Complex Integration (2A) Young Won Lim 1/29/13

Transcription

1 omplex Integration (2A)

2 opyright (c) 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free ocumentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-over Texts, and no Back-over Texts. A copy of the license is included in the section entitled "GNU Free ocumentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 ontour Integrals defined at points of a smooth curve a smooth curve is defined by x = x(t) a t b y = y (t) The contour integral of f along dz = b = a (u +iv)(dx +i dy) = u dx v dy + i v dx +udy [u x'(t) v y '(t)]d t + i[v x '(t) +u y '(t)] dt b dz = a b = (u +iv)(x '(t) +i y '(t))dt a f (z (t))z '(t)dt z (t) = x(t) +i y (t) z '(t) = x '(t) +i y '(t) a t b omplex Integration (2A) 3

4 onnected Region onnected omains Simply onnected oubly onnected Triply onnected losed Paths simple closed path not simple closed path not simple closed path omplex Integration (2A) 4

5 ontour Integration Evaluation (1) Indefinite Integration of Analytic Functions : analytic in a simply connected domain = F '(z) There exists an indefinite integral in : an analytic function F (z) z 1 dz = F(z 1 ) F (z 0 ) z 0 for every path in between z 0 and z 1 (2) Integration by the Use of the Path : a piecewise smooth path represented by z = z (t) (a t b) a continuous function on b dz = a f [ z(t)] z '(t) dt omplex Integration (2A) 5

6 ontour Integration Evaluation f(z) = 1/z (1) Indefinite Integration of Analytic Functions z 1 = z 0 z 1 f (z ) dz = F ( z 1 ) F ( z 0 ) = 0 z 0 But f (z ) = 1 z not analytic at z = 0 cannot apply this method (2) Integration by the Use of the Path : the unit circle z (t) = cost + isin t = e i t (0 t 2π) z '(t) = sint + icost = i e it 2π it f (z ) dz = ie 0 e it 2 π dt = i dt = 2πi 0 omplex Integration (2A) 6

7 ontour Integration Evaluation f(z) = z m (1) Indefinite Integration of Analytic Functions z 1 = z 0 z 1 f (z ) dz = F ( z 1 ) F ( z 0 ) = 0 z 0 But not analytic at f (z ) = z m z = 0 for m < 0 cannot apply this method (2) Integration by the Use of the Path : the unit circle z (t) = cost + isin t = e i t (0 t 2π) z '(t) = sint + icost = i e it 2π f (z ) dz = 0 2 π e mit i e it dt = ie i(m+1)t dt 0 2 π = i[ 0 2 π cos((m+1)t) dt + i sin((m+1)t) dt] 0 z m dz = 2πi 0 (m = 1) (m 1) omplex Integration (2A) 7

8 auchy's Integral Theorem (1) f '(z) : analytic in a simply connected domain : continuous in a simply connected domain for every simple closed contour in dz = 0 dz = (u+iv)(dx+i dy) = udx v dy + iv dx+u dy = ( v x u ) y d A + i ( u x v ) y d A = 0 u y = v x u x = v y omplex Integration (2A) 8

9 auchy's Integral Theorem (2) f '(z) : analytic in a simply connected domain : continuous in a simply connected domain for every simple closed contour in dz = f (z ) dz = f (z ) dz = 0 f (z ) dz = 0 omplex Integration (2A) 9

10 auchy-goursat Theorem (1) auchy-goursat Theorem : analytic in a simply connected domain for every simple closed contour in dz = 0 auchy Theorem f '(z) : analytic in a simply connected domain : continuous in a simply connected domain for every simple closed contour in dz = 0 omplex Integration (2A) 10

11 auchy-goursat Theorem (2) : analytic in a simply connected domain for every simple closed contour in dz = 0 f '(z) : continuous in a simply connected domain simple closed curve a continuously turning tangent except possibly at a finite number of points allow a finite number of corners (not smooth) omplex Integration (2A) 11

12 auchy-goursat Theorem (3) : analytic in a multiply connected domain for every simple closed contour in dz = 0 doubly connected domain simply connected domain ' 1 cut ' 1 ' 1 ccw dz + dz = 0 cw 1 ccw dz = ccw 1 dz omplex Integration (2A) 12

13 auchy-goursat Theorem (4) triply connected domain simply connected domain ' cuts 2 ' ' ccw cw 1 cw 2 f ( z) dz + f ( z) dz + f ( z) dz = 0 1 f ( z ) dz = f (z ) dz + 2 f ( z) dz omplex Integration (2A) 13

14 Independence of the Path z 0, z 1 : points in a domain For all contours in with an initial point z 0 and a terminal point z 1 The value of its contour integral is the same dz : independence of the path z 1 z z 0 1 z 0 1 f (z ) dz = 2 f (z ) dz 1 f (z ) + 2 f ( z) dz = 0 omplex Integration (2A) 14

15 Analyticity Path Independence : analytic in a simply connected domain dz : independence of the path eformation of ontours z z z 1 2 ' ' 1 z omplex Integration (2A) 15

16 Principle of eformation of Path Impose a continuous deformation of the path of an integral As long as deforming path always contains only points at which f(z) is analytic, the integral retains the same value z 2 z 2 m 1 z 1 z 1 (z z 0 ) m dz = 0 z 2 continuous deformation : impossible z 2 m = 1 not necessarily zero (z z 0 ) m dz = 0 z 1 z 0 z 1 (z z 0 ) m dz = 2π i omplex Integration (2A) 16

