Riemann Integration and Uniform Convergence of Functions, Lecture notes of Calculus

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MA

Analysis 3 - Definitions

and Theorems

Revision Guide

Written by Sean Middlehurst

ii MA244 Analysis III

Contents

1 Riemann Integral 1 1.1 Defining the Integral....................................... 1 1.2 Properties of the Integral.................................... 2 1.3 The Fundamental Theorem of Calculus............................. 3 1.4 Improper Integrals........................................ 4 1.5 The Cantor Set.......................................... 4

2 Sequences and Series of Functions 5 2.1 Pointwise and Uniform Convergence.............................. 5 2.2 Series of Functions........................................ 6 2.3 Absolute Continuity....................................... 7

3 Complex Analysis 7 3.1 C Review............................................. 7 3.2 Power Series............................................ 8 3.3 Complex Integration and Contour Integrals.......................... 10

Introduction

This revision guide for MA244 Analysis III has been designed as an aid to revision, not a substitute for it. This guide is useful for revising through key definitions, theorems and some shorter proofs found in the course. However, a lot of the calculation methods and practical applications of the content of this module are omitted, in which it would be best to refer to the lectures and the online notes for said techniques.

Disclaimer: Use at your own risk. No guarantee is made that this revision guide is accurate or complete, or that it will improve your exam performance.

Authors

Written by Sean Middlehurst. Based upon the lecture notes for MA244 Analysis III, written by Jose Rodrigo at the University of Warwick. Any corrections or improvements should be reported by email to comms@warwickmaths.org.

Corollary 1.9. Given f : [a, b] → R bounded, we have:

L(f ) ≤ U (f )

Theorem 1.10. Let f : [a, b] → R be a bounded function. Then f is integrable if and only if for ∀ ε > 0 , ∃ a partition P of [a, b] such that:

U (f, P ) − L(f, P ) < ε

Theorem 1.11. Let f : [a, b] → R be a bounded function. f is integrable if and only if ∃ Pn, a sequence of partitions such that: lim n→∞ U (f, Pn) − L(f, Pn) = 0

Definition 1.12. Given f : [a, b] → R, we say that f is continuous at x if ∀ ε > 0 ∃ δ > 0 such that:

y ∈ [a, b] and |x − y| < δ =⇒ |f (y) − f (x)| < ε

Definition 1.13. Given f : [a, b] → R, we say that f is uniformly continuous if ∀ ε > 0 ∃ δ > 0 such that: x, y ∈ [a, b] and |x − y| < δ =⇒ |f (y) − f (x)| < ε

Theorem 1.14. Let f : [a, b] → R. f is continuous implies f is uniformly continuous.

Theorem 1.15. Let f : [a, b] → R be a monotonic function. Then it is Riemann integrable.

1.2 Properties of the Integral

Theorem 1.16. Let f, g : [a, b] → R be Riemann integrable functions, and c ∈ R. Then f + g, cf are Riemann integrable functions, such that:

∫ (^) b

a

cf = c

∫ (^) b

a

f

∫ (^) b

a

(f + g) =

∫ (^) b

a

f +

∫ (^) b

a

g

Theorem 1.17. Let f, g : [a, b] → R be Riemann integrable functions such that f ≤ g. Then:

∫ (^) b

a

f ≤

∫ (^) b

a

g

Note: We can now see that if f = 0, then clearly g ≥ 0 =⇒

g ≥ 0.

Corollary 1.18. Let f : [a, b] → R be integrable. Let m = inf f and M = sup f. Then:

m(b − a) ≤

∫ (^) b

a

f ≤ M (b − a)

Corollary 1.19. Let f : [a, b] → R be a continuous function. Then ∃ c ∈ [a, b] such that:

f (c) =

b − a

∫ (^) b

a

f

Note: We use the value (^) b−^1 a

∫ (^) b a f^ to correspond to the average of^ f^. Replacing^ f^ with the constant given by (^) b−^1 a

∫ (^) b a f^ gives the same value for the integral for both functions.

Theorem 1.20. Let f : [a, b] → R be an integrable function. Then |f | is integrable, such that:

∫ (^) b

a

f | ≤

∫ (^) b

a

|f |

Theorem 1.21. Let f : [a, b] → R, c ∈ (a, b). Then f is Riemann integrable on [a, b] if and only if it is Riemann integrable on [a, c] and [c, b]. Moreover:

∫ (^) c

a

f +

∫ (^) b

c

f =

∫ (^) b

a

f

Theorem 1.22. Let f : [a, b] → R be a bounded, Riemann integrable function and φ : R → R a continuous function. Then φ ◦ f is Riemann integrable.

