MAT3223

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Course Catalog

MAT 3223. Complex Variables. (3-0) 3 Credit Hours.

Prerequisites: MAT 2214 and MAT 3213. An introduction to complex variables, including elementary functions, line integrals, power series, residues and poles, and conformal mappings. Generally offered: Spring. Differential Tuition: $150.

Week # Sections Topics Prerequisite Skills Student Learning Outcomes
1

1.1 & 1.2

Propositional Logic

  • Recognize propositional formulas built from atoms using connectives.
  • Correctly interpret propositional formulas using truth tables.

2

1.3 & 1.4

  • Tautologies and Deductions.
  • Quantifiers.
  • Propositional Logic.
  • Establish whether a propositional formula is a tautology.
  • State De Morgan's Laws of logic.
  • Recognize conditional tautologies as laws of deduction.
  • Express conditionals in disjunctive form.
  • Express the negation of a conditional in conjunctive form.
  • Identify the direct and contrapositive forms of a conditional.
  • Recognize the non-equivalence of a conditional and its converse.
  • Recognize a biconditional as the conjunction of a conditional and its converse.
  • Identify the domain of interpretation of a quantified statement.
  • Correctly interpret quantified statements.
  • Correctly negate quantified statements.

3

1.5 & 1.6

  • Sets.
  • Set Operations.
  • Introduction to proofs of universal statements in set theory
  • Disproving universal statements via counterexamples.
  • Tautologies and Deductions.
  • Quantifiers.
  • Recognize and interpret set equality and set inclusion.
  • Recognize set operations and state their formal definitions.
  • Recognize formal proofs as processes of logical deduction of conclusions from assumptions.
  • Prove basic universal statements pertaining to set inclusion and set operations.
  • Correctly identify false universal statements in set theory and disprove them with appropriate counterexamples.
  • Correctly use propositional and quantified tautologies as deductive laws.

4

2.1

  • Divisibility of integers.
  • The Division Algorithm.
  • Proofs and Counterexamples.
  • Propositional Logic.
  • Quantifiers.
  • Recognize the notion of integer divisibility via its formal definition, examples and counterexamples.
  • Correctly state and apply the Division Algorithm of integers.
  • Prove basic facts pertaining to divisibility and the division algorithm.

5

2.2 & 2.3

  • Greatest Common Divisor.
  • Bèzout's Identity: GCD(a,b) = au + bv for some u,v∊ℤ.
  • Coprime integers.
  • The Extended Euclidean Algorithm.
  • Divisibility of integers.
  • The Division Algorithm.
  • Compute the GCD of two integers using the Euclidean algorithm.
  • Express the GCD of two integers as a linear combination thereof using the extended Euclidean algorithm.

6

2.5

  • Primes.
  • Euclid's proof of the infinitude of primes by contradiction.
  • Euclid's Lemma: for p prime, p|ab implies p∣a or p∣b.
  • Unique factorization and the Fundamental Theorem of Arithmetic.
  • Divisibility of integers.
  • The Extended Euclidean Algorithm.
  • Greatest Common Divisor.
  • Coprime integers.
  • Define prime numbers and state their basic properties.
  • Prove the infinitude of primes.
  • Prove Euclid's Lemma using Bèzout's identity.
  • Prove uniqueness of prime factorization using Euclid's Lemma.
  • Characterize divisibility and GCD of integers in terms of their prime factorizations.

7

3.1–3.3

  • Arithmetic congruences and basic modular arithmetic.
  • Tests of divisibility.
  • Divisibility of integers.
  • The Division Algorithm.
  • Use arithmetic congruences to interpret the remainder of integer division.
  • Use congruences to compute remainders of divisions where the quotient is large or irrelevant.
  • Prove basic divisibility criteria by 2, 3, 5, 9 and 11 for number in base 10, using modular arithmetic.

8

3.4

  • Modular rings ℤₙ.
  • Modular fields ℤₚ.
  • Fermat's Little Theorem.
  • Primes.
  • Arithmetic congruences and basic modular arithmetic.
  • Recognize the modular rings ℤₙ as number systems.
  • Evaluate sums, differences, negations and products in ℤₙ.
  • Identify invertible and non-invertible elements of ℤₙ.
  • Find the inverse (when defined) of a given element of ℤₙ.
  • Prove that the modular ring ℤₚ is a field if and only if p is prime.
  • Correctly state Fermat's Little Theorem, both as a theorem in modular arithmetic modulo a prime p, and as a theorem for the finite field ℤₚ.
  • Apply Fermat's Little Theorem to solve arithmetic problems.

