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Unformatted text preview: PUI)’ IIUIIIIGID. Let G be a region and let f and g be analytic functions on G such that
z)g(z) = O for all z in G. Show that either/5 0 or g E 0. 9. Let U: C —+ R be a harmonic function such that U(z) 2 0 for all z in C;
prove that U is constant. 10. Show that if fand 9 are analvtic functian nn 2 rpoinn r: cnnh that fa ic I “,5yn n uuuu m u yuuu u v— 1]; WIN! "Ky, u) : K. Let p(z) be a polynomial of degree n and let R > 0 be sufficiently large
so that p never vanishes in {z : lzl 2 R}. If y(t) = Re“, 0 s t s 21:, show
that p (Z) 7 PU) dz = 27in. nXCl kl”: ®Suppose f: G——>C is analytic and define (p : G X G—>C by tp(z,w)=[f(z)
—f(w)](z — w)" if z 9% w and ¢p(z,z) =f’(z). Prove that (p is continuous and
for each fixed w, z—up(z.w) is analytic.
2. Give the details of the proof of Theorem 5.6.
3. Let B: = Bung), G = 3(0; 3) ~ (3, UB_). Let y,,~,v2,~,g, be curves
whose traces are z — l= l, l: + ll= l. and z =2, respectively. Give vhyz,
and y, orientations such that n(y.;w)+n(72;w)+n(73;w)=0 for all w in
C— G.
4. Show that the Integral Formula follows from Cauchy’s Theorem.
5. Let y be a closed rectifiable curve in C and (1% {y}. Show that for n 2 2
[7(2 ~ a)""dz =0.
6. Letfbe analytic on D= 8(0; l) and suppose f(z)l Sl for z < 1. Show
'(0) s I.
Let y(!)= l +e” for OSISZ'IT. Find ;,( tegers n. 8. Let G be a region and suppose L:G—>C is analytic for each n2 l.
Suppose that {f,,} converges uniformly to a function f: G—aC Show thatf
IS analytic. 9. Show that if f :C—>C is a continuous function such that f is analytic off
[— 1.1] then] is an entire function. Use Cauchy‘s Integral Formula to prove the CayleyHamilton Theo
rem: If A is an n X n matrix over C and f(z)=det(z  A) is the characteris
tic polynomial of A then f(A)=0. (This exercise was taken from a paper
by C. A. McCarthy, Amer. Math. Monthly, 82 (1975), 390—391). 2
z—l )"dz for all positive in e G) Let G be a region and let 01,02: [0,1] —> G be the constant curves
01(t) E a, 02(1) .=— b. Sh0w that if y is a closed rectifiable curve in G and
Y ~ 01 then y ~ 02. (Hint: connect a and b by a curve.) 2 Show that if we remove the requirement “F(0, t) = PO, 1) for all I"
from Deﬁnition6.1 then the curve 310(1) = e"“, 0 s t _<_ l, is homotopic to
the constant curve y,(t) E l in the region G = C— {0}. Met % = all rectiﬁable curves in Gjoining a to b and show that Deﬁnition 6.” gives an equivalence relation on %.
Let G = C— {0} and show that every closed curve in G is homotopic
to a closed curve whose trace is contained in {2: [2 = 1}. , :1
Let y(0) = 0e” for 0 s 6 s 271' and y(0) = 41r—0 for 211 S 0 g 4,,_
d
Evaluate J; z zz+‘n’2 ' 'Y .
7. Let f(::) = [(z—§—i)(z—l——%i)(z—l—§)(2.'%—i)]'1 and let 7 be the
polygon [0, 2, 2+2i, 21', 01‘ Find Lf. 8. Let G = C— {(1, b}, a 9:9 b, and let )1 be the curve in the ﬁgure below. (a) Show that n(y; a) = n(y; b) = 0. (b) Convince yourself that y is not homotopic to zero. (Notice that the
word is “convince” and not “prove”. Can you prove it?) Notice that this
example shows that it is possible to have a closed curve y in a region such that n(y; :) = 0 for all : not in G without y being homotopic to zero. That
is. the converse to Corollary 6.10 is false. 9. Let G be a region and let yo and y, be two closed smooth curves in 6. Suppose yo~yl and F satisfies (6.2). Also suppose that y,(s)=F(s,t) is smooth for each 1. If w 6 (3—6 define h(t)=n(y,;w) and show that h: [0. l]—»Zis continuous. Find all possible values of f d: where y is any closed rectifiable
Y I + 22
curve in C not passing through i i.
, AZ ,, m: @Let f be analytic in B(a; R) and suppose that f(a) = 0. Show that a is a
zero of multiplicity m iﬂ‘ f ""‘”(a) = . . . = f(a) = 0 and f ("’(a) 9!: 0. @Suppose that f: G —> C is analytic and oneone; show that f’(z) aé 0 for
any 2 in G. ...
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 Spring '08
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