functions of one complex variable (math30040)mathsci.ucd.ie/~rsmith/pdfs/linalg/intro.pdf · the...

Functions of One Complex Variable

(MATH30040)

Dr Richard Smith

maths.ucd.ie/~rsmith

Dr Richard Smith (maths.ucd.ie/~rsmith) Functions of One Complex Variable (MATH30040) Semester 1 2013 – 2014 1 / 8

Introduction Module Details

Module DetailsLecturesMondays 12.00 – 12.50, A105 Arts, all weeksWednesdays 11.00 – 11.50, EP 128 Science Centre North, all weeks.

Tutorials and class testsThursdays 14.00 – 14.50, B101 Arts, most weeks, starting 19th September.

Lecturer – Dr Richard SmithEmail: richard.smith@maths.ucd.ieWeb: maths.ucd.ie/~rsmith

Lecture notes, slides, homework etc.Blackboard: elearning.ucd.ie

School of Mathematical Sciences LocationMy current location is Room 3.52 A, Radio House, Belfield Office Park.

School of Mathematical Sciences LocationThe Maths Sciences school office is Room 533, Library Building.

Introduction Module outline and schedule

Module outline and scheduleThis is a first course on the theory of functions of one complex variable. Aswell as being a rich and beautiful theory in its own right, it has innumerableapplications in physics, engineering and other areas of mathematics.

Topics coveredComplex numbers, functions defined on the complex plane, complex dif-ferentiability and the Cauchy-Riemann equations.Integration along paths in the complex plane, Cauchy’s Theorem andCauchy’s Integral Formula.Power series, Taylor’s Theorem, Liouville’s Theorem, the FundamentalTheorem of Algebra and the Identity Theorem.Laurent’s Theorem, singularities, Cauchy’s Residue Theorem.Applications of the residue calculus to contour integration.

Topics coveredComplex numbers, functions defined on the complex plane, complex dif-ferentiability and the Cauchy-Riemann equations.

Integration along paths in the complex plane, Cauchy’s Theorem andCauchy’s Integral Formula.Power series, Taylor’s Theorem, Liouville’s Theorem, the FundamentalTheorem of Algebra and the Identity Theorem.Laurent’s Theorem, singularities, Cauchy’s Residue Theorem.Applications of the residue calculus to contour integration.

Topics coveredComplex numbers, functions defined on the complex plane, complex dif-ferentiability and the Cauchy-Riemann equations.Integration along paths in the complex plane, Cauchy’s Theorem andCauchy’s Integral Formula.

Power series, Taylor’s Theorem, Liouville’s Theorem, the FundamentalTheorem of Algebra and the Identity Theorem.Laurent’s Theorem, singularities, Cauchy’s Residue Theorem.Applications of the residue calculus to contour integration.

Topics coveredComplex numbers, functions defined on the complex plane, complex dif-ferentiability and the Cauchy-Riemann equations.Integration along paths in the complex plane, Cauchy’s Theorem andCauchy’s Integral Formula.Power series, Taylor’s Theorem, Liouville’s Theorem, the FundamentalTheorem of Algebra and the Identity Theorem.

Laurent’s Theorem, singularities, Cauchy’s Residue Theorem.Applications of the residue calculus to contour integration.

Topics coveredComplex numbers, functions defined on the complex plane, complex dif-ferentiability and the Cauchy-Riemann equations.Integration along paths in the complex plane, Cauchy’s Theorem andCauchy’s Integral Formula.Power series, Taylor’s Theorem, Liouville’s Theorem, the FundamentalTheorem of Algebra and the Identity Theorem.Laurent’s Theorem, singularities, Cauchy’s Residue Theorem.

Applications of the residue calculus to contour integration.

Topics coveredComplex numbers, functions defined on the complex plane, complex dif-ferentiability and the Cauchy-Riemann equations.Integration along paths in the complex plane, Cauchy’s Theorem andCauchy’s Integral Formula.Power series, Taylor’s Theorem, Liouville’s Theorem, the FundamentalTheorem of Algebra and the Identity Theorem.Laurent’s Theorem, singularities, Cauchy’s Residue Theorem.Applications of the residue calculus to contour integration.

Introduction Prior Learning

Prior learningTopics used in this module

Complex numbersFundamental arithmetic, polar form. De Moivre’s Theorem, Euler’s formulaand roots of unity.