Note: The composition of two Riemann integrable functions is not necessarily integrable.

Theorem 1.23. Let f, g : [a, b] → R be Riemann integrable functions. Then the product f g is Riemann integrable. If in addition (^1) g is bounded, then fg is Riemann integrable.

1.3 The Fundamental Theorem of Calculus

Theorem 1.24. Let F : [a, b] → R be a continuous function that is differentiable on (a, b) with F ′^ = f. Assume that f : [a, b] → R is an integrable function. Then:

∫ (^) b

a

f (x)dx = F (b) − F (a)

Theorem 1.25. Let f : [a, b] → R be an integrable function and define the function F : [a, b] → R by:

F (x) :=

∫ (^) x

a

f (t)dt

Then F is continuous on [a, b]. If f is continuous at c ∈ [a, b] then F ′(c) = f (c).

Theorem 1.26. Let f : [a, b] → R be an integrable function on [a, b] and continuous (from the right) at a. Then:

lim h→ 0

h

∫ (^) a+h

a

f (t)dt = f (a)

Similarly, if f is continuous (from the left) at b:

lim h→ 0

h

∫ (^) b

b−h

f (t)dt = f (b)

We can also consider a family of intervals Ih such that x ∈ Ih, |Ih| → 0 and (assuming f is continuous at x):

lim h→ 0

|Ih|

Ih

f (t)dt = f (x)

Theorem 1.27. Let f, g : [a, b] → R be continuous functions on [a, b] that are differentiable on (a, b), and such that f ′^ and g′^ are integrable on [a, b]. Then:

∫ (^) b

a

f (x)g′(x)dx = f (b)g(b) − f (a)g(a) −

∫ (^) b

a

f ′(x)g(x)dx

Theorem 1.28. Let f : [a, b] → R be a differentiable function such that f ′^ is integrable on [a, b]. Let g be a continuous function on f ([a, b]), the image of [a, b] under the map f. Then:

∫ (^) b

a

g(f (x))f ′(x)dx =

∫ (^) f (b)

f (a)

g(t)dt

Definition 1.34. The Devil’s Staircase function is the function defined as follows:

• Set f 0 (x) = x.
• Set f 1 as follows:

f 1 (x) :=

3 2 x^ x^ ∈^ [0,^

1 3 ] 1 2 x^ ∈^ [^

1 3 ,^

2 3 ] 1 2 +^

3 2 (x^ −^

2 3 )^ x^ ∈^ [^

2 3 ,^ 1]

• Inductively, given fk, set fk+1 as follows:

fn(x) :=

1 2 fn(3x)^ x^ ∈^ [0,^

1 3 ) 1 2 x^ ∈^ [^

1 3 ,^

2 3 ) 1 2 +^

1 2 fn(3x^ −^ 2)^ x^ ∈^ [^

2 3 ,^ 1]

Theorem 1.35. The limit of the sequence (fn) exists and is continuous with f (0) = 0 and f (1) = 1.

2 Sequences and Series of Functions

2.1 Pointwise and Uniform Convergence

Definition 2.1. Let (fn)∞ n=1 be a sequence of functions, with fn : Ω → R. We say that (fn) or fn converges pointwise to f : Ω → R if and only if for every x ∈ Ω, we have limn→∞ fn(x) = f (x). Pointwise convergence is denoted by fn → f.

Note: Pointwise limits of sequences of continuous functions need not be continuous.

Definition 2.2. Let fn : Ω → R be a sequence of functions. We say that (fn) converges uniformly to f : Ω → R if and only if ∀ ε > 0 ∃ N ∈ N such that |fn(x) − f (x)| < ε ∀ x ∈ Ω and ∀ n > N. Uniform convergence is denoted by fn ⇒ f.

Definition 2.3. We define the following norm:

‖f ‖∞ = sup x∈Ω

|f (x)|

As a norm, ‖ · ‖inf ty holds the following properties:

• ‖f ‖∞ ≥ 0 ,
• ‖λ‖∞‖ = |λ|‖f ‖∞, ∀λ ∈ R,
• |f + g‖∞ ≤ ‖f ‖∞ + ‖g‖∞.

