Exam June 2015, questions and answers PDF

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MATH201601 This question paper consists of 4 printed pages, each of which is identified the reference MATH201601. All calculators must carry an approval sticker issued the School of Mathematics. c University of Leeds School of Mathematics 2015 MATH201601 Analysis Time Allowed: 2 hours 30 minutes Ans...

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MATH201601

This question paper consists of 4 printed pages, each of which is identified by the reference MATH201601.

All calculators must carry an approval sticker issued by the School of Mathematics.

University c of Leeds School of Mathematics May/June 2015 MATH201601 Analysis Time Allowed: 2 hours 30 minutes Answer no more than 4 questions. If you attempt 5, only the best 4 will be counted. All questions carry equal marks.

1

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MATH201601 1. (a)

i. State the ε − δ definition of continuity of a function f at a point c. ii. Using the ε − δ definition of continuity show that  2 + x, x ≥ 0, f (x) = 2 − x3 , x < 0, is continuous at 0. iii. Using the ε − δ definition of continuity or some other method show that    1  , x 6= 0, cos f (x) = x  0, x = 0, is not continuous at 0.

(b)

i. Define what it means for a function to be differentiable at a point a. ii. Determine whether or not the following function is differentiable on R:   sin2 x , x 6= 0 f (x) = 0 x x = 0.

iii. Show that f : [a, b] → R is differentiable at a if and only if there exists ε(x) a function ε : [a, b] → R such that → 0 as x → a and f (x) = x−a f (a) + β(x − a) + ε(x) for some β ∈ R. 2. (a)

i. Define what is meant by dissection and regular dissection. ii. Define what is meant by the upper and lower Riemann sums of a function f : [a, b] → R over a dissection, these are denoted u(D, f ) and l(D, f ) for the dissection D . iii. Define what is meant by the upper and lower Riemann sums of a function f , these are denoted u(f ) and l(f ).

(b) Prove that u(f ) and l(f ) exist for all bounded functions f : [a, b] → R.

(c) Calculate l(f ) for the function f (x) = 2 − 5x2 on [1, 2] for a regular dissection. P 1 You may use ni=1 i2 = n(n + 1)(2n + 1). Explain why u(f ) and l(f ) are equal 6 for this function. Verify your answer by applying the Fundamental Theorem of Calculus.

(d) Give examples (without proof) of functions f , g, h and k on the interval [0, 1] such that i. f is bounded and integrable, R1 ii. g is continuous and 0 g = 0 but g is not identically zero, R1 iii. 0 ≤ h(x) ≤ 1 for 0 ≤ x ≤ 1 and 0 h = 0 but h is not identically zero, iv. k is not integrable. Z x2 ′ cos (π/t) dt, x > 0. (e) Calculate the derivative F (x) for F (x) = x

2

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MATH201601 3. (a) Define what is meant by the words contour integral and winding number. (b) On a single diagram draw the following contours indicating their orientation: α(t) = 1 + 3i + e−it, it

π ≤ t ≤ 2π,

β(t) = 3 + 3i + 3e , −π ≤ t ≤ π/2, γ(t) = (3 + 6i) − (1 + 3i)t, 0 ≤ t ≤ 1, δ(t) = 2 + 3i − 2t2 ,

0 ≤ t ≤ 1.

(c) Calculate, where possible, the winding numbers of the following curves around the 7 point 1 + i: 2 i. α + δ , ii. α − δ , iii. β + γ − α, iv. β + γ + δ , v. β + γ + δ + α + δ + α + δ . (d) Calculate, where possible, the following integrals using the contours above. Explain your reasoning. R i. δ Re(z) dz. Z 1 ii. dz. z − (2 + 7i)/2 Z α+δ 1 iii. dz. −α−δ z − (2 + 7i)/2 Z 3z 3 iv. dz, (z − 20)2 Zα−γ−β 2 cos(z) dz. v. β

4. (a)

i. ii. iii. iv.

State (but do not prove) the Cauchy-Riemann equations. Define what it means for a function to be harmonic. Show that the function u(x, y) = −2xy + 3x is harmonic. Find the harmonic conjugate to u above.

(b) Suppose f : C → C is analytic at each point of C and for all z ∈ C, Re (f (z)) = 0. Show that f (z) = ci for some c ∈ R, for all z ∈ C.

