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9.2. Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.2.1. Elementary Notions in Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.2.2. Curvature of Curves on Surface .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 392
9.2.3. Intrinsic Geometry ofSurface .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 395
References for Chapter 9 . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 397
10. Functions of Complex Variable .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 399
10.1. Basic Notions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 399
10.1.1. Complex Numbers. Functions of Complex Variable . . ... .. .. .. .. .. .. .. .. 399
10.1.2. Functions of Complex Variable . . . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 401
10.2. MainApplications .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 419
10.2.1. Conformal Mappings .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 419
10.2.2. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
References for Chapter 10 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 433
11. Integral Transforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 435
11.1. General Form of Integral Transforms. Some Formulas . .. .. .. ... .. .. .. .. .. .. .. .. 435
11.1.1. Integral Transforms and Inversion Formulas . . . . . . . . ... .. .. .. .. .. .. .. .. 435
11.1.2. Residues. Jordan Lemma .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 435
11.2. Laplace Transform . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 436
11.2.1. Laplace Transform and the Inverse Laplace Transform . .. .. .. .. .. .. .. .. . 436
11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some
Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 437
11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series
andAsymptotic Expansions .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 440
11.3. Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.3.1. Mellin Transform and the Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin,
Laplace, and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
11.4. Various Forms of the Fourier Transform .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 443
11.4.1. Fourier Transform and the Inverse Fourier Transform ... .. .. .. .. .. .. .. .. 443
11.4.2. Fourier Cosine and Sine Transforms .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 445
11.5. Other Integral Transforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 446
11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified
Bessel Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 446
11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral
Transforms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 448
References for Chapter 11 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 451
12. Ordinary Differential Equations .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 453
12.1. First-Order Differential Equations .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 453
12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 453
12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration . . . . 456
12.1.3. Exact Differential Equations. Integrating Factor . . . . . ... .. .. .. .. .. .. .. .. 458
12.1.4. Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
12.1.5. Abel Equations of the First Kind . . . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 462
12.1.6. Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
12.1.7. Equations Not Solved for the Derivative . .. .. .. .. .. ... .. .. .. .. .. .. .. .. 465
12.1.8. Contact Transformations .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 468
12.1.9. Approximate Analytic Methods for Solution of Equations . . . . . . . . . . . . . . . . 469
12.1.10. Numerical Integration of Differential Equations .. .. ... .. .. .. .. .. .. .. .. 471