COMPLEX VARIABLES
4 7 THE POISSON INTEGRAL FORMULA AND ITS APPLICATIONS 4 8 THE JENSEN AND POISSON-JENSEN FORMULAS 4 9 ANALYTIC CONTINUATION CHAPTER 5: FAMILIES OF ANALYTIC FUNCTIONS 5 1 THE SPACES A(?) AND C(?) 5 2 THE RIEMANN MAPPING THEOREM 5 3 EXTENDING CONFORMAL MAPS TO THE BOUNDARY CHAPTER 6: FACTORIZATION OF ANALYTIC FUNCTIONS 6 1 IN?NITE PRODUCTS 6 2 WEIERSTRASS PRODUCTS 6 3 MITTAG-LE?ER’S THEOREM AND APPLICATIONS CHAPTER 7: THE PRIME NUMBER THEOREM 7 1 THE RIEMANN ZETA FUNCTION 7 2 AN EQUIVALENT VERSION OF THE PRIME NUMBER THEOREM 7 3 PROOF OF THE PRIME NUMBER THEOREM CHAPTER [AJ, AJ+1], OFTEN ABBREVIATED AS [A1, , AN+1] COMPLEX VARIABLES DE?NITIONS A REGION IN C IS AN OPEN CONNECTED SUBSET OF C A SET E ?C IS CONVEX IF FOR EACH PAIR OF POINTS A, B ?E, WE HAVE [A, B] ?E; E IS STARLIKE IF THERE IS A POINT A ?E (CALLED A STAR CENTER) SUCH THAT [A, Z] ?E FOR EACH Z ?E NOTE THAT ANY NONEMPTY CONVEX SET IS STARLIKE AND THAT STARLIKE SETS ARE POLYGONALLY CONNECTED COMPLEX VARIABLES AND F2 ARE ANALYTIC ON ?, SO ARE F1 + F2, F1 ?F2, KF1 FOR K ?C, F1F2, AND F1/F2 (PROVIDED THAT F2 IS NEVER 0 ON ?) FURTHERMORE, (F1 + F2)? = F ? 1 + F ? 2, (F1 ?F2)? = F ? 1 ?F ? 2, (KF1)? = KF ? ? = F2F ? 1 ?F1F ? COMPLEX VARIABLES REAL-DI?ERENTIABILITY AND THE CAUCHY-RIEMANN EQUA- TIONS LET F : ??C, AND SET U = RE F, V = IM F THEN U AND V ARE REAL-VALUED FUNCTIONS ON ? AND F = U + IV IN THIS SECTION WE ARE INTERESTED IN THE RELATION BETWEEN F AND ITS REAL AND IMAGINARY PARTS U AND V FOR EXAMPLE, F IS CONTINUOUS AT A POINT Z0 I?BOTH U AND V ARE CONTINUOUS AT Z0 RELATIONS INVOLVING DERIVATIVES WILL BE MORE SIGNI?CANT FOR US, AND FOR THIS IT IS CONVENIENT TO BE ABLE TO EXPRESS THE IDEA OF DI?ERENTIABILITY OF REAL-VALUED FUNCTION OF TWO REAL VARIABLES BY MEANS OF A SINGLE FORMULA, WITHOUT HAVING TO CONSIDER PARTIAL DERIVATIVES SEPARATELY WE DO THIS BY MEANS OF A CONDITION ANALOGOUS TO (5) OF (1 3 1) COMPLEX VARIABLES