complex analysis ahlfors

[...]... nf a complex variable, complex functions of a real variable, and complex functions of a complex variable As a practical matter we that the letters and ID ,ball always denote complex variables; thus, to indicate a complex function of a complex variable we use the notation ID = !(z) t The notation 11 - J(z) will be Jlsed in a neutral manner with the UIlderstMding that z and 1/ can be either rea! or complex. .. my eolleague Lynn Loomis, who kindly let me share student reaction to a recent based on my book LaTif V AM/Drs COMPLEX ANALYSIS • PLEX NU 1 • L THE ALOE.ItA OF COMPLEX NU • • EItI It is fundamental that real and complex numbers obey the sarne basic laws of arithmetic We begin Our study of complex fWletion theory by stressing and implementing this analogy '; 1.1 4ritiunetJe Operations From elementary... toa correct definition of the trigonometric functions, and to a computational proof of the addition theorems.The reason we do not follow this path is that complex analysis, as Reliltlrk: PlO w, vector multiplication COMPLEX NUMBERS l' opposed to real analysis, offers a much more direet approach The clue lies in a direet oonneetion between the exponential function and the trigonometrie functions, to be... imporlant in the theory of complex t:;.- bnmbu detbjmine a.ll of equality in (11) In (10) the equality holds if and only if ali ~ 0 (it is convenient to let c > 0 indicate that ill is real and~) If b pi 0 , oondition can be written in the : form Ibll(a/b) ~ 0, and it ill bellee!!CtiliWlleDt to alb ;; O In ' ' " I: _ Let\JJl t', r f "1 ' ", ", ," , ' .- '- 10 COMPLEX ANALYSIS we proceed 88 follows:... of the product of two complex numbers we introduce polar coordinafAls If the polar coordinafAls of the point (a,/J) are (r,'P), we know that a=rCOll'P fl ~ rain 'P • Hence we can write a = a + ifl = r{c08 'P + i RiD ",) In thia trigonometric fO''IIl of a complex nllm ber r is alway8 £; 0 and equal to the modulua lal The polar angle", is called the argument or amp/iltMie of the complex number, and we... and ime.giuary part of a complex number a will also be denoted by Re a, 1m a In deriving the rules for complex addition and multiplication we used only the fact that i! ~ -1 Since -i has the same property, all rules must remain valid if i is evelY where replaced by - i Direct verification shows that this is indeed so The trlVlBformation which replaces + i/3 by a - ifl is called complex Clmjugation, and... In algebraic form it can be rewritten as (z - a){l! - 4) ~ r" The fact that this equation is invariant under complex conjugation is an indication that it repreeents a single real equation A Bkaight line in the complex plane can be given by a parametric equation z '" a + bt, where a and b are complex numbers and II ,& 0; the parameter t nIDS through all real values Two equations II '" II bt and II ~... -, ' "'-t' - , ' • #.: - :.;' " " , - .,.-., - , , ' ' :,-,," '" _", ,,'.-.,• ,'., ,'-', :,- '."" • ,;,t .~.,.'·1,~ -""'~-·' . based on my book. LaTif V. AM/Drs COMPLEX ANALYSIS • 1 PLEX NU • L THE ALOE.ItA OF COMPLEX NU •• EItI It is fundamental that real and complex numbers obey the sarne basic. Numerical Analysis H l1IC1elwld#r: The Numerical Treatment of a Single Nonlinear Equation Kal_. fi'alh. aM Arbib: Topics in Mathematical Systems Theory LaB Vector and Tensor Analysis. R<JbiMtJJitz: A Firet Course in Numerical Analysis RilIF and ROle: Difiaential Equations with Applications Rudin: Principles of Mathematical Analysis Shapiro: Introduction to Abetmet