Diola Bagaoko (1 ELEMENTARY FNCTIONS OFA COMPLEX VARIABLES I Basic Defiitio of a Fuctio of a Comple Variable A applicatio of a subset S of C oto aother S' defies a fuctio [f(] of the comple variable z The word applicatio, i the above cotet, has a ver uique mathematical meaig: A applicatio of a subset S oto aother subset S' is defied if ad ol if to ever elemet i S there correspods oe ad ol oe elemet i S' Notatio: f( = W If f( is comple W = Re W + ii mw Re W ad I mw deped o both ad : Re W = (, ; I mw = V (, So whatever f(, it ca be writte as f( = (, + iv(, Eamples of fuctios of a comple variable: f( = z, =, V = P( = a z + a z + 1 + a z, a polomial P( f( =,[ Q( i the domai of defiitio] is a ratioal fuctio Q( if P(, Q( are polomials I1 Notios of Limits Basic defiitios ad properties of limits for fuctios of a comple variable are ver similar to those for fuctios of a real variable (see pre-calculus or itroductor calculus tetbooks lim f ( = w ε >, δ > such that < z - z < δ f ( z w < ε Please read some of the ma eamples worked out i the tetbook } { THEOREMS: Let f( = (z, + iv(, : z = +, W = + iv, the lim f ( = w lim (, = ad lim V (, (, (, (, (, 1 = V Let lim f ( = w ad lim g( = W, the lim[ f ( + g( ] = w + W, lim[ f ( g( ] = ww f ( w z z ad lim = if W ( g z W f ( w Note: ad are defied for the values of z such that g(, ad g( W W These results follow from the first theorem ad the theorems of real aalsis Also, lim f ( = W lim f ( z = W 1
Diola Bagaoko (1 That the limit w of f( eists for z goig to z is oe thig which does ot mea that f(z = w This basic poit leads to the differece betwee the eistece of a limit ad the otio of cotiuit of a fuctio! I Cotiuit a fuctio Defiitio f ( cotiuous at z f ( cotiuous at z Limf ( for z ε >, δ > ad goig to z z z eists ad is equal to < δ f ( f ( + F( cotiuous at z f( ad F( cotiuous at z f ( * F( cotiuous at z f ( cotiuous at z if F( F( The compositio of fuctio preserves cotiuit: G(f ad f( respectivel cotiuous at f ad z G[ f ( ] cotiuous at z f( cotiuous i a closed ad bouded regio R, the f( is bouded i R ad f( reaches a maimum value i R f( cotiuous i a closed bouded regio R is uiforml cotiuous i R iform cotiuit meas that δ is idepedet of z The above basic defiitios ad theorems are to be mastered The are mostl restatemets of similar results for fuctios of a real variable (Provig them is ot our aim here, kowig them ad correctl applig them i Phsics is our objective I3 Differetiatio of a Fuctio of a Comple Variable-Derivatives f ( f ( z The derivative of f( at z, if it eists, is f '( z = lim z z If the above limit eists, the f( is said to be differetiable at differetiabilit cotiuit [Note: f '( z V, V,, ] d (costat =, d d [ cf ( ] = cf '(, [ f ( + g( ] = f '( + g'( z d f ( F( f '( f ( F'( d 1 = ; ( = ( z z F z [ F( ] The Cauch-Riema Coditio (Theorem { if f( = (, + iv(, is differetiable at z, f '( z eists} (,, (,, V (,, V (, eist ad satisf { z f ( z f ( z
Diola Bagaoko (1 f '( = (, + iv(, = V ad V = with f '( = V (, i (, V V ad where =, V = =, V = are partial derivatives z Homework: 1 Show that z, z z, e are ot differetiable awhere i the comple plae sig = r cos(θ, = r si(θ show that the Cauch Riema Relatios i polar 1 1 coordiates are r ( r, θ = Vθ ( r, θ, ( r, θ = Vr ( r, θ r r We should ot eed to uderscore the fact that differetiabilit for fuctios of a comple variable is ot similar to that for a fuctio of a real variable This poit is clearer below I4 Aaltic (holomorphic Fuctios Defiitio: (See previous otes for the defiitio of a eighborhood {f( is aaltic at z } f ( eist at z ad at all poits iside a eighborhood of z } Note well that a fuctio ma be differetiable at z ad ot be aaltic at z A fuctio aaltic at all z C is said to be ad etire fuctio Eample: P( = a + a z + a z + + a z 1 is a etire fuctio Sigular poits for a fuctio: { z is a (regular sigular poit for f(} f ( is ot aaltic at z but is aaltic at some poits i ever eighbourhood of z Attetio: Make certai to distiguish differetiabilit ad aalticit The eistece of, V,, V ad satisfig the Cauch-Riema Coditios (CRC are ecessar ad sufficiet for differetiabilit but ot for aalticit For aalticit, the are ecessar but ot sufficiet Theorem: The sum, product ad ratio (deomiator of aaltic fuctios are aaltic Theorem: If F(f( is aaltic at f( z, ad f( is aaltic at z, the F(f( is aaltic at z I other words, the compositio of fuctios preserves aalticit Harmoic Fuctios 3
