrical of H0 { z:im( ) 0, b 1}, i.e. b when a = 0 and b 1. In this case, write z r(cos isin ) and bcos isin, then z r[cos( ) isin( )]. b a L a b Now, z H0 sin( ) 0 ( z Im( ) 0) b 0 H 0 L is the half plane lying to the left of line L passing through origin, if one walks along L in the direction of b.

2 Now, a w a w wh0 awha ( Im( ) Im( ) 0). b b Therefore, H a is the half plane lying to the left of line L a passing through a and in the direction of b (as we walk in the direction of b ). Ellipses in terms of Complex Numbers za za r represents an ellipse for r > 0 and a r z r a 0 a za za r represents the interior of the above ellipse za za r represents the exterior of the above ellipse. Annulus r1 za r a r 1 r z

3 3 Half Plane in terms of Complex Numbers H0 { z: z1 z1} represents the right half plane. z 1 i 1 Another way to represent H 0 is z H0 z:im 0 z:rez 0 i Hyperbola in terms of Complex Numbers The hyperbola is represented by the parametric equation 1 1 z t i ( x t, y xy 1) t t

4 Interior Point of a Set, Open Set, Connected Set and Domain of Complex Numbers: Interior Point, Exterior Point of a Set: A point is called interior point of a set A C if an open disk centered at this point and contained in A can be found. A point is called exterior point of a set A if it is an interior c point of A (complement of A). Example: Every point of A { z: za za r} is an interior * point of set A. Every point of A { z: za za r} is an exterior point of set A. Open Set: A set G C is called an open set if every point of G is an interior point of G. Example: The set A { z: za za r} is an open set. Connected Set: A set A C is called connected, if for every pair of points in A, a continuous curve contained in A can be found that joins these points. Example: (i) The set A { z: za za r} is connected (ii) union of two disjoint disks is not a connected set. Domain: A set A C which is both open and connected is called domain. 4

5 Example: (i) The set A { z: za za r} is a domain (ii) union of two disjoint disks is not a domain (iii) The set A { z: za za r} is not a domain. 5 Convergent Sequences of Complex Numbers. A sequence z n of complex numbers is said to be convergent if, for some z0 and every 0, there exists a non negative integer n such that is 0 z z for alln n (*) n 0 0 z 0 is called limit of the sequence and we use the notation z0 lim z n. n It is easily seen that zn xn i yn converges to z0 x0 i y0 iff and only if x n converges to x 0 and y n converges to y 0 (use x x z z, y y z z ) n 0 n 0 n 0 n 0

6 6 All the results as well as their proofs about convergence of sequences of complex numbers are analogous to corresponding results and their proofs for convergence of sequences of real numbers. For example, it can be easily shown by arguments similar to those for real sequences that the sequence { z n } converges to complex number 0 as n if z 1 (using (*)), while, for n z 1, the sequence { z } does not converge, since cosn isin n, real, does not converge (since, the sequence cosn of its real part does not converge as n ).

7 7 Continuous Functions. A function f : is called continuous at z0, if for any 0, there exists a 0 such that f( z) f( z ) for zz. 0 0 The function f is said to be continuous in a set A if f is continuous at all the points of A. The definition of continuity is meaningful only if f is defined in some neighbourhood of z 0(i.e. a disk centered at z 0). Example 1: Let xy z Im( ) if ( x, y) (0,0) f( x, y) x y z 0 if ( x, y) (0,0) m Then, f ( x, y) along the line y mx 1 m Consequently, f is not continuous at (0, 0). as ( xy, ) 0,0.

8 8 Example : Let x y if ( x, y) (0,0) f( x, y) 4 x y 0 if ( x, y) (0,0) f ( x, y) 0 along the line y mx as ( x, y) 0,0 1 but f( x, y) along y x as ( x, y) (0,0). Consequently, f is not continuous at (0, 0). Then,

9 9 Proposition 1. TFAE ( The following are equivalent) (i) f is continuous at z 0 (ii) zn z0 f( zn) f( z0) Proof. Equivalence of (i) and (ii) () i ( ii) : Let for any 0, there exists a 0 such that f( z) f( z0) for zz0. (1) Now, zn z0 zn zz0 for all n n0 f ( zn ) f ( z0) for all n n0 f ( z ) f( z ) n 0 ( ii) ( i) : Let zn z0 f( zn) f( z0). Let f be not continuous at z 0. Then, 0 such that, for every natural number n, 0 zz 1/ n contains a point satisfying 0 * n 0 0. * * n 0 n 0 f( z ) f( z ) z z but f ( z ) f ( z ). * z n

10 Proposition. The functions f g, f, fg, f / g ( g 0) are continuous whenever f and g are continuous. Converse need not be true. Proposition 3. If f is continuous in A and g is continuous in the range of f, then g f is continuous in A. The proofs of above propositions are analogous to corresponding proofs for real valued functions of real variables. 10

11 11 Examples. (i) Any polynomial in z is continuous in (use Proposition ). (ii) f is continuous if and only if Re( f ) and Im( f ) are continuous (use Proposition and Re( f ) f, Im( f ) f ). (iii) f is continuous if and only if f is continuous (use f f ). (iv) f is continuous, then f is continuous. Converse need not be true, e.g. consider, f ( z) u( z) iv( z), where, if z has rational coordinates uz ( ), vz ( )., otherwise

12 1 Differentiable Functions. A function f : 0 is called differentiable at z 0 if f( z0 ) f( z0) lim (1) exists finitely. In that case, the limit in (1) is called the derivative of f(z) at the point z 0 and is denoted by f ( z0). Note that f ( z0) can also be written as f f ( z0 z) f( z0) f( z) f( z0) ( z0) lim lim z zz z 0 zz0 ( take z z z) 0 0 Remark. All the results on differentiability of functions f : are true for differentiability of functions f : and can be proved analogously.

13 13 Note: If f is diff. it is cont. but the converse need not be true. (Ex. f ( z) z is cont. everywhere but is diff. only at 0) z0 z z0 z0 z z0zz0 z z0 lim lim z 0 z z 0 z z lim ( z0 zz0 ) z 0 z = 0 if z0 0 (limit does not exist if z0 0)

14 14 Analytic Function. A function f is said to be analytic at z 0 if f (i) f is differentiable at z 0 and (ii) f is differentiable in some neighbourhood of z 0 (i.e. in a disk centered at z 0). The function f is said to be analytic in a set A, if f is analytic at all points of A. It is clear that if f is differentiable in any open set G, then f is analytic in G. Converse holds obviously. Examples: (i) z is not analytic anywhere. (ii) Any polynomial is analytic at all points of.