Complex variables for engineers
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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$1,162 · Complex variables for engineers · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle or Venmo.
Topics typically covered
Standard syllabus
Complex arithmetic and functions
- Complex numbers, polar form, and Euler's formula
- Complex functions and mappings in the plane
- Limits, continuity, and analyticity (engineering emphasis)
- Cauchy–Riemann equations and harmonic conjugates
- Elementary functions: exp, log, sin, cos in the complex plane
Integration and series
- Contour integrals and Cauchy's integral theorem
- Cauchy's integral formula and derivatives
- Taylor and Laurent series; residues at poles
- Residue theorem for evaluating real integrals
- Inverse Laplace transforms via contour integration (introduction)
Conformal mapping and applications
- Conformal maps and angle preservation
- Möbius transformations and standard mappings
- Potential theory and 2D steady-state problems
- Joukowski transformation and airfoil profiles (introduction)
- Branch cuts in transform methods
STEM / applied
Signal and systems applications
- Frequency response and analytic signal (Hilbert transform overview)
- Stability analysis using complex plane geometry
- Filter design and pole-zero plots
- Laplace and z-transform connections to complex analysis
- MATLAB/Python visualization of complex mappings
Engineering case studies
- Heat conduction and electrostatics via harmonic functions
- Fluid flow around cylinders (potential flow overview)
- Control theory: root locus and Nyquist criteria (mathematical background)
- AC circuit analysis with phasors and complex impedance
- Numerical conformal mapping tools (introduction)
Notes
Topics reflect common complex variables courses for engineering majors at US universities. Proof depth is typically less than in a mathematics-major complex analysis course.