Statistics 203: Introduction to Regression and Analysis of Variance
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Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
- p. 1/20 Statistics 203: Introduction to Regression and Analysis of Variance Experimental design Jonathan Taylor
- bonferroni correction for simultaneous inference for residuals
- powerful enough
- confidence bands
-
Voir
Queen Mary, University of London
B. Sc. Examination (Sample)
MTH6111 ComplexAnalysis Duration: 2hours Date and time:
AnswerALLquestions.
Calculators are NOT permitted in this examination.The unauthorised use of a calculator constitutes an examination offence.
You are not allowed to start reading the question paper until instructed to do so by the invigilator.
You must not remove the examination paper from the examination room.
cQueen Mary, University of London, 2012
MTH6111
1
TURN OVER
Question 1(a)Let Ω be a domain inCand letf: Ω−→Cbe a function. Explain what is meant by saying thatfhas aderivativeat a point in Ω, and thatfisholomorphicin Ω. Letf: Ω−→Cbe a holomorphic function satisfyingf(Ω)⊂iR. Show thatfis constant.[10]
(b)State, but do not prove,Cauchy’s deformation theoremfor a holomorphic functionfdefined on a domain Ω⊂C. [5]
∗ Letγ: [0,2π]−→Cbe a closed path whose imageγis the ellipse
2 2 x y + =1 (>a, b0). 2 2 a b By showing that Z Z −1−1 z dz=z dz γΓ for a suitable circle Γ, deduce that Z 2π 1 2π dθ=. 2 2 22 0acosθ+bsinθ ab
[10]
Question 2(a)State, but do not prove, theMaximum Modulus Principlefor a holomorphic functionfdefined on a bounded domain Ω inC.
Given a holomorphic functionfsatisfying|f(z)|=|z|for|z|<1, show that there is a complex numberαwith|α|= 1 such thatf(z) =αzfor|z|<1. [10]
(b)Locate and classify the isolated singularities of the following functions: z z−sinz1e , ,. 3 13 z(z−1) cos z [15]
MTH6111
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Question 3(a)LetD={z∈C:|z|<1}be the open unit disc centred at the origin, with boundaryT={z∈C:|z|= 1}.
For eachα∈D, let the functiongα:D∪T−→Cbe defined by z−α gα(z() =|z| ≤1). 1−αz Show thatgαis a bijection ontoD∪T, with inverseg−α, and satisfies gα(T) =T. [10]
(b)Letfbe an entire function and let|f(z)| ≤Mfor|z−a|=rthat. Show (n) then-th derivativef(a) offatasatisfies n!M (n) |f(a)| ≤. n r 4 Given that|f(z)| ≤1 +|z|for allz∈C, show thatfis a polynomial of degree at most 4. [15]
Question 4(a)Let Ω be a region inCand letf: Ω−→Cbe a function. Explain what is meant by saying thatfis ameromorphicfunction.
Letf: Ω−→Cbe a meromorphic function with simple zeros ata1, . . ., an, and simple poles atb1, . . ., bmin Ω.Given a closed pathγin Ω not passing througha1, a, . . .n, b1, . . ., bmthat. Show Z0n m X X f(z) dz= Indγ(ak)−Indγ(bk) γf(z) k=1k=1 where Indγ(z) denotes the index ofγatz. [15]
(b)State, but do not prove,Liouville’s theoremfor an entire function onC.
Letfbe an elliptic function onCwith primitive periodsπandπi. Given thatfhas no pole in the fundamental period-parallelogram, show thatfis constant. [10]
MTH6111
3END OF EXAMINATION