Note: From this notation, we have:

fn ⇒ f ⇐⇒ ∀ ε > 0 , ∃ N such that ‖fn − f ‖∞ < ε ∀ n > N

Definition 2.4. A sequence (fn) of functions in Ω is called uniformly Cauchy if and only if ∀  > 0 ∃ N ∈ N such that ‖fn − fm‖∞ ∀ n, m > N.

Theorem 2.5. A sequence (fn) is uniformly convergent if and only if it is uniformly Cauchy.

Theorem 2.6. Let (fn) be a sequence of continuous functions in Ω that converges uniformly to f : Ω → R. Then f is continuous.

Theorem 2.7. (Cb; ‖ · ‖∞) is a complete space, i.e., every Cauchy sequence converges to a continuous bounded function.

Theorem 2.8. Let (fn), f : [a, b] → R be Riemann integrable functions that converges uniformly to f : [a, b] → R. Then f is Riemann integrable and

fn →

f.

Definition 2.9. Given f : Ω ⊂ R^2 → R, we say that f is continuous at x if ∀ ε > 0 ∃ δ > 0 such that:

y ∈ Ω and |x − y| < δ =⇒ |f (y) − f (x)| < ε

Theorem 2.10. Given f : Ω ⊂ R^2 → R, we say that it is uniformly continuous if ∀ ε > 0 ∃ δ > 0 such that: x, y ∈ Ω and |x − y| < δ =⇒ |f (y) − f (x)| < ε

Theorem 2.11. Let f : Ω ⊂ R^2 → R be a continuous function. Assume that Ω is closed and bounded. Then it is uniformly continuous.

Theorem 2.12. Let f : [a, b] × [c, d] → R be a continuous function. Define:

I(t) :=

∫ (^) b

a

f (x, t)dx

is a continuous function in [c, d].

Theorem 2.13. Let f, ∂f∂t be continuous functions on [a, b] × [c, d]. Then, for t ∈ (c, d):

d dt

∫ (^) b

a

f (x, t)dx =

∫ (^) b

a

∂f ∂t

(x, t)dx

Theorem 2.14. Let f : [a, b] × [c, d] → R be a continuous function. Then:

∫ (^) b

a

( ∫ (^) d

c

f (x, y)dy

dx =

∫ (^) d

c

( ∫ (^) d

a

f (x, y)dx

dy

Theorem 2.15. Let (fn) be a sequence of C^1 functions on [a, b]. Assume fn → f pointwise and that f (^) n′ converges uniformly to g. Then f is C^1 and g = f ′^ or f (^) n′ → f ′.

2.2 Series of Functions

Definition 2.16. Let (fk) be a sequence of functions fk : Ω → R. Let (Sn) be the sequence of partial sums, with Sn : Ω → R defined by:

Sn(x) :=

∑^ n

k=

fk(x)

Then the series: (^) ∞ ∑

k=

fk(x)

converges pointwise to S : Ω → R on Ω if Sn → S pointwise on Ω and uniformly to S on Ω if Sn ⇒ S uniformly on Ω.

Theorem 2.17. Let (fk), with fk : [a, b] → R, be a sequence of integrable functions. Assume that Sn =

∑n k=1 fk^ converges uniformly. Then^

k=1 fk^ is Riemann integrable and: ∫ (^) ∑∞

k=

fk =

∑^ ∞

k=

fk

Theorem 2.18. Let (fk), with fk : [a, b] → R, be a sequence of C 1 functions such that Sn =

∑n k=1 fk converges pointwise. Assume that

∑n k=1 f^ ′ k converges uniformly. Then: ( (^) ∑∞

k=

fk(x)

∑^ ∞

k=

f (^) k′(x)

Theorem 2.19. The Weierstrass M-test: Let (fk) be a sequence of functions fn : Ω → R, and assume that ∀ k ∃ Mk > 0 such that |fk(x)| ≤ Mk ∀ x ∈ Ω and

k=1 Mk^ <^ ∞. Then^

k=1 fk^ converges uniformly on Ω.

Definition 3.4. A set K ⊂ C is sequentially compact if and only if every sequence (xj )j∈N ⊂ K has a convergent subsequence (xj(t))l∈N whose limit is in K.

Definition 3.5. Given f : Ω ⊂ C → C we say that it is continuous at z 0 ∈ Ω if and only if ∀ ε > 0 ∃ δ such that |z − z 0 | < δ, with z ∈ Ω implies that |f (z) − f (z 0 )| < ε.

