Complex Series (3A) 8/7/3
Copyright (c) 202, 203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave. 8/7/3
Power and Taylor Series A power series Taylor series n=0 c n (z a) n n=0 f (n) (a) n! (z a) n = c 0 + c (z a) + c 2 ( z a) 2 + always converges if z a < R can also be differentiated = f (a) + f '(a)(z a) + f ''(a) 2 ( z a)2 + = n=0 f (n) (a) n! (z a) n only valid if the series converges Complex Series (3A) 3 8/7/3
Cauchy's Formula and Taylor Series Cauchy's Formula Taylor series = 2πi f (w) w z dw n=0 f (n) (a) n! (z a) n if point f '(z) exists in the neighborhood of a a is infinitely differentiable in that neighborhood can be expanded in a Taylor series about a that converges inside a disk whose radius is equal to the distance between a and the nearest singularity of = f (a) + f '(a)(z a) + f ''(a) 2 ( z a)2 + = n=0 f (n) (a) n! (z a) n only valid if the series converges f (n) (z) = n! 2πi C f (w) (w z) n+ dw Complex Series (3A) 4 8/7/3
Analyticity f '(z) = d f d z = lim Δ z 0 Δ f Δ z Δ f = f (z+δ z) Δ z = Δ x + i Δ y : analytic in a region : analytic at a point z = a has a (unique) derivative at every point of the region has a (unique) derivative at every point of some small circle about z = a Regular point of a point at which is analytic Singular point of a point at which is not analytic Isolated singular point of a point at which is analytic everywhere else inside some small circle about the singular point Complex Series (3A) 5 8/7/3
Isolated Singularities f (z ) : singular at z = is not analytic at z = but every neighborhood of z = contains points at which f (z ) is analytic z = : isolated singularity has neighborhood without further singularities of tan(z) z = π 2 + nπ n = ±,±2,±3, :isolated singularities tan ( z ) z = ( π 2 + nπ ) z = π 2 + n π :isolated singularities Complex Series (3A) 6 8/7/3
Derivatives = u(x, y) + i v(x, y) : analytic in a region R derivatives of all orders at points inside region f '( ), f ''( ), f (3) ( ), f (4) ( ), f (5) ( ), Taylor series expansion about any point inside the region The power series converges inside the circle about This circle extends to the nearest singular point Complex Series (3A) 7 8/7/3
Power Series A power series in powers of ) non-negative powers n=0 (z ) n = a 0 + a ) + a 2 ) 2 + converges for z < R termwise differentiation termwise integration the same radius of convergence R A power series in powers of z ( = 0) non-negative powers n=0 z n = a 0 + a z + a 2 z 2 + Complex Series (3A) 8 8/7/3
Cauchy's Integral Formula : analytic on and inside simple close curve C f (a) = 2π i z a d z the value of at a point z = a inside C ccw C dz z a = ccw C ' dz z a C z = a ρe iθ dz = iρe i θ dθ d z z a = iρei θ dθ ρe iθ a C' ccw C f ( z) dz z a 2π = 0 f ( z)i dθ = 2π i f (a) d d z f (z ) = d d z { 2π i f (w) w z dw } f '(z ) = 2πi f (w) (w z) 2 dw Complex Series (3A) 9 8/7/3
Taylor Series () A power series in powers of ) non-negative powers n=0 (z ) n = a 0 + a ) + a 2 ) 2 + Conversely, every analytic function can be represented by power series. The Taylor series of a function non-negative powers = n=0 (z ) n = n! f (n) ( ) converges for all z in the open disk with center and radius generally equal to the distance from to the nearest singularity of Complex Series (3A) 0 8/7/3