(c) State (but do not prove) the Estimation Lemma. Hence, show that  Z   e5iz πR    z 2 − 7 dz  ≤ R2 − 7 , γ √ it where γ(t) = Re , 0 ≤ t ≤ π and R > 7. (d) State (but do not prove) a formula for finding the residue of a function with a pole of order N at w. Hence, deduce that if q is an analytic function with a zero of multiplicity 1 at w, then 1 res(1/q, w) = ′ . q (w) [Hint: Write q as a power series first.] 3

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MATH201601

5. (a)

1 . + 25)(z 2 + 4) ii. Calculate the residues of the poles in the upper half-plane, i.e., z ∈ C such that Im (z) ≥ 0. iii. Let γR (t) = t for −R ≤ t ≤ R and CR+ (t) = Reit for 0 ≤ t ≤ π. Draw the graph of g : (0, ∞)\{2, 5} → R given by   Z  1   dz . g(r) =  2 2 γr +Cr+ (z + 25)(z + 4) i. Locate all the poles of f (z) =

(z 2

iv. Hence, use the calculus of residues to show that Z ∞ π 1 = . 2 2 70 −∞ (x + 25)(x + 4) (b)

i. Calculate the residues of the function f (z) =

3z 2

1 + 10z + 3

at all its poles. ii. Hence, using the calculus of residues, calculate Z 2π 1 dθ. 5 + 3 cos(θ) 0

4

End.

IMPORTANT NOTE The attached check-sheet contains the final answers to some, not necessarily all questions on the exam. Answers to questions requiring longer answers, for example proofs, are not given.

Please note. In the exam, students are expected to show their full work on the exam script, not just final answers.

Advice. Use this check-sheet to check your answers AFTER you have worked through the exam as if you were in an exam situation, i.e. without access to notes, books, answers to exercises, etc. This way you will test whether you can tackle the problems without any help as in the exam.

MATH2016 ANALYSIS May/June 2015 Checksheet 1.

(a) (i). — √ (ii). δ = min{1, 3 ε} will do. (iii). Use sequential continuity. Let 1/xn = 2πn for all n ∈ N (b) (i). — (ii). f is differentiable on all of R. (iii). Hint (if needed) For x 6= a f(x) = f(a) + β(x − a) + ε(x) ⇐⇒

2.

f(x) − f (a) ε(x) . =β+ x−a x−a

(a) — (b) Bring together separate proofs from course. 29 (c) l(Dn , f ) = − . Here u(f) = l(f) as f is continuous. 3 (d) Many answers possible. (i). Anything continuous, f(x) = 0 for eg, (ii). g(x) = x − 21 .   0, (iii). h(x) =  1,   0, (iv). k(x) =  1,

x 6= 0 x = 0. x is rational, x is irrational.

(e) By the FTC

F ′ (x) = 2x cos

1

π π − cos 2 x x

3.

(a) — (b) Im

6i

γ

α 3i

1+ 7 2i

β

δ

3

6

Re

(c) (i). −1, (ii). Can’t be defined as α − δ is not a contour, (iii). 1, (iv). 0, (v). −2 (d) (i) −2,

(ii) −2πi,

(iii) 2πi,

(iv) 0,

2i sinh(3). 4.

(a) (iv) x2 − y2 + 3y (can add any constant). (b) Use Cauchy-Riemann. (c) —

2

(v) 2(sin(3) cosh(6) + i cos(3) sinh(6)) +

(d) Let q(z) = a0 + a1 (z − w) + a2 (z − w)2 + . . . . Then a0 = 0 and a1 = q ′ (w) and   1 res(f, w) = lim (z − w) z→w (z − w)(a1 + a2 (z − w) + . . . ) 1 1 1 = ′ . = lim = z→w a1 + a2 (z − w) + . . . q (w) a1 5.

(a) (i). Poles at (z 2 + 25)(z 2 + 4) = 0, i.e., z = ±5i and z = ±2i. Only 2i and 5i are in the upper half-plane. (ii). res(f, w) =

1 . Thus 4w3 + 58w res(1/f, 2i) = −

i 84

and

res(1/f, 5i) =

i . 210

(iii). The graph of f is: g(r) π 42 π 70

r

(iv). — (b) (i). Let f(z) = 3z 2 + 10z + 3. Then f(z) = 0 ⇐⇒ z = −1/3 or z = −3. For the residues: res(1/f, w) =

1 . 6w + 10

Thus, res(1/f, −1/3) = (ii).

Z

0

2π

π 1 dθ = . 5 + 3 cos θ 2

3

1 8

and

1 res(1/f, −3) = − . 8...