17 Antiderivative : continuous in a domain = F '(z) for every z in a domain F (z) : antiderivative of F (z) F (z) F (z) : antiderivative of has a derivative at every z in a domain analytic at every z in a domain continuous at every z in a domain ifferentiability implies continuity Fundamental Theorem of alculus b a f (x) dx = F (b) F (a) omplex Integration (2A) 17

18 Fundamental Theorem (1) : continuous in a domain = F '(z) for every z in a domain F (z) : antiderivative of Fundamental Theorem for ontour Integrals F (z) : continuous in a domain : antiderivative of F '(z) = for every z in a domain For any contour in with an initial point z 1 and a terminal point z 2 dz = F(z 2 ) F (z 1 ) omplex Integration (2A) 18

19 Fundamental Theorem (2) F (z) : continuous in a domain : antiderivative of F '(z) = for every z in a domain For any contour in with an initial point z 1 and a terminal point z 2 dz = F(z 2 ) F (z 1 ) b dz = a b = a f (z(t)) z '(t) dt F '(z(t))z '(t) dt b = a d dt F (z(t)) dt = F (z(b)) F (z(a)) = F (z 2 ) F (z 1 ) omplex Integration (2A) 19

20 Antiderivative and Path Independence F (z) : continuous in a domain : antiderivative of F '(z) = for every z in a domain For any contour in with an initial point z 1 and a terminal point z 2 dz = F(z 2 ) F (z 1 ) F (z) : continuous in a domain : antiderivative of F '(z) = for every z in a domain dz : independence of the path omplex Integration (2A) 20

21 Existence of an Antiderivative : analytic in a simply connected domain F (z) : antiderivative of f(z) F '(z) = for every z in a domain z = e w (z 0) w = ln z (z 0) z = x + i y = e u + iv = e u (cosv+isinv) = e u cosv + i e u sinv > 0 0 d d z L n z = 1 z principal value : multiply connected domain : simple closed path L n z : not analytic in 1 L n z is not an antiderivative of z in L n z 1 z dz = 0 1 z dz = 2π i is not continuous on the negative real axis branch cut omplex Integration (2A) 21

22 auchy's Integral Formula (1) f (z ) analytic in ( z z 0 ) not analytic in : simply connected domain ': multiply connected domain z 0 z 0 f (z 0 ) dz = 0 The value of an analytic function f at any point z 0 in a simply connected domain can be represented by a contour integral (z z 0 ) dz = 0 not necessarily zero (z z 0 ) dz = 2πi f (z 0) f (z 0 ) = 1 2π i (z z 0 ) dz omplex Integration (2A) 22

23 auchy's Integral Formula (2) : analytic on and inside simple close curve f (a) = 1 2π i z a d z the value of at a point z = a inside a a ' dz = 0 ccw dz z a + cw ' dz z a = 0 omplex Integration (2A) 23

24 auchy's Integral Formula (3) : analytic on and inside simple close curve f (a) = 1 2π i z a d z the value of at a point z = a inside ccw dz z a + cw ' dz z a = 0 ' ccw dz z a = ccw ' dz z a a omplex Integration (2A) 24

25 auchy's Integral Formula (4) ccw dz z a = ccw ' dz z a = 2π i f (a) along ' z a = ρe iθ as z a ρ 0 ' z = a ρe iθ dz = iρe i θ dθ z a '' d z z a = iρei θ dθ ρe iθ a ccw f ( z) dz z a 2π = 0 f ( z)i dθ = 2π i f (a) omplex Integration (2A) 25

26 auchy's Integral Formula (5) d z ( z a) = iρeiθ dθ 2 (ρe i θ ) 2 along ' z a = ρe iθ ccw f ( z) dz (z a) 2 2 π = 0 f ( z )i dθ ρe i θ a z = a ρe iθ dz = iρe i θ dθ 2 π = 0 f ( z ) ρ ie i θ dθ = [ f (z ρ ) e i θ ] 2 π 0 = f ( z) ρ (e i2 π e i 0 ) = 0 d z = iρe iθ dθ ccw 2 π f ( z) dz = 0 = [ f (z )ρe i θ ] 0 2 π f ( z)iρe i θ dθ = f ( z)ρ(e i2 π e i0 ) = 0 (z a) d z = ρ e iθ iρe iθ dθ ccw 2 π = 0 2 π ( z a) f ( z) dz = 0 f ( z)ρ 2 ie i 2θ dθ = f ( z) ρ 2 (e i4 π e i 0 ) = 0 f ( z)i(ρe iθ ) 2 dθ = [ f (z ) ρ 2 ei2 θ ] 0 2 π omplex Integration (2A) 26

27 auchy's Integral Formula (6) : analytic on and inside simple close curve f (a) = 1 2π i z a d z the value of at a point z = a inside = 1 2π i f (w) w z dw omplex Integration (2A) 27

28 auchy's Integral Formula for erivatives : analytic on and inside simple close curve f (a) = 1 2π i z a d z the value of at a point z = a inside f (n) (a) = n! 2π i (z a) n+1 d z omplex Integration (2A) 28

29 References [1] [2] [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] E. Kreyszig, Advanced Engineering Mathematics [5]. G. Zill, W. S. Wright, Advanced Engineering Mathematics