Glossary of Terms from Complex Variable Theory and Analysis accumulation point Let at, a2, ... be points in the complex plane. A point b is an accumulation point of the sequence {aj} if the aj get arbitrar­ ily close to b. More formally, we require that for each t > °there exists an N> °such that when j > N, then laj - bl < t. §§3.2.1. analytic continuation The procedure for enlarging the domain of a holo­ morphic function. §§1O.1.1, §§1O.1.2. analytic continuation of a function If(ft, Ut}, ... , (!k, Uk) are function elements and if each (Ii, Uj ) is a direct analytic continuation of (1i-1l Uj-t}, j = 2, ... , k, then we say that (fk, Uk) is an analytic continuation of (ft, Ut ). §§1O.1.5. analytic continuation of a function element along a curve An an­ alytic continuation of (f, U) along the curve, is a collection of function elements (ft, Ut), t E [0,1], such that 1) (fo, Uo) = (f, U). 2) For each t E [0,1], the center of the disc Ut is ,(t), °::; t ::; 1. 3) For each t E [0,1], there is an t > °such that, for each t' E [0,1] with It' - tl < t, it holds that (a) ,(t') E Ut and hence Ut' nUt =I- 0; (b) It == ItI on Ut' nUt [s<'> that (ft, Ut ) is a direct analytic continuation of (ft' , UtI )]. §§1O.2.1. annulus A set of one of the forms {z E C: °< Izi < R} or {z E C: r < Izi < R} or {z E C: r < Izi < co}. §§4.2.3. 231 232 Glossary area principle If f is schlicht and if 1 1 00 . h(z) = f(z) = ~ +~ bjzJ-L then 00 2 Ljlbj l :s 1. §§12.1.4 j=l argument If z = rei9 is a complex number written in polar form, then () is the argument of z. §§1.2.6. argument principle Let f be a function that is holomorphic on a domain that contains the closed disc D(P, r). Assume that no zeros of f lie on aD(P, r). Then, counting the zeros of f according to multiplicity, 1 f'(() .. -2. i f(/') d( = # zeros of f mSlde D(P,r). §§5.1.5 1TZ 8D(P,r) ." argument principle for meromorphic functions Let f be a holomor­ phic function on a domain U ~ C. Assume that D(P, r) ~ U, and that f has neither zeros nor poles on aD(P,r). Then where nil n2, ... ,np are the multiplicities of the zeros Zil Z2, ,zp of f in D(P, r) and m1, m2, . .. , m q are the orders of the poles Wil W2, , wq of f in D(P, r). §§5.1.7. associative law If a, b, C are complex numbers, then (a + b) + c = a + (b + c) (Associativity of Addition) and (a· b) . c = a . (b· c). (Associativity of Multiplication) §§1.1.2, 1.1.6. assumes the value (3 to order n A holomorphic function assumes the value (3 to order n at the point P if the function f(z) - {3 vanishes to order n at P. §§5.1.3. Glossary 233 barrier Let U ~ C be an open set and P E aU. We call a function b : U --+ IR a barrier for U at P if 1. b is continuous; 2. b is subharmonic on U; ~ 3. bl au OJ 4. {z E aU : b(z) = O} = {Pl. §§7.7.9. beta function If Re z > 0, Re w > 0, then the beta function of z and w is 1 1 B(z, w) = 1e- (1 - t)W-1 dt. §§13.1.11. Bieberbach conjecture This is the problem of showing that each coeffi­ cient aj of the power series expansion of a schlicht function satisfies laj I ~ j. In addition, the Kobe functions are the only ones for which equality holds. §§12.1.2. biholomorphic mapping See conformal mapping. Blaschke condition A sequence of complex numbers {aj} ~ D(O, 1) sat­ isfying 00 ~)l-lajl) < 00 j=l is said to satisfy the Blaschke condition. §§9.1.5. Blaschke factor This is a function of the form z-a Ba(z) = --_­ I-az for some complex constant a of modulus less than one. See also Mobius tmnsformation. §§9.1.1. Blaschke factorization If f is a bounded holomorphic function or, more generally, a Hardy space function on the unit disc, then we may write ;~1 f(z) = zm. {tJ H.,(Z)} .F(z) 234 Glossary Here m is the order of the zero of I at z = 0, the points aj are the other zeros of I (counting multiplicities), the Ba; are Blaschke factors, and F is a non­ vanishing Hardy space function. §§9.1.7, §§12.3.7. Blaschke