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Study Briefs

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All the information you need in one place! Each Study Brief is a summary of one specific subject; facts, figures, and explanations to help you learn faster.

Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Moduli Complex Conjugates Exponential Form Products and Quotients in Exponential Form Roots of Complex Numbers Examples Regions in the Complex Plane

Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle

Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions

Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Examples Upper Bounds for Moduli of Contour Integrals Antiderivatives Examples Cauchy-Goursat Theorem Proof of the Theorem Simply and Multiply Connected Domains Cauchy Integral Formula Derivatives of Analytic Functions Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle

Series Convergence of Sequences Convergence of Series Taylor Series Examples Laurent Series Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series

Residues and Poles Residues Cauchy's Residue Theorem Using a Single Residue The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of f Near Isolated Singular Points

Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouchï¿½'s Theorem Inverse Laplace Transforms Examples

Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions

Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions

Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder

The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate

Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz

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