(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0
15.12 Suppose that R>0 is given. Prove that, if N is sufficiently large, Σχη/n!*0 for all z ED(0; R). n-0 15.12 Suppose that R>0 is given. Prove that, if N is sufficiently large, Σχη/n!*0 for all z ED(0; R). n-0
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo (12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0 definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0
Suppose that (a-r, a) C E or (a, a + r) C E, f : E → R, L E R, and (1) Prove that there exist numbers 0 < δ < r and M > 0 such that If(x)| < M for all (2) Prove that if L is nonzero, then there exist numbers 0 < δ < r and η > 0 such that limx→af(x) = L xEEwith 0 < |x-a| < δ. If(x)| > η for all...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C? u(20) for all z...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
(4) w e suppose j is a measure, f E L1(μ),dom(f)-R, f 0 and EnaER: f(x) 2 n). a) Prove that limn-100 JE, fdy -0 (b) Prove that limn-0o n(En)-0 (4) w e suppose j is a measure, f E L1(μ),dom(f)-R, f 0 and EnaER: f(x) 2 n). a) Prove that limn-100 JE, fdy -0 (b) Prove that limn-0o n(En)-0
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C?
Here you are asked to prove the Fundamental Theorem of Algebra a different way by using Rouché's Theorem. Where n E N, consider the polynomial n-1 Pn (z)z" k-0 Using the circular contour C-[z : zR with R appropriately chosen, (a) prove that pn(2) has (counting multiplicity) precisely n zeros in the open disc D(0, R); (b) also show that Pn(z) has no zeros in C \ D(0, R) Here you are asked to prove the Fundamental Theorem of Algebra...