9

4.1

  • Mathematical Induction.
  • Inductive proofs.
  • Basic proofs.
  • Tautologies and Deductions.
  • Quantifiers.
  • Divisibility of integers.
  • State the Principle of Mathematical Induction (PMI).
  • Prove elementary algebraic and arithmetic statements by induction.
  • Prove elementary algebraic and arithmetic statements by strong induction.

10

4.2 & 4.3

  • Recursion.
  • The Binomial Theorem (Binomial Expansion Formula).
  • Mathematical Induction.
  • Inductive proofs.
  • Factorials.
  • Recognize recursive definitions of sequences and sets.
  • Prove elementary properties of recursively defined sets and sequences (Fibonacci and geometric sequences).
  • Recursively construct successive rows of Pascal's triangle.
  • Identify the entries in Pascal's Triangle as Binomial Coefficients.
  • State and apply the Binomial Expansion Formula.
  • Compute individual binomial coefficients using the quotient-of-falling powers formula (n𝑪k) = n(n−1)…(n−k+1)/k!

11

5.1 & 5.2

  • The rational number system 𝐐.
  • The real number system 𝐑.
  • Fractional powers and roots of real numbers.
  • Rational and irrational numbers. Existence of irrationals.
  • Divisibility of integers.
  • Unique factorization and the Fundamental Theorem of Arithmetic.
  • Decimals and decimal expansions.
  • Roots and fractional powers of real numbers.
  • Identify the set 𝐐 of rational numbers as a number system (a field).
  • Identify the set 𝐑 of real numbers as a number system (a field extending 𝐐).
  • Prove the irrationality of √2 and, more generally, of √p for p prime.
  • Prove that fractional powers xm/n of real x>0 are well defined and unique.
  • Informally interpret the convergence of decimal expansions as the completeness of 𝐑.
  • Informally recognize that the universal existence of roots ⁿ√x and fractional powers xm/n of real numbers x>0 relies on the completeness of 𝐑.

12

8.1–8.4

  • Introduction to complex numbers and their operations.
  • The complex number system 𝐂.
  • The complex plane.
  • The real number system 𝐑.
  • Fractional powers and roots of real numbers.
  • Represent complex numbers algebraically in Cartesian form.
  • Represent complex numbers geometrically as points on a plane.
  • Carry out arithmetic operations with complex numbers.
  • Interpret the geometric meaning of addition, subtraction and complex conjugation.
  • Identify the set 𝐂 of complex numbers as a field extending the real number system 𝐑.

13

8.5–8.7

  • Polar form of complex numbers.
  • Geometric meaning of complex multiplication and division.
  • Powers and roots of complex numbers. De Moivre’s Theorem.
  • The complex number system 𝐂.
  • The complex plane.
  • Roots and fractional powers of real numbers.
  • Represent complex numbers in polar form.
  • Algebraically relate the Cartesian and polar forms of a complex number.
  • Use the identities cis(𝜃+ɸ) = cis𝜃∙cisɸ and (cis𝜃)n = cis(n𝜃) (De Moivre's formula) for the complex trigonometric function cis𝜃 = cos𝜃 + i∙sin𝜃 to evaluate products and powers both algebraically and geometrically.
  • Evaluate all n-th roots of a given complex number both in trigonometric and (when possible) in algebraic closed form, and represent them geometrically.

14

8.8–9.2

  • Roots and factors of polynomials. The Remainder Theorem.
  • Real and complex roots.
  • The Fundamental Theorem of Algebra.
  • The complex number system 𝐂.
  • Powers and roots of complex numbers. De Moivre’s Theorem.
  • Polynomials: arithmetic operations, long division, and factorizations.
  • State and prove the Remainder Theorem.
  • Identify roots with linear factors of a polynomial.
  • Factor given simple polynomials into irreducible factors over ℚ, ℝ and ℂ.
  • State the Fundamental Theorem of Algebra.
  • Use the Fundamental Theorem of Algebra to prove that irreducible real polynomials are linear or quadratic.