AN INTRODUCTION TO COMPLEX ANALYSIS FOR ENGINEERS
CHAPTER YOU KNOW ALSO THAT THE BEGINNING IS THE MOST IMPORTANT PART OF ANY WORK PLATO S REPUBLIC THIS CHAPTER SETS NOTATION AND REVIEWS VARIOUS SUBJECTS THAT IT IS HOPED THE READER HAS SEEN BEFORE FOR SOME LIKE CALCULUS THE BASIC MATERIAL IS REALLY A PREREQUISITE OTHERS LIKE COVERING SPACES ARE ONLY USED IN A FEW PLACES AND ENOUGH BACKGROUND IS GIVEN TO PERHAPS TIDE OVER THE STUDENT WITH LESS PREPARATION ESSENTIALLY SECTIONS 1 1 1 4 ARE USED OFTEN AND THE 1 1 NOTATION AND TERMINOLOGY A FOOLISH CONSISTENCY IS THE HOBGOBLIN OF LITTLE MINDS IS IT SO BAD THEN TO BE MISUNDERSTOOD PYTHAGORAS WAS MISUNDERSTOOD AND SOCRATES AND JESUS AND LUTHER AND COPERNICUS AND GALILEO AND NEWTON AND EVERY PURE AND WISE SPIRIT THAT EVER TOOK ESH TO BE GREAT IS TO BE MISUNDER RALPH WALDO EMERSON FOR A REAL NUMBER A WE WILL USE THE TERMS POSITIVE AND STRICTLY POSITIVE FOR A 0 AND A 0 RESPECTIVELY IT IS NOT SO MUCH THAT WE ND NONNEGATIVE BAD BUT THE PHRASE MONOTONE NONDECREASING FOR X Y F X F Y IS DOWNRIGHT CONFUSING SO WE USE MONOTONE INCREASING AND STRICTLY MONO TONE INCREASING AND THEN FOR CONSISTENCY POSITIVE AND STRICTLY POSITIVE SIMILARLY FOR MATRICES WE USE POSITIVE DE NITE AND STRICTLY POSITIVE DE NITE WHERE OTHERS MIGHT USE POSITIVE SEMI DE NITE AND POSITIVE DE NITE BASIC RINGS AND FIELDS R REAL NUMBERS C COMPLEX NUMBERS X IY X Y R WITH THEIR SUMS AND PRODUCTS FOR Z X IY C WE USE RE Z X IM Z Y Z X2 Y2 1 PRODUCTS XN N TUPLES OF POINTS IN X WITH INDUCED VECTOR SPACE AND OR ADDITIVE STRUCTURE IN PARTICULAR RN QN ZN CN SUBSETS OF C AND ONE SUPERSET C C AS A ONE POINT COMPACTI CATION IN SECTION 7 2 WE LL PUT A COMPLEX STRUCTURE NEAR C UPPER HALF PLANE Z IM Z H RIGHT HALF PLANE Z RE Z Z N Z N 0 1 2 N 0 Z 0 1 D Z C Z 1 D Z C Z D Z0 Z C Z Z0 FOR Z0 C C C 0 D D 0 R R IN GENERAL Y INVERTIBLE ELEMENTS OF Y AR R ANNULUS Z R Z R FOR 0 R R S SECTOR R COS I SIN R 0 FOR AND 2 FOR EXAMPLE S C K A COMPACT HAUSDOR SPACE C K CONTINUOUS COMPLEX VALUED FUNC TIONS ON K FOR X R X GREATEST INTEGER LESS THAN X THAT IS X Z X X X X X FRACTIONAL PARTS OF X A NUMBER OF ELEMENTS IN A SET A RAN F RANGE OF A FUNCTION F LOG Z NATURAL LOGARITHM THAT IS LOGARITHM TO BASE E IF Z IS COMPLEX PUT THE CUT ALONG 0 I E LOG Z LOG Z I ARG Z WITH REGION OPEN CONNECTED SUBSET OF C FOR SETS A B SUBSETS OF X A B UNION A B INTERSECTION AC COMPLEMENT OF A IN X A B A BC A B A B B A FOR MATRICES M12 MEANS ROW ONE COLUMN TWO F K RESTRICTION OF A FUNCTION TO K A SUBSET OF THE DOMAIN OF F INTEGER NOTATION SEE SECTION 1 3 FOR MODULAR ARITHMETIC THIS IS MAINLY RELEVANT IN CHAPTER 13 OF PART 2B ALL LETTERS IN THIS SUBSECTION ARE IN Z D N MEANS D IS A DIVISOR OF N THAT IS N DK FOR K Z A B MOD M M A B ZK CONJUGACY CLASSES MOD K K RELATIVELY PRIME CONJUGACY CLASSES WE USE BIG OH LITTTLE OH NOTATION E G F X O X OR F X O IF THE READER IS NOT FAMILIAR WITH IT SEE SECTION 1 1 2 COMPLEX NUMBERS BECAUSE ALL POSSIBLE NUMBERS WHICH ONE CAN THINK OF ARE EITHER GREATER OR LESS THAN 0 OR ELSE 0 