MA 2580 Mathematical Analysis III 2020
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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 北(log(1(1)/北))2 dz.
c) Show that the improper integral 0/ z一2/3e一北 dz is convergent.
d) State conditions on the function 大 : (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北 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 (大π g) = 01 大 (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 大 : [0 π 1] → R is a regulated function?
Show that if 大 is regulated then so is the function I大 (z)I.
Define
大 (z) =
Explain why 大 is not a regulated function and why g is a regulated function.
b) Briefly explain how the integral 01 大 (z)dz is defined for a regulated function 大 : [0 π 1] → R.
c) Use integration by parts on the integral
北
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 大 : [0 π 1] → R is regulated and let θ (z) = 0北 大 (t)dt for z e [0 π 1].
Prove that if 大 is continuous at z e (0 π 1) then θ is differentiable at z and
θ \ (z) = 大 (z).
Give an example of a regulated 大 : [0 π 1] → R and z e (0 π 1) for which θ is not
differentiable at z.
b) Integrate the geometric series 1北 = 1 - z +z2 - z3 + 尸 尸 尸 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 大n : [aπ b] → R converge uniformly to 大 : [aπ b] → R as n → o. Suppose that 大n are continuous for all n 2 1. Give the proof that 大 is continuous. Explain why g(z) = 2一k cos(kz) defines 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) = π一π 大 (t)g(t)dt. Show that the functions 大n (z) = sin(nz) satisfy (大n π 大m ) = 0 for any pair of integers nπ n with n n. Suppose that 大 = ak sin(kz). Show that |大 |2(2) = 尸 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 大\ on [0 π 1]} define |大 | = sup I大 (z)I + sup I大\ (z)I尸 北←[0,1 北←[0,1 Show that this defines a norm. Show that >1 [0 π 1] is complete in this norm. |
2022-09-06