## Vocabulary

A list of important terms and theorems for this course, listed by chapter and section of your textbook. Please make sure you understand the meaning of each of these terms. They form the building blocks for the conceptual framework of the subject.

**Chapter 1****Sections 1-11**- complex number
- complex plane
- real axis
- imaginary axis
- real part
- imaginary part
- additive inverse; multiplicative inverse
- vector interpretation of complex numbers
- modulus
- triangle inequality
- polynomial of degree n
- conjugate
- polar form
- argument
- principal value of arg z
- Euler's formula
- exponential form
- de Moivre's formula
- roots of complex numbers
- principal
*n*^{th}root

**Section 12**- neighborhood; open disk
- deleted neighborhood; punctured disk
- interior point
- exterior point
- boundary point
- boundary
- open set
- closed set
- closure
- connected
- domain
- region
- annulus; annular region
- bounded
- unbounded

**Chapter 2****Sections 13-14**- function
- domain of definition
- rational function
- multiple-valued functions
- mapping
- image
- inverse image
- range

**Sections 15-18**- limit of a complex function
- basic limit laws
- point at infinity; extended complex plane
- Riemann sphere
- neighborhood of infinity
- continuity of a complex function

**Sections 19-27**- complex differentiable, derivative
- complex differentiability implies continuity
- rules for differentiation
- Cauchy-Riemann equations
- necessary condition for differentiability
- sufficient condition for differentiability
- polar form of the Cauchy-Riemann equations
- analytic in an open set; analytic at a point
- entire function
- singular point of
*f* - Laplace's equation
- harmonic function
- harmonic conjugate

**Chapter 3****Sections 30-39**- complex exponential function
- periodic with period
*T* - complex logarithm
- principal value of
*log z* - branch cut; principal branch
- identities involving logarithms
- complex power function; principal value of a complex power
- complex trigonometric functions
- complex hyperbolic functions

**Chapter 4****Sections 41-42**- derivative of a complex-valued function of a real variable
- integral of a complex-valued function of a real variable
- the fundamental theorem of calculus

**Sections 43-47**- arc; simple arc
- simple closed curve
- positively oriented curve
- differentiable arc; smooth arc
- contour (piecewise smooth arc); simple closed contour
- Jordan curve theorem
- contour integral
*ML*-inequality

**Sections 48-53**- independent of path
- antiderivative; fundamental theorem for contour integrals
- Cauchy-Goursat theorem
- simply connected domain; multiply connected domain
- principle of deformation of paths

**Sections 54-59**- Cauchy integral formula
- Cauchy integral formula for derivatives
- Morera's theorem
- Cauchy's inequality
- Liouville's theorem
- the fundamental theorem of algebra
- maximum modulus principle

**Chapter 5****Sections 60-68**- sequence; limit of a sequence; convergent; divergent
- series; sum; partial sums; convergent; divergent
- absolutely convergent series
- remainder
- power series
- Taylor's theorem; Taylor series; Maclaurin series
- Maclaurin series expansions for
*1/(1-z)*,*exp(z)*,*sin z*,*cos z*,*sinh z*,*cosh z* - Laurent's theorem; Laurent series

**Sections 69-73**- unifrom convergence
- circle of convergence
- absolutely convergent series
- term by term differentiation
- term by term integration
- Uniqueness of Taylor and Laurent series representations
- Multiplication and division of power series

**Chapter 6****Sections 74-78**- isolated singular point
- residue
- Cauchy's residue theorem
- principal part
- removable singularity
- essential singularity
- pole; simple pole; pole of order
*m*