Collection of Problems on the Theory of Functions of a Complex Variable

[704/39%]

M. I. Shabunin, E. S. Polovinkin, M. I. Karlov

Chapter 1Introduction[124/34%]

Section 1Complex Numbers[32/31%]

Section 2Sequences and Series of Complex Numbers. Complex-Valued Functions of a Real Variable. Curves and Domains in the Complex Plane[50/22%]

Section 3Limit and Continuity of a Function of a Complex Variable. Integration of a Function of a Complex Variable[27/59%]

Section 4Uniform Convergence. Power Series[15/33%]

Chapter 2Regular Functions[151/17%]

Section 5Differentiability of Functions. Harmonic Functions[19/37%]

Section 6Cauchy's Theorem. The Cauchy-Type Integral.[43/14%]

Section 7Taylor Series[17/53%]

Section 8Sequences and Series of Regular Functions. Integrals Depending on a Parameter.[18/6%]

Section 9The Uniqueness Theorem. Regular Continuation.[23/9%]

Section 10The Maximum Principle[31/0%]

Chapter 3Laurent Series. Singular Points. Residues[89/39%]

Section 11Laurent Series[15/80%]

Section 12Isolated Singular Points of Single-Valued Character[31/29%]

Section 13Computing Residues[18/33%]

Section 14Computing Integrals Over a Closed Contour[10/50%]

Section 15The Argument Principle. Rouché's Theorem[15/20%]

Chapter 4Multivalued Analytic Functions[193/55%]

Section 16The Increment of the Argument of a Function Along a Curve[8/88%]

Section 17Extracting Regular Branches[15/0%]

Section 18Computing Values of Regular Branches of Multivalued Functions. Laurent Series for Regular Branches.[50/82%]

Section 19Integrals of Regular Branches[50/94%]

Section 20Analytic Continuation. Complete Analytic Functions.[23/4%]

Section 21Singular Points of Complete Analytic Functions[47/21%]

Chapter 5Applications of Residue Theory[33/24%]

Section 22Expansion of Meromorphic Functions into Series of Simplest Fractions and into Infinite Products[8/0%]

Section 23Computing Improper Integrals[11/73%]

Section 24Integrals Reducible to the Gamma Function[14/0%]

Chapter 6Conformal Mappings[114/49%]

Section 25The Geometric Meaning of the Derivative[11/55%]

Section 26Definition and General Properties of Conformal Mappings[19/11%]

Section 27Fractional-Linear (Möbius) Mappings[17/59%]

Section 28Conformal Mappings by Elementary Functions[22/68%]

Section 29The Symmetry Principle[20/5%]

Section 30Mapping of Polygons[11/73%]

Section 31Application of Conformal Mappings to Solving Boundary Value Problems for Harmonic Functions[10/100%]

Section 32The Laplace Transform (Operational Calculus) and its Application to Solving Differential Equations[4/100%]