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MA 2580

SECOND YEAR  EXAMINATION: April 2020

Mathematical Analysis III

COMPULSORY QUESTION

1.    a) Find the derivative of 打(z) =   1ez (log(t))2 dt for z e R.

b) Find the value of the integral   01/2 (log(1(1)/北))2 dz.

c)  Show that the improper integral   0/ z2/3e一北 dz is convergent.

d)  State conditions on the function  : (aπ b) x [0 π 1] → R that ensure that

d     1                                   1  / 

dz  0                                   0    /z

Apply this result to d北(d)    01 t dt when z 0 and hence calculate   01 t1/2 ln(t)dt.

e)  State the Cauchy Schwarz inequality for the space >[0 π 1] with the inner product  (π g) =  01 (t)g(t)dt.

f)  Give an example of a sequence of functions 大n   :  [0 π 1] → R for n =  1 π 2 π 尸 尸 尸 for which 大n (z) → 0 as n o for all z e [0 π 1] and where   01 大n (z)dz →\ 0. (Explain why your example has these properties).

g)  On Rn , show that (zπ y) =     k(n)=1 k2 zk yy  is an inner product.

OPTIONAL QUESTIONS

2.    a) What does it mean that : [0 π 1] → R is a regulated function?

Show that if  is regulated then so is the function I (z)I.

Define

 (z) =

Explain why  is not a regulated function and why g is a regulated function.

b) Briefly explain how the integral   01 (z)dz is defined for a regulated function 大 : [0 π 1] → R.

c) Use integration by parts on the integral

d(n) =       t(ln(t))m dt   for n = 0 π 1 π 2 π 尸 尸 尸

0

to show that d(n) =2(北2) (ln z)m - d(n - 1).

Deduce the value of d(1) and d(2).


3.    a)  Suppose that : [0 π 1] → R is regulated and let θ (z) =  0 (t)dt for z e [0 π 1].


Prove that if  is continuous at z e (0 π 1) then θ is differentiable at z and

θ \ (z) = (z).

Give an example of a regulated : [0 π 1] → R and z e (0 π 1) for which θ is not

differentiable at z.  

b) Integrate the geometric series 1  = 1 - z +z2 - z3 + term by term to deduce the power series for ln(1 + z).

What is the radius of convergence R for this power series?  Does it converge when z = 土R?

Use term by term integration again to deduce that

1 ln(1 + z)                 1       1       1

0             z                     22        32        42

Explain why this term by term integration is justified.    

 

4.    a)

 

 

 

b)

Suppose that n  : [aπ b] R converge uniformly to  : [aπ b] R as n o. Suppose that n are continuous for all n 2 1. Give the proof that is continuous.

Explain why g(z) =      2k cos(kz) denes a continuous function g : R R. Show that g has a continuous derivative.

Calculate   0(π/)2 g(z)dz.

(State carefully all results from the module that you are using).

 

5.    a)

 

 

 

 

b)

Consider the space >[-尸π 尸] with the inner product (大π g) =   ππ (t)g(t)dt.      Show that the functions n (z) = sin(nz) satisfy (大n π 大m )  = 0 for any pair of integers nπ n with n  n.

Suppose that  =      ak sin(kz). Show that |大 |2(2)  =     ak(2) . What does it mean that a normed space (vπ | . |) is complete?    For the vector space

>1 [0 π 1] = {大 : [0 π 1] Rπ 大 has a continuous derivative \  on [0 π 1]}

define

|大 | =  sup  I大 (z)I + sup  I大\ (z)I尸

[0,1                      [0,1

Show that this denes a norm.

Show that >1 [0 π 1] is complete in this norm.