Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
2a) Let a, b e R with a < b and let g [a, bR be continuous. Show that g(x) cos(nx) dx→ 0 n →oo. as Hint: Let ε > 0, By uniform continuity of g, there exists δ > 0 such that 2(b - a Choose points a = xo < x1 < . . . < Xm such that Irh-1-2k| < δ. Then we may write rb g (z) cos(nx) dx = An + Bn where 7m (g(x)...
Notation: In what follows, let D = {z:z<1} and H = {z: Im(2) >0}, 6. Let a CD be nonzero. Show that there is a unique automorphism f of D such that f(a) = 0 and f(0) = a. (Hint: Use Theorem 2.2, Chapter 8, Section 2.1 of Stein- Shakarchi (page 220.)
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.
0 Let (f.) be a group, show that (ly) G where Gly); = Lael | ag= ga & gely is the center of G. (So, show that cly)< ; & cgjat. ) @ let y be a group, gel & Haf. Prove that Ks4 where us Ki Cig) := {acly I ag = gay is the centralizer of g inily, and K: N (H): = hatly I aH=Hay in the normalizen of Henly.
1. Consider the following two probability density functions: f(3) = 2053 } for a <I<02 and g() = where ci and ca are finite real numbers. 265. for <y<02, (a) Show that f(r)dx = 9(r)dt = 1. (b) Find the cumulative distribution functions F(x) and Gu). (d) Show that if X-f(x), then 1-X g(x). (e) Show that if X h(x) = 21, for 0 <<1, then Y = c +(2-c)X ~f. (h) Show that if Uſ and U2 are two...
4. Let Z ~ N(0,1) be a standard normal variable. Calculate the probability (a) P(1 <Z < 2). (b) P(-0.25 < < < 0.8). (c) P(Z = 0). (d) P(Z > -1).
Let X be a random variable with CDF z<0 G()=/2 0 <IS2 z>2 1 Suppose Y = X2 is another random variable, find (a) P(1/2 X 3/2), (b) P(1s X< 2) (c) P(Y X) (d) P(X 2Y). (f) If Z VX, find the CDF of Z. (d) P(X+Y 3/4)