Differentiation of functions of 1 variableDefinition of derivative in terms of limit. Derivatives of polynomials, trigono-metric, hyperbolic, exponential and logarithmic functions. Differentiationrules e.g. product, quotient and chain rule. Mean Value Theorem.Partial derivatives, line/path integrationPartial derivatives of functions of 2 variables, Green’s Theorem in theplane (see Dr O’Naraigh’s notes ACM_20150_sept_2013.pdf).Continuity of functions of 1 variableDefinition of continuity in terms of limit. A continuous function on a closedbounded interval is bounded (and attains its bounds).

Complex numbersFundamental arithmetic, polar form. De Moivre’s Theorem, Euler’s formulaand roots of unity.Differentiation of functions of 1 variableDefinition of derivative in terms of limit. Derivatives of polynomials, trigono-metric, hyperbolic, exponential and logarithmic functions. Differentiationrules e.g. product, quotient and chain rule. Mean Value Theorem.

Partial derivatives, line/path integrationPartial derivatives of functions of 2 variables, Green’s Theorem in theplane (see Dr O’Naraigh’s notes ACM_20150_sept_2013.pdf).Continuity of functions of 1 variableDefinition of continuity in terms of limit. A continuous function on a closedbounded interval is bounded (and attains its bounds).

Complex numbersFundamental arithmetic, polar form. De Moivre’s Theorem, Euler’s formulaand roots of unity.Differentiation of functions of 1 variableDefinition of derivative in terms of limit. Derivatives of polynomials, trigono-metric, hyperbolic, exponential and logarithmic functions. Differentiationrules e.g. product, quotient and chain rule. Mean Value Theorem.Partial derivatives, line/path integrationPartial derivatives of functions of 2 variables, Green’s Theorem in theplane (see Dr O’Naraigh’s notes ACM_20150_sept_2013.pdf).

Continuity of functions of 1 variableDefinition of continuity in terms of limit. A continuous function on a closedbounded interval is bounded (and attains its bounds).

Complex numbersFundamental arithmetic, polar form. De Moivre’s Theorem, Euler’s formulaand roots of unity.Differentiation of functions of 1 variableDefinition of derivative in terms of limit. Derivatives of polynomials, trigono-metric, hyperbolic, exponential and logarithmic functions. Differentiationrules e.g. product, quotient and chain rule. Mean Value Theorem.Partial derivatives, line/path integrationPartial derivatives of functions of 2 variables, Green’s Theorem in theplane (see Dr O’Naraigh’s notes ACM_20150_sept_2013.pdf).Continuity of functions of 1 variableDefinition of continuity in terms of limit. A continuous function on a closedbounded interval is bounded (and attains its bounds).

Introduction Assessment and grading

Assessment and gradingTutorials and class tests (30%)

Eight 50-minute tutorials, starting 19th September.

Three 50-minute tests, worth 10% each, will take place at 2pm, Thursdayweeks 5, 8 and 11 (10th, 31st Oct, 21st Nov), in C006 Health Sciences.

Final Exam (70%)The final 2-hour written exam will take place at 9am, Tuesday 17th De-cember, in the RDS Shelbourne Hall, Anglesea Road.

MATH modules have their own mark-to-grade conversion table.

A+ 90 – 100%A 80 – 89.99%A- 70 – 79.99%

See maths.ucd.ie/tl/grading/en02 for the full table.

Eight 50-minute tutorials, starting 19th September.Three 50-minute tests, worth 10% each, will take place at 2pm, Thursdayweeks 5, 8 and 11 (10th, 31st Oct, 21st Nov), in C006 Health Sciences.

A+ 90 – 100%A 80 – 89.99%A- 70 – 79.99%

Introduction Follow-on modules

Follow-on modulesModules that make use of the material in this module include:

Partial Differential Equations (ACM30220) (Semester 1)

Electromagnetism and Gauge Theory (ACM40010) (Semester 1)

Introduction Follow-on modules

Follow-on modulesModules that make use of the material in this module include:

Partial Differential Equations (ACM30220) (Semester 1)Electromagnetism and Gauge Theory (ACM40010) (Semester 1)

Introduction Final remarks

Final remarksIf you have any queries about the material, ask during the lectures or after-wards, or send me an email.

If you want to meet me to discuss aspects of the module (homework problemsetc.), then send me an email and we can arrange a time and location.

If you miss a class test due to illness, then I can compensate you, provided yousupply a doctor’s sick note.

functions of one complex variable (math30040)mathsci.ucd.ie/~rsmith/pdfs/linalg/intro.pdf · the...

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