Taylor Series (2) A power series in powers of ) non-negative powers n=0 (z ) n = a 0 + a ) + a 2 ) 2 + The Taylor series of a function non-negative powers = n=0 (z ) n = n! f (n) ( ) f (n) (z ) = n! 2π i C f (w) (w z) n+ dw = n! f (n) ( ) = 2π i C f (w) (w ) n+ dw Complex Series (3A) 8/7/3
Maclaurin Series A power series in powers of z non-negative powers n=0 z n = a 0 + a z + a 2 z 2 + The Maclaurin series of a function non-negative powers = n=0 z n = n! f (n) (0) f (n) (z) = n! 2π i C f (w) (w z) n+ dw = n! f (n) (0) f (n) (0) = 2π i C f (w) w n+ dw Complex Series (3A) 2 8/7/3
Laurent's Theorem : analytic in the region R between circles C, C 2 centered at C C 2 = a 0 + a ) + a 2 ) 2 + + b ) + b 2 ) 2 + Principal part : convergent in the region R the above equation is valid in some deleted neighborhood of z0 (punctured open disk) for all z near 0 < z < R Complex Series (3A) 3 8/7/3
Laurent's Series : analytic in the region R between circles C, C 2 C C 2 centered at = n=0 ) + n= b n ) n = a 0 + a ) + a 2 ) 2 + + b ) + b 2 ) 2 + Principal part = 2πi C f (w) (w z) n+ dw b n = 2πi C (w z) n f (w) dw Complex Series (3A) 4 8/7/3
Laurent's Theorem - Region of Convergence : analytic in the region R between circles C, C 2 C C2 centered at = a 0 + a ) + a 2 ) 2 + For this a series to converge, the ROC must be in the form z < const C 2 Inside of C 2 Principal part + b ) + b 2 ) 2 + For this b series to converge, the ROC must be in the form z < const C Outside of C Complex Series (3A) 5 8/7/3
Laurent's Theorem Coefficients : analytic in the region R between circles C, C 2 R C C2 centered at = a 0 + a ) + a 2 ) 2 + + b ) + b 2 ) + : convergent in the region R 2 = 2π i C b n = 2π i C d z ) n+ d z ) n+ Complex Series (3A) 6 8/7/3
Laurent's Theorem Coefficients f (z ) = a 0 + a ) + a 2 ( z ) 2 + + ) n + + ( z ) n+ + +2 ) n+2 + b ( z ) + b 2 ) 2 + f ( z) ( z ) n+ = a 0 ) n+ + a ( z ) n + a 2 ( z ) n + + ) + + + +2 ) + + b ( z ) n+2 + b 2 ( z ) n+3 + ) n ) n+ ) = 2π i C d z ) n+ Complex Series (3A) 7 8/7/3
Laurent's Theorem Coefficients b n f (z ) = a 0 + a ) + a 2 ( z ) 2 + + b ( z ) + b 2 ) 2 + + b n ) n + b n ) n + b n+ ( z ) n+ + f (z ) ( z ) n+ = a 0 ( z )n + a ( z ) n + a 2 ) n+ + + b ( z ) n+2 + b 2 ( z ) n+3 + + b n + b n ( z ) + b n+ ) 2 + b n ) n ) n+ b n ) b n = 2π i C d z ) n+ Complex Series (3A) 8 8/7/3
Laurent's Theorem b n = a -n ) n ) n+ ) b n ) n ) n+ b n ) = 2π i C d z ) n+ b n = 2π i C d z ) n+ b n = a n n =, 2, 3, a n ) n ) n+ a n ) a n = 2π i C d z ) n+ Complex Series (3A) 9 8/7/3
Laurent's Theorem Coefficients a k : analytic in the region R between circles C, C 2 R C C2 centered at = a 0 + a ) + a 2 ) 2 + + b ) + b 2 ) + : convergent in the region R 2 = 2π i C d z b ) n+ n = C 2π i d z ) n+ = n= a k (z ) k a k = 2π i C d z ) k+ Complex Series (3A) 20 8/7/3
Residue : analytic in the region R R C C2 between circles C, C 2 centered at = n= a k (z ) k a k = 2π i C d z ) k+ coefficient a of ) a = Res(, ) : residue of the function at the isolated singularity C d z = 2 πi Res(, ) a = 2π i C d z Complex Series (3A) 2 8/7/3
