Contents xi
6.5 The Cauchy-Riemann Equations 317
6.6 Laplace Equation. Harmonic and Subharmonic Functions 320
6.7 The Concept of the Differential of a Complex Function.
Complex Differential Operators 326
6.8 Properties of the Complex Differential Operators 8: and 82 328
Exercises 6.1 331
6.9 The Complex Directional Derivative 3386.10 Expression of the Directional Derivative in Terms off z, f-z,
and (^0 )
6.11 The Kasner Circle 349
Exercises 6.2 350
6.12 The Finite Increments Formula 355
6.13 Existence of a Local Single-Valued Inverse Function to
w = f(z) 356
6.14 The Directional Derivative and the Kasner Circle for the
Inverse Function to w = f ( z) 362
6.15 Conformal mappings 363
6.16 Nonconformal Mappings. The Magnification Ratio p as a
Function of 0 366
6.17 Nonconformal Mappings. The Angular Distortion 7/J as a
Function of (^0 )
6.18 Nonconformal Mappings. Tissot's Theorem for the Plane 377
6.19 Some Transformations Induced by the Directional Derivative 379
Exercises 6.3 380
6.20 The Maximum and Minimum Modulus Theorems for
Functions of Class 'D( A) 381
Exercises 6.4 384
6.21 The Fundamental Topological Properties of Analytic
Functions^384
6.22 Continuity of the First Derivative of an Analytic Function^385
6.23 On the Analyticity Conditions 388
6.24 Generalized Analytic Functions 390
Bibliography 401
7 Integration^409
7.1 Introduction 409
7.2 Integral of a Complex Function of a Real Variable 409
7.3 Integral of a Complex Function of a Complex Variable
Along a Continuously Differentiable Arc^413
7.4 Integral of a Complex Function of a Complex Variable
Along a Piecewise Differentiable Arc^415