Definition 3.6. Let Ω ⊂ C be an open set and z 0 ∈ Ω. We say that f is complex differentiable at z if and only if the limit:

lim h→ 0

f (z + h) − f (z) h

exists. We denote this limit by f ′(z).

Definition 3.7. We say that f : Ω → C is analytic/holomorphic in a neighbourhood U of z if it is complex differentiable everywhere in U. We say that f is entire if it is analytic in the whole of C.

Theorem 3.8. Let f : Ω ⊂ C → C with Ω open. f is complex differentiable of z = a + ib ∈ Ω if and only if f , when considered as a map from Ω ⊂ R^2 → R^2 (f (z) = u(z) + iv(z), where u and v are real valued functions) has a differential at the point (a, b) that satisfies the Cauchy-Riemann equations:

ux = vy uy = −vx

Theorem 3.9. Let f, g : Ω ⊂ C → C be complex differentiable functions. Then (assuming g 6 = 0 in the third expression, we have the familiar expressions:

(f + g)′^ = f ′^ + g′^ (f g)′^ = f ′g + f g′

f g

f ′g − f g′ g^2

(f (g))′^ = f ′(g)g′

3.2 Power Series

Definition 3.10. The series

n=0 an^ ∈^ C^ is^ convergent^ if and only if the sequence^ SN^ =^

n=0 is convergent in C.

Definition 3.11. The series

∑∞^ n=0^ an, with^ an^ ∈^ C^ is^ absolutely convergent^ if and only if the series n=0 |an|^ is convergent.

Theorem 3.12. Ratio Test: Consider

n=0 an. Then:

• If lim sup |a|na+1n| |< 1, then

n=0 an^ is convergent.

• If lim sup |a|na+1n| |> 1, then

n=0 an^ is divergent.

Theorem 3.13. Root Test: Consider

n=0 an. Then:

• If lim sup |an|^1 /n^ < 1, then

n=0 an^ converges.

• If lim sup |an|^1 /n^ > 1, then

n=0 an^ diverges.

Theorem 3.14. Given (an)∞ n=0, ∃ R ∈ [0, ∞] such that

n=0 anz n (^) converges ∀ |z| < R and diverges

∀ |z| > R. Furthermore:

R =

lim sup |an|^1 /n

Theorem 3.15. Let an 6 = 0 ∀ n ≥ N , and assume that lim |a|na+1n| |exists. Then

n=0 an^ has radius of convergence R = lim (^) |a|ann+1||.

Theorem 3.16. Let

n=0 anz

n (^) have radius of convergence R. Then for |z| < R the function f (z) := ∑∞ n=0 anz

n (^) is differentiable and:

f ′(z) =

∑^ ∞

n=

nanzn−^1

Corollary 3.17. Let

n=0 anz

n (^) be a power series with radius of convergence R > 0. Then f (z) := ∑∞ n=0 anz

n (^) is infinitely differentiable and moreover:

f (n)(0) = ann! n = 0, 1 , 2 ,...

Theorem 3.18. Let

n=0 anz

n (^) be a power series with radius of convergence R > 0. Then for every

r < R the sequence of functions:

fk :=

∑^ k

n=

anzn

converges uniformly in |z| ≤ r.

Definition 3.19. We define the following power series for z ∈ C:

• ez^ :=

n=

1 n! z

n

• cos(z) :=

n=

(−1)n (2n)! z

2 n

• cosh(z) :=

n=

1 (2n)! z

2 n

• sin(z) :=

n=

(−1)n (2n+1)! z

2 n+

• sinh(z) :=

n=

1 (2n+1)! z

2 n+

Proposition 3.20. The following identities hold ∀ z ∈ C:

cos(z) =

eiz^ + e−iz 2

cosh(z) =

ez^ + e−z 2

sin(z) =

eiz^ − e−iz 2

sinh(z) =

ez^ − e−z 2

Note: The following relationships are evident:

cos(iz) = cosh(z) cosh(iz) = cos(z) sin(iz) = i sinh(z) sinh(iz) = i sin(z)

Theorem 3.21. The exponential function ez^ satisfies the following properties:

• ez+w^ = ez^ ew^ ∀ z, w ∈ C.
• ez^6 = 0 ∀z ∈ C.
• ez^ = 1 if and only if z = 2kπi for k ∈ Z, and as a result, ez+w^ = ez^ if and only if w = 2kπi, k ∈ Z.

