| Week |
Sections |
Topics |
Student Learning Outcomes
|
| 1
|
2.1 & 2.2
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Introduction to complex numbers, their operations and geometry.
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- Complex numbers and the complex plane.
- Elementary operations on complex numbers (addition, subtraction, multiplication, division, conjugation, modulus, argument).
- Complex numbers in Cartesian and polar forms.
- Complex operations: Elementary algebraic identities and inequalities.
- Geometric meaning of complex arithmetic operations.
- DeMoivre's Formula.
|
|
2
|
3.1, 3.2, 3.3
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Topology of the complex plane. Continuous complex functions.
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- Essential analysis concepts: sequences, series, limits, convergence, completeness.
- Basic topology of the complex plane: open, closed and punctured discs, open and closed sets, neighborhoods.
- Continuous functions and operations on them.
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|
3
|
4.1
|
Complex differentiation
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- Definition of complex derivative at a point.
- Cauchy-Riemann equations.
- Examples of differentiable and non-differentiable complex functions.
- Holomorphic functions.
|
|
4
|
4.2
|
Power (Taylor) series of holomorphic functions.
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- Taylor coefficients and Taylor series of a holomorphic function.
- Radius of convergence.
- Differentiation of Taylor series.
- Taylor series of rational functions.
- The complex exponential, trigonometric and hyperbolic functions and their Taylor series.
|
|
5
|
4.3, 4.5 & 4.5
|
Complex natural logarithms. Multivalued holomorphic functions. Singularities.
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- Definition of the multivalued complex natural logarithm, its principal branch, and other branches.
- Derivatives of inverse functions. Derivative of the complex natural logarithm.
- Complex powers via logarithms.
- Definition of branch point and branches.
- Functions holomorphic in punctured neighborhoods. Poles and other singularities.
- Examples of branching and singularities (complex logarithms, inverse trigonometric/hyperbolic functions, and complex powers).
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|
6
|
None
|
Review. First midterm exam.
|
|
|
7
|
4.5,
|
- 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.
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|
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.
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|
9
|
4.1
|
- Mathematical Induction.
- Inductive proofs.
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- 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!
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|
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.
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|
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.
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|
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.
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|
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.
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