Math 120A Fall 2025
Terminology and Theorems

Updated 7/30/25

Note: The following list is a minimal collection of the important terms and theorems for this course. You simply must be familiar with them and how they are used: in the case of theorems, you should be familiar with how they are proved.

This basic knowledge is analogous to the vocabulary of a language: it is impossible to speak the language without it.

You are encouraged to make flashcards and review them frequently.

• Chapter I
  • complex number z = x + i y ↔ (x,y)
  • real part, imaginary part
  • complex plane
  • absolute value, modulus
  • complex conjugate
  • fundamental theorem of algebra
  • polar representation of z
  • arg(z), the argument of z
  • Arg(z), the principal value of arg(z)
  • stereographic projection
  • extended complex plane
  • slit, branch cut
  • branch
  • the exponential function e^z = exp(z)
  • log(z), the logarithm of z
  • Log(z), the principal value of log(z)
  • phase factor
  • phase change lemma
• Chapter II
  • converges, convergent
  • Cauchy sequence
  • limit
  • continuous, continuous function
  • open, open set
  • domain
  • closed
  • bounded
  • compact
  • (complex) differentiable
  • (real) differentiable
  • complex derivative
  • (real) derivative, Jacobian matrix
  • analytic (on an open set U)
  • Cauchy-Riemann equations
  • Laplace's equation, Laplacian operator
  • harmonic function
  • harmonic conjugate
  • tangent vector to a curve
  • angle between two curves
  • conformal, conformal mapping
  • fractional linear transformation
  • affine transformation
  • translation
  • dilation
  • inversion
• Chapter III
  • path
  • simple path
  • closed path
  • smooth path
  • piecewise smooth path
  • curve
  • line integral
  • piecewise smooth boundary
  • Green's theorem
  ------------------------ Exam 1 ------------------------
  • Fundamental Theorem of Calculus, Part I (for line integrals)
  • Fundamental Theorem of Calculus, Part II (for line integrals)
  • differential
  • exact differential
  • closed differential
• Chapter IV
  • |dz| = ds, arclength differential (infinitesmal arclength)
  • ML-estimate
  • (complex) primitive
  • Cauchy's theorem
  • Cauchy integral formula
  • Liouville's theorem
• Chapter V
  • series convergence
  • comparison test
  • geometric series
  • converge absolutely
  • converges pointwise
  • converges uniformly
  • Weierstrass M-test
  • converges normally
  • power series
  • radius of convergence
  • ratio test
  • root test
  • analytic at z = ∞
  • simple zero
  • double zero
  • zero of order N
  • zero at z = ∞ of order N
  • isolated point
  • uniqueness principle
  • principle of permanence of functional equations
• Chapter VI
  • Laurent decomposition
  • Laurent series expansion
  ------------------------ Exam 2 ------------------------
  • isolated singularity
  • removable singularity
  • Riemann's theorem on isolated singularities
  • pole
  • order of a pole
  • principal part of f(z) at a pole
  • simple pole
  • double pole
  • pole of order N
  • meromorphic
  • essential singularity
  • Casorati-Weierstrass theorem
  • isolated singularity of f(z) at z = ∞
  • removable singularity of f(z) at ∞
  • essential singularity of f(z) at ∞
  • principle part of f(z) at ∞
  • meromorphic in extended complex plane C^*
  • partial fractions decomposition
• ChapterVII
  • residue
  • residue theorem
  • contour integration
  • contour integration with branch points