ITSELF IT IS CLEAR THAT THE SQUARE ROOTS OF NEGATIVE NUMBERS CANNOT BE COUNTED AMONG THE POSSIBLE NUMBERS CONSEQUENTLY WE MUST SAY THAT THESE ARE IMPOSSIBLE NUMBERS L EULER AS TRANSLATED BY U BOTTAZZINI WE VE ALREADY SAID SOMETHING ABOUT COMPLEX NUMBERS AND THEIR RECTAN GULAR COORDINATES HERE WE WANT TO SAY SOMETHING ABOUT THE POLAR REPRESEN TATION WE SUPPOSE THE READER KNOWS THE GEOMETRIC BASICS OF SIN X AND COS X AS WELL AS THE FACT THAT THEY BOTH SOLVE F X F X AND INDEED THAT ANY SOLUTION HAS THE FORM F X COS X SIN X WHERE F 0 F IN PARTICULAR SIN AND COS HAVE THE TAYLOR EXPANSIONS SEE SECTION 2 3 FOR 1 N 2N EI COS I SIN SO Z REI LATER IN SECTION 2 3 WE DE NE EZ BY A POWER SERIES AND 1 2 2 WILL BECOME A DERIVED EQUALITY RATHER THAN A DE NITION THE GEO METRIC PICTURE OF COMPLEX NUMBERS IN A PLANE SEE FIGURE 1 2 1 WILL PLAY AN IMPORTANT ROLE IN MANY PLACES IN THIS VOLUME ARITHMETIC OPERATIONS HAVE GEOMETRIC MEANING ADDITION IS VIA PARALLELOGRAM MULTIPLICATION BY Z REI IS UNIFORM SCALING BY A FACTOR R FOLLOWED BY ROTATION BY ANGLE Z X IY R EI X R COS Y R SIN FIGURE 1 2 1 POLAR AND RECTANGULAR COORDINATES NOTES AND HISTORICAL REMARKS AS THE TERMS IMAGINARY AND COM PLEX ATTEST ACCEPTANCE OF COMPLEX NUMBERS WAS A DI CULT PROCESS ONE MIGHT HAVE GUESSED GIVEN THE FORMULA FOR SOLUTIONS OF THE QUADRATIC EQUA TION THAT SQUARE ROOTS OF NEGATIVE NUMBERS WOULD HAVE GONE BACK TO ANTIQ UITY BUT SCHOLARS WERE CONTENT TO THINK THAT SOME EQUATIONS LIKE X2 1 HAVE NO SOLUTIONS RATHER COMPLEX NUMBERS CAME THROUGH CUBIC EQUATIONS WHERE X3 AX B 0 WAS RST STUDIED IN 1515 SCIPIONE DEL FERRO 1526 FOUND BUT DID NOT PUBLISH THE SOLUTION EVEN IF B2 27 0 THE EQUATION HAS REAL SOLUTIONS FOR A B REAL THE FORMULA WAS NALLY PUBLISHED BY GIROLAMO CARDANO 1501 76 IN 1545 INTERESTINGLY ENOUGH WHILE CARDANO S BOOK HAS EXAMPLES WHERE B2 27 0 HE NEVER APPLIED THE FORMULA IN SUCH CASES IN 1572 RAFAEL BOMBELLI 1526 72 PUBLISHED A BOOK 62 WHICH SPELLED OUT RULES OF ARITHMETIC FOR COMPLEX NUMBERS AND USED THEM IN CARDANO S FORMULA FOR NDING REAL SOLUTIONS OF CUBICS KEY LATER WORK IS BY WALLIS AND EULER 178 SEE THE NOTES TO SECTION 9 2 FOR A CAPSULE BIOGRAPHY IN PARTICULAR EULER CLARI ED COMPLEX ROOTS OF UNITY AND FOUND THE MULTIPLE ROOTS REPRESENTED BY 1 2 3 IT TOOK ALMOST ANOTHER CENTURY BEFORE MATHE MATICIANS AS A COMMUNITY FULLY ACCEPTED COMPLEX NUMBERS THUS MUCH OF CAUCHY S WORK WAS DONE IN A MILIEU WHERE MANY EVEN SOMETIMES CAUCHY HIMSELF WERE NOT TOTALLY COMFORTABLE WITH COMPLEX NUMBERS THE GEOMETRIC VIEW OF COMPLEX NUMBERS AS A PLANE WAS INTRODUCED BY JEAN ROBERT ARGAND 1768 1822 AND CASPAR WESSEL 1745 1818 AND CHAM PIONED BY GAUSS NEITHER ARGAND NOR WESSEL WERE PROFESSIONAL MATHEMATI CIANS WESSEL WAS A DANISH SURVEYOR AND PRESENTED THE GEOMETRIC INTERPRE TATION TO THE ROYAL DANISH ACADEMY IN 1797 IT WAS PUBLISHED IN 1799 AND PROMPTLY FORGOTTEN UNTIL REDISCOVERED AND PUBLICIZED BY TWO DANISH MATH EMATICIANS IN 1895 SOPHUS CHRISTIAN JUEL AND THE FAMOUS SOPHUS LIE SO WESSEL S WORK HAD NO IMPACT