Diola Bagaoko (1 st d The1 ad partial derivatives of are cotiuous i D (so the eist (, harmoic i a domai D} ad (, + (, = (Laplace's equatio Theorem: If f( = (, + iv(, is aaltic i a domai D, the (, ad V(, are harmoic i D Theorem: If (,, V(, are harmoic i a domai D ad their derivatives satisf the CRC the V is the harmoic cojugate of Attetio: The above two theorems are to be cosidered together, ie, V the harmoic cojugate of if f( = (, + iv(, V is the harmoic cojugate of does ot mea that is the harmoic cojugate of V!" II Elemetar Fuctios of a Comple Variable II1 Polomials: P( Polomials are sums of moomials, ie, epressios of the form az, "a" is a costat (which could be zero ad "" is a positive iteger or zero, with "a" idepedet of "" Eamples are P ( = 5, P1 ( = az, ad P ( = a + a1z + a z + a z Polomials are etire fuctios, ie, the are ot ol differetiable, but also aaltic everwhere i the comple plae To illustrate this propert, let us take the case of P( = z, with z = + i P( = ( + i ( + i = + i + i + ii = + i P( = (, + iv(,, with (, = ad V(, = = =, =, V =, ad V = We ca see from the lie above that the Cauch-Riema coditios for differetiabilit are satisfied everwhere i the comple plae (ie, = V ad = V sig the above eample, oe ca easil demostrate that a polomial is a etire fuctio Due to the differetiabilit everwhere, we coclude that for a give poit z, p( will be differetiable at z ad at ever poit iside a eighborhood of z Hece, it is aaltic at z, b defiitio II Epoetial Fuctios: Ep( = e z Ep( = e z = e e +i, or ep( = e e i = e [cos( + isi(] Ep( = (, + iv(,, with (, = e cos( ad V(, = e si( 4
Diola Bagaoko (1 Applig the Cauch-Riema coditios shows that ep( is also a etire fuctio π! A ver importat propert of this fuctio cosists of its periodicit with a period T = i Ideed, ep( = e [cos( + isi(] = e [cos( + π + isi( + π ]= e z+(iπ Hece, the periodicit of ep( is directl tied to that of the cosie ad sie fuctios Note well, however, that the period of ep( is iπ while that for sie or cosie is π! The differece is uderstood b otig that e z = e z+i π = e [cos( + π + si( + π ] O the other had, e z ad e z+ π are ot equal! Oe of the ver importat applicatios of this propert is i the polar represetatio of comple umbers: i( θ + π If z = re, the z = e = e Please see our method of solutios for a equatio of the tpe z 1 = z After writig z i polar form as z = re, where θ = ta ( + kπ, k = m 1 θ + mπ i or 1, oe sets z = r e, m = to (-1 Assigmet: draw the graph of ep( versus z! Accordig to the fudametal theorem of algebra, z = z ideed has roots or solutios! II3 Cosie Hperbolic ad Sie Hperbolic of z Cosh( ad sih( are defied as follows z z z z e + e e e cosh( = ad sih( = Clearl, the properties of these fuctios are directl determied b those of ep( I particular, cosh( ad sih( are both periodic fuctios of period T = iπ The above defiitios of cosh( ad of sih( are similar to those of cosh( ad sih( for a real variable II4 Cosie ad Sie Fuctios of z iz iz iz iz e + e e e B defiitio, Cos( = ad Si( = i The above defiitios are best derived (or recalled from the Moivre-Euler formula, ie, 5
Diola Bagaoko (1 θ e i θ = cos( θ + i si( θ or e i = cos( θ i si( θ We get, b addig or subtractig these e + e e e two equatios, cos( θ = ad si ( θ = These two relatios show that the i forms of cos ad sie fuctios of θ, for a real variable θ, are their respective forms for a comple variable z The periods of cos( ad of si( are π II5 Other Fuctios It is critical to uderstad that ma importat applicatios of fuctios of a comple variable utilize, ad etesivel so, the above simple properties of elemetar fuctios Thoroughl read our tetbook o the subject I particular, make our otes for the properties of the elemetar fuctio L( Simple relatios eist betwee some of the above elemetar fuctios These relatios are easil established b usig the defiitios ad properties oted above 6