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Math 120A Summer Session II 2021
Terminology and Theorems
Updated 7/27/21
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 might try making flashcards and reviewing 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 ez = 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
- 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
------------------------ Exam 3 ------------------------
- contour integration with branch points
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