product If {aj} satisfies the Blaschke condition, then the infi­ nite product converges uniformly on compact subsets ofthe unit disc to define a holomor­ phic function B on D(O, 1). The function B is called a Blaschke product. §§9.1.6. Bohr-Mollerup theorem Suppose that cp : (0,00) -t (0,00) satisfies 1. log cp(x) is convex; 2. cp(x + 1) = x· cp(x), all x> 0; 3. cp(l) = 1. Then cp(x) == r(x), where r is the gamma function of Euler. Thus r is the only meromorphic function on C satisfying the functional equation zr(z) = r(z + 1),r(l) = 1, and which is logarithmically convex on the positive real axis. §§13.1.l0. boundary maximum principle for harmonic functions Let U ~ C be a bounded domain. Let u be a continuous function on U that is har­ monic on U. Then the maximum value of u on U (which must occur, since U is closed and bounded-see [RUDl], [KRA2]) must occur on au. §§7.2.3. boundary maximum principle for holomorphic functions Let U ~ C be a bounded domain. Let I be a continuous function on U that is holomor­ phic on U. Then the maximum value of Ilion U (which must occur, since U is closed and bounded-see [RUDl], [KRA2]) must occur on au. §§5.4.2. boundary minimum principle for harmonic functions Let U ~ C be a bounded domain. Let u be a continuous function on U that is har­ monic on U. Then the minimum value of u on U (which must occur, since U is closed and bounded-see [RUDl], [KRA2]) must occur on au. §§7.2.3. Glossary 235 boundary minimum principle for holomorphic functions Let U ~ C be a bounded domain. Let I be a continuous function on U that is holo­ morphic on U. Assume that I is non-vanishing. Then the minimum value of Ion U (which must occur, since U is closed and bounded-see [RUDl], [KRA2]) must occur on aU. §§5.4.2. boundary uniqueness for harmonic functions Let U ~ C be a bounded domain. Let Ul and U2 be continuous functions on U that are harmonic on U. If Ul = U2 on aU then Ul = U2 on all of U. §§7.2.5. bounded on compact sets Let :F be a family of functions on an open set U ~ C. We say that :F is bounded on compact sets if for each compact set K ~ U, there is a constant M = M K such that for all I E :F and all z E K we have II(z)1 ~ M. §§8.4.3. bounded holomorphic function A holomorphic function I on a domain U is said to be bounded if there is a positive constant M such that II(z)1 ~ M for all z E U. §§9.1.4. Caratheodory's theorem Let tp : 0 1 --+ O2 be a conformal mapping. If aOb 002 are Jordan curves (simple, closed curves), then tp (resp. tp-l) extends one-to-one and continuously to 001 (resp. 002 ), §§12.2.2. Casorati-Weierstrass theorem Let I be holomorphic on a deleted neigh­ borhood of P and supposed that I has an essential singularity at P. Then the set of values of I is dense in the complex plane. §§4.1.6. Cauchy estimates If I is holomorphic on a region containing the disc D(P,r) and if III ~ M on D(P,r), then Ok I(P)I<_ M. k! . §§3 1 2 Iaz k r k .•. Cauchy integral formula Let I be holomorphic on an open set U that it contains the closed disc D(P,r). Let ')'(t) = P + re . Then, for each z E D(P,r), I(z) = ~ 1 I(() d(. 21l"t lr (- z 236 Glossary See §§2.3.1. The formula is also true for certain more general curves (§§2.3.3). Cauchy integral formula for an annulus Let f be holomorphic on an annulus {z E C: r < Iz - PI < R}. Let r < s < S < R. Then for each Z E D(P, S) \ D(P, s) we have f(z) = _1 1 f() d( __1 1 f() d(. 2rri JlC;-PI=s (- P 2rri JlC;-PI=s (- P §§4.2.5. Cauchy integral theorem If f is holomorphic on a disc U and if 'Y : [a, b] ---. U is a closed curve, then i f(z)dz = O. §§2.3.2. The formula is also true for certain more general curves (§§2.3.3). Cauchy-Riemann equations If u and v are real-valued, continuously differentiable functions on the domain U, then u and v are said to satisfy the Cauchy-Riemann equations on U if au av av au -=-ax ay and ax = -ay· §§1.3.2.