Isolated Singularity Classification () When Laurent expansion is valid R C for the punctured open disk 0 < z < M z M 2 0 around : isolated singularity in of f Depending on the number of terms of the principal part An isolated singular point is called An removable singularity A simple pole A pole of order n An essential singularity no principal part one term in the principal part n terms in the principal part infinite terms in the principal part = n= a k (z ) k = n= a k (z ) k + n=0 principal part a k (z ) k Complex Series (3A) 22 8/7/3
Isolated Singularity Classification (2) : analytic in the region R R C C2 b : residue of between circles C, C 2 centered at b = Res(, ) f ( z) = a 0 + a ) + a 2 ( z ) 2 + f ( z) = a 0 + a ) + a 2 ( z ) 2 + regular point simple pole + b ( z ) one term f ( z) = a 0 + a ) + a 2 ( z ) 2 + pole of order n + b ( z ) + b 2 ) 2 + + b n ( z ) n n terms f ( z) = a 0 + a ) + a 2 ( z ) 2 + essential singularity + b ( z ) + b 2 ) 2 + + b n ( z ) n + infinite terms Complex Series (3A) 23 8/7/3
Laurent Expansion Example () = z(z ) Complex Series (3A) 24 8/7/3
Laurent Expansion Example (2) = z(z ) 0 < z < z > ( 0 < z < z > z < ) ( z < ) z z ( z ) ( z ) +z+z 2 + + z + z 2 + +(z )+(z ) 2 + + (z ) + ( z ) 2+ Complex Series (3A) 25 8/7/3
Laurent Expansion Example (3) z < z > ( z < ) z z +z+z 2 + + z + z 2 + = z(z ) = z ( z) = z(z ) = z 2 ( z ) = z [ + z + z2 + z 3 + ] = z 2 [ + z + z 2 + z 3 + ] = z + z + z2 + = z 2 + z 3 + z 4 + A simple pole z = 0 Not annular domain Complex Series (3A) 26 8/7/3
Laurent Expansion Example (4) 0 < z < z > ( z < ) (z ) (z ) +(z )+(z ) 2 + + (z ) + ( z ) 2+ = = = z(z ) = (z ) ( + (z )) (z ) [ (z ) + (z )2 ] (z ) + (z ) (z )2 + A simple pole z = = = = z(z ) = 2( (z ) + (z )) (z ) 2 [ (z ) + (z ) 2 ] (z ) 2 (z ) + 3 (z ) 4 Not annular domain Complex Series (3A) 27 8/7/3
Laurent Expansion Example (5) = z = 2 z(z ) = z + z Not an isolated singular point < z 2 < 2 2 z = + z 2 = (z 2) ( + z 2) z 2 < = z 2 [ ( z 2) + (z 2) 2 ( z 2) 3 ] 2 z = 2 + z 2 = = 2 [ ( z 2) 2 2 ( + z 2 2 ) + (z 2)2 2 2 ( z 2)3 2 3 ] z 2 < 2 z 2 2 < 2 Complex Series (3A) 28 8/7/3
Laurent Expansion Example (6) = z = 2 z(z ) = z + z Not an isolated singular point 2 < z + 2 < 2 z = z+2 3 = 3 ( z+2) 2 z + 2 < = 3 [ + ( z+2) + (z+2) 2 + ( z+2) 3 ] z = z+2 2 = = 2 [ + ( z+2) 2 2 ( z+2 2 ) + (z+2)2 2 2 + ( z+2)3 2 3 ] 2 z + 2 < 2 z + 2 2 < Complex Series (3A) 29 8/7/3
Laurent Expansion Example (7) = z = 2 z(z ) = z + z Not an isolated singular point < z + 2 < 2 z = z ( z ) = z [ + z + z 2 + z 3 ] z < z = z < 2 = z Complex Series (3A) 30 8/7/3
Several Isolated Singularities z 2 z A Laurent series converges between two circle, if it converges at all. Several isolated singularites Several annular rings Several different Laurent series for each rings Complex Series (3A) 3 8/7/3
Singularities of Order n A zero of a function f f ( ) = 0 f : analytic at z = A zero of order n of a function f f : analytic at z = f ( ) = 0, f '( ) = 0, f ''( ) = 0,, f (n) ( ) = 0 A pole of order n of a function F (z) = g(z) / g( ) 0 f, g : analytic at z = f ( ) = 0, f '( ) = 0, f ''( ) = 0,, f (n) ( ) = 0 Complex Series (3A) 32 8/7/3
References [] http://en.wikipedia.org/ [2] http://planetmath.org/ [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] E. Kreyszig, Advanced Engineering Mathematics [5] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [6] T. J. Cavicchi, Digital Signal Processing 8/7/3