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MAST30021 Complex Analysis
创建日期：2024-05-13 17:05:42
所属分类：数学mathematics
标签： MAST30021
Complex Analysis

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Mathematics and Statistics

MAST30021 Complex Analysis
Written assignment 3 and Cover Sheet
Student Name Student Number
Submit your assignment solutions together with this coversheet via the MAST30021 Gradescope
website before Tuesday 20 September (noon AET) sharp. No extensions will be granted! Only
exemptions for suitably justified reasons can be granted.
• This assignment is worth 5% of your final MAST30021 mark.
• Assignments must be neatly handwritten unless you have a medical exception.
• Full working must be shown in your solutions.
• Marks will be deducted in every question for incomplete working, insufficient justification of steps and
incorrect mathematical notation.
• You must use methods taught in MAST30021 Complex Analysis to solve the assignment questions.
• The first task is mandatory for everyone, meaning everyone must write a summary of three indicated
lectures of not less than half and not more than one entire A4 page for each lecture.
• The second and third tasks are mandatory for everyone, meaning everyone must completely answer these
questions.
• The fourth and fifth tasks are optional, meaning you have to decide which of the two you want to answer.
In the case you answer both, indicate which of the two questions should be marked! If you do not do that the
tutor has free choice to decide which is going to be marked.
• There are in total 40 points to achieve. You cannot get 50 points!
• Begin your answer for each question on a new page!
Please, turn the page for the other questions!
Page 1 of 5
1. Mandatory Summary 10 points.
Write a summary of three lectures chosen from Lecture 14 to Lecture 19. Note, that any Theorem and
Definition, especially those with a name of a renowned mathematician, are worth mentioning. Use the
space below. Clearly indicate which lecture you are writing about.
Lecture :
Please, turn the page for the other questions!
Page 2 of 5
Lecture :
Please, turn the page for the other questions!
Page 3 of 5
Lecture :
Please, turn the page for the other questions!
Page 4 of 5
2. Mandatory Question (simple computation) 10 points.
Find all zeros and singularities of the following functions and classify those (isolated or not, essential,
removable or the order of the poles and zeros). Give an explanation of your classification. Moreover
compute the residues at all isolated singularities (removable singularities are excluded). You are allowed
to use the fact that all zeros of sin(z) in the complex plane lie at z = pin with n ∈ Z. Simplify your
results as much as possible (fractions and factors of pi remain as they are)!)
(a) f(z) =
sin2(z)
sinh2(z)
(make use of Landau symbols when computing the residue),
(b) f(z) =
4− z2
cos(pi/z)
(make use of the limit formula when computing the residues).
3. Mandatory Question (simple proof) 10 points.
Consider the real function
f(x) = ln(x2 + 1), with x ∈ R.
(a) Compute the Taylor series at any x0 ∈ R and show that its radius of convergence is R(x0) =√
1 + x20. When computing the radius of convergence make use of the theorems and corollaries of
Lecture 13 (say which you use and why you can apply those!) so that no −N criterion is required.
(Hint: you can use the fact that the sequence {| cos(jϕ)|}j∈N is divergent for any ϕ 6= npi/2 with
n ∈ Z.)
(b) Prove that the union of the open discs of convergence D(x0, R(x0)) of these Taylor series is equal
to the set
S = {z ∈ C|Re(z) 6= 0} ∪ {iy ∈ C| − 1 < y < 1}.
4. Optional Question (advanced computation) 10 points.
Compute the Laurent series at any point z0 ∈ C and any open annulus centred at z0 of the function
f(z) =
ez
z
.
What are the inner and outer radii of convergence Ri < Ro (you only need to give a simple explanation
via holomorphicity of the function why these are the radii)? (Hint: it is sometimes helpful to make use
of the contour integral formula to find the Laurent coefficients!)
5. Optional Question (advanced proof) 10 points.
Prove the following statement from Lecture 15: If f(z) is entire and there are R,K > 0 with |f(z)| ≥ K
whenever |z| ≥ R, then f(z) is a polynomial.
(Hint: Combine the theorems and corollaries in Lectures 14 and 15. Mention those explicitly!)
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EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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