Definition 3.22. We define, for z 6 = 0:

arg(z) = {θ ∈ R : z = |z|eiθ^ }

Proposition 3.23. Properties of arg(z):

• arg(αz) = arg(z) ∀ α > 0,
• arg(αz) = arg(z) + π = {θ + π : θ ∈ arg(z)} ∀ α > 0,
• arg(z) = −arg(z) = {−θ : θ ∈ arg(z)},
• arg( (^1) z ) = −arg(z),
• arg(zw) = arg(z) + arg(w) = {θ + φ : θ ∈ arg(z), φ ∈ arg(w)}.

Theorem 3.30. Assume that F : Ω ⊂ C → C is analytic and set f (z) = dFdz. Let γ : [a, b] → Ω be a curve. Then: (^) ∫

γ

f dz = F (γ(b)) − F (γ(a))

Proposition 3.31. For a function f = u + iv and a curve γ(t) = γ 1 (t) + iγ 2 (t), we have: ∫

γ

f dz = circulation(f ) + iflux(f )

Where we have the vector field f = (u, −v) and the definitions for circulation and flux are taken from MA259 Multivariable Calculus.

Theorem 3.32. A set Ω ⊂ C is connected if it cannot be expressed as a union of non-empty open sets Ω 1 and Ω 2 such that Ω 1 ∩ Ω 2 = ∅. An open, connected set Ω ⊂ C is called simply connected if every closed curve in Ω can be continuously deformed to a point. To put it simply, a simply connected domain has ’no holes’.

Theorem 3.33. Cauchy’s Theorem: Let f : Ω → C be an analytic function, with Ω an open, simply connected domain. Let γ be a C^1 closed curve in Ω. Then: ∫

γ

f (z)dz = 0

Theorem 3.34. Let Ω ⊂ C be a region bounded by two simple curves γ 1 (the exterior curve) and γ 2 (the interior). Assume they are oriented positively, and let f be an analytic function in Ω ∪ γ 1 ∪ γ 2. Then: (^) ∫

γ 1

f dz +

γ 2

f dz = 0

Similarly, if we denote by γ 2 − the anti-clockwise parameterisation, then the result can be rephrased as: ∫

γ 1

f dz =

γ− 2

f dz

Definition 3.35. Given a simple closed C^1 curve γ we denote by I(γ) the interior region to γ. We denote by O(γ) the exterior region to γ.

Theorem 3.36. Let γ : [a, b] → C be a positively oriented simple closed C^1 curve. Assume that f is analytic in γ and on the interior of γ, I(γ). Then:

f (z) =

2 πi

γ

f (w) w − z

dw ∀ z ∈ I(γ)

Theorem 3.37. Let γ : [a, b] → C be a positively oriented simple closed C^1 curve. Assume that f is analytic in γ and on the interior of γ, I(γ). Then f (n)(z) exists ∀ n ∈ N and the derivative is given by:

f (n)(z) =

n! 2 πi

γ

f (w) (w − z)(n+1)^

dw ∀ z ∈ I(γ)

Theorem 3.38. Taylor Series Expansion: Let f be an analytic function on BR(a) for a ∈ C, R > 0. Then ∃! constants cn, n ∈ N such that:

f (z) =

∑^ ∞

n=

cn(z − a)n^ ∀ z ∈ BR(a)

Moreover, the coefficients cn are given by:

cn =

2 πi

γ

f (w) (w − a)n+^

dw =

f (n)(a) n!

where γ is any positively oriented simple closed curve that is contained in BR(a).

Theorem 3.39. Liouville’s Theorem: Let f : C → C be an analytic, bounded function. Then f is constant.

Theorem 3.40. Fundamental Theorem of Algebra: Every non-constant polynomial p on C has a root, that is ∃ a ∈ C such that p(a) = 0.

Theorem 3.41. Let fn : Ω → C be a sequence of analytic functions on an open set Ω. If fn converges uniformly to f , then f is analytic.

ANOTHER DISCLAIMER: Please note that this guide is a list of the key definitions and theorems you should be able to recall and utilise throughout the module. Proofs, calculations and examples, whilst omitted, are also very important for not just preparing for the exam but for preparing you for later modules in your degree. We highly recommend that you don’t just use this guide as your sole revision resource. Please refer back to the printed lecture notes and your own notes for a full comprehension of the module, as this guide is more for helping factual recall, rather than understanding.