MAT290 — Advanced Engineering Mathematics

Course discusses differential equations, complex analysis, and the Laplace transform.¹ Follows MAT187 — Calculus II. We are using the Advanced Engineering Mathematics textbook (7th edition) by Dennis Zill.

The textbook answers (odd numbers only) can be found here.

Concepts covered

Complex analysis

Foundations

• Complex number
• De Moivre’s theorem
• Exponential function, logarithmic function, trigonometric function, and hyperbolic function
• Point-set topology
  • Singularity
  • Zeroes and poles
• Differentiation from first principles
• Differentiability and continuity
• Epsilon-delta definition (only briefly, not testable)
• Cauchy-Riemann equations
• Analytic function
• Harmonic function
• Laplacian

Integration

• Contour integral
  • Vector field
  • Circulation
  • Flux
• Cauchy-Goursat theorem
  • Stokes’ theorem
  • Divergence theorem
• Fundamental theorem of contour integrals
  • Independence of path
• ML inequality
• Cauchy’s integral formula
• Fundamental theorem of algebra
• Cauchy inequality
• Liouville’s theorem
• Principal value integral
• Jordan’s lemma
• Indentation of contours
• Trigonometric integral

Sequences and series

• Sequence and series and power series
  • Series convergence
    • Ratio test
  • Taylor series
  • Laurent series
• Residue
  • Cauchy’s residue theorem
• Great Picard theorem (not testable)

Differential equations

Differential equation

• Homogeneous ODE
• Non-homogeneous ODE
  • Method of undetermined coefficients
  • Variation of parameters
    • Cramer’s rule
• First-order ODE
  • Integrating factor (linear first-order ODEs)
  • Direction field
  • Phase portrait
  • Equilibrium solution
• Second-order ODE
  • Reduction of order (homogeneous, linear)
  • Series RLC circuit
• Higher-order ODE
• Initial-value problem
• Solutions to differential equations (interval of definition)
• Solutions to differential equations
  • Linear independence
  • Wronskian determinant

Laplace transform

• Laplace transform
• Inverse Laplace transform
• Partial fraction decomposition (cover-up method)
• Translation theorem (first and second)
• Convolution
• Dirac delta function

Final exam preparation

Complex analysis looks pretty funky. Genuinely having a stroke doing each problem — running into minor computational mistakes so frequently.

Need to work on:

• Liouville’s theorem
• Laurent series

Things to memorise:

• Cauchy’s integral formula
• Liouville’s theorem + proof
• Cauchy inequality
• Residues (order $m$) - simple pole should be fine

Things to remember:

• Existence and uniqueness (continuity of function and partials)
• What the interval of definition is
• Integration by parts
• That if the Wronskian is 0, the set of solutions is linearly dependent
• Don’t forget constant when using integrating factor
• That underdamped = imaginary roots, overdamped = distinct real
• Harmonic functions
• Circulation: Real of conjugate; flux: imaginary of conjugate
• Jordan’s lemma — what is and isn’t a valid semi-circle
• Complex integration of real integrals
• Check for orientation of contours

Footnotes

1. ”[MAT290] will provide you with the mathematical foundation for advanced control systems, robotics, and intelligent systems.” - Prof Stickel ↩