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Glossary of Terms from Complex Variable Theory and Analysis Accumulation Point Let At, A2, 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.
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