Complex variables and applications / Ruel V. Churchill.
Editor: New York : McGraw-Hill, 1960Edición: 2nd edDescripción: ix, 297 p. : il. ; 24 cmTítulos uniformes: Introduction to complex variables and applications Otra clasificación: 30-01CONTENTS Preface. v 1. Complex Numbers [1] Definition. Further Properties. Geometric Representation. Complex Conjugates. Absolute Values. The Polar Form. Products, Powers, and Quotients. Extraction of Roots. Regions in the Complex Plane. 2. Analytic Functions [19] Functions of a Complex Variable. Mapping. Limits. Theorems on Limits. Continuity. The Derivative. Differentiation Formulas. The Cauchy-Riemann Conditions. Sufficient Conditions. Analytic Functions. Harmonic Functions. 3. Elementary Functions [46] The Exponential Function. Other Properties of exp z. The Trigonometric Functions. Further Properties of Trigonometric Functions. Hyperbolic Functions. The Logarithmic Function. Branches. Properties of Logarithms. Complex Exponents. Inverse Trigonometric Functions. 4. Mapping by Elementary Functions [65] Linear Functions. The Functions zn. The Function 1/z. The Point at Infinity. The Linear Fractional Transformation. Special Linear Fractional Transformations. The Function z1/2. Other Irrational Functions. The Transformation w = exp z. The Transformation w = sin z. Successive Transformations. Table of Transformations of Regions. 5. Integrals [95] Definite Integrals. Contours. Line Integrals. Examples. The Cauchy-Goursat Theorem. A Preliminary Theorem. Proof of the Cauchy-Goursat Theorem. Simply and Multiply Connected Domains. Indefinite Integrals. The Cauchy Integral Formula. Derivatives of Analytic Functions. Morera’s Theorem. Maximum Moduli of Functions. The Fundamental Theorem of Algebra. 0. Power Series [129] Taylor’s Serios. Observations and Examples, Laurent’s Series. Properties of Series. Uniform Convergence. Integration and Differentiation of Power Serios. Uniqueness of Representations by Power Series. Multiplication and Division. Examples. Zeros of Analytic Functions. 7. Residues and Poles [153] Residues. The Residue Theorem. Poles. Quotients of Analytic Functions. Evaluation of Improper Real Integrals. Another Example. Improper Integrals Involving Trigonometric Functions. Definite Integrals of Trigonometric Functions. Integration around a Branch Point. 8. Conformal Mapping [174] Rotation of Tangents. Conformal Mapping. Examples. Conjugate Harmonic Functions. Inverse Functions. Transformation of Harmonic Functions. Transformation of Boundary Conditions. 9. Applications of Conformal Mapping [189] Steady Temperatures. Steady Temperatures in a Wall. Temperatures in a Quadrant with Part of One Boundary Insulated. Electric Potential. Potential in a Cylindrical Space. Two-dimensional Fluid Flow. The Stream Function. Flow around a Corner. Flow around a Cylinder. 10. The Schwarz-Christoffel Transformation [218] Mapping the Real Axis onto a Polygon. The Schwarz-Christoffel Transformation. Triangles and Rectangles. Degenerate Polygons. The Infinite Strip. Fluid Flow in a Channel through a Slit. Flow in a Channel with an Offset. Electrostatic Potential about an Edge of a Conducting Plate. 11. Integral Formulas of Poisson Type [242] Poisson’s Integral Formula. A Dirichlet Problem for the Circle. Related Boundary Value Problems. Integral Formulas for the Half Plane. A Dirichlet Problem for the Half Plane. Neumann Problems for Circular Regions. A Neumann Problem for the Half Plane Further Theory of Functions [259] Analytic Continuation [259] Conditions under Which f(z) ≡ [0.] Permanence of Forms of Functional Identities. Uniqueness of Analytic Continuation. Examples. The Principle of Reflection. B. Singular Points and Zeros [268] Poles and Zeros. Essential Singular Points. The Number of Zeros and Poles. C. Riemann Surfaces [273] A Surface for the Function log z. A Surface for the Function z1/2. Surfaces for Other Irrational Functions. Appendixes [281] 1. Bibliography [281] 2. Table of Transformations of Regions [284] Index [293]
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1a. ed. publicada en 1948 bajo el título: Introduction to complex variables and applications.
Bibliografía: p. 281-283.
CONTENTS --
Preface. v --
1. Complex Numbers [1] --
Definition. Further Properties. --
Geometric Representation. --
Complex Conjugates. --
Absolute Values. --
The Polar Form. --
Products, Powers, and Quotients. --
Extraction of Roots. --
Regions in the Complex Plane. --
2. Analytic Functions [19] --
Functions of a Complex Variable. --
Mapping. --
Limits. --
Theorems on Limits. Continuity. --
The Derivative. --
Differentiation Formulas. --
The Cauchy-Riemann Conditions. --
Sufficient Conditions. --
Analytic Functions. --
Harmonic Functions. --
3. Elementary Functions [46] --
The Exponential Function. --
Other Properties of exp z. --
The Trigonometric Functions. --
Further Properties of Trigonometric Functions. --
Hyperbolic Functions. --
The Logarithmic Function. --
Branches. --
Properties of Logarithms. --
Complex Exponents. --
Inverse Trigonometric Functions. --
4. Mapping by Elementary Functions [65] --
Linear Functions. --
The Functions zn. --
The Function 1/z. --
The Point at Infinity. --
The Linear Fractional Transformation. --
Special Linear Fractional Transformations. --
The Function z1/2. --
Other Irrational Functions. --
The Transformation w = exp z. --
The Transformation w = sin z. --
Successive Transformations. --
Table of Transformations of Regions. --
5. Integrals [95] --
Definite Integrals. --
Contours. Line Integrals. --
Examples. --
The Cauchy-Goursat Theorem. --
A Preliminary Theorem. --
Proof of the Cauchy-Goursat Theorem. --
Simply and Multiply Connected Domains. --
Indefinite Integrals. --
The Cauchy Integral Formula. --
Derivatives of Analytic Functions. --
Morera’s Theorem. --
Maximum Moduli of Functions. --
The Fundamental Theorem of Algebra. --
0. Power Series [129] --
Taylor’s Serios. --
Observations and Examples, Laurent’s Series. --
Properties of Series. Uniform Convergence. --
Integration and Differentiation of Power Serios. --
Uniqueness of Representations by Power Series. --
Multiplication and Division. --
Examples. --
Zeros of Analytic Functions. --
7. Residues and Poles [153] --
Residues. --
The Residue Theorem. --
Poles. --
Quotients of Analytic Functions. --
Evaluation of Improper Real Integrals. --
Another Example. --
Improper Integrals Involving Trigonometric Functions. --
Definite Integrals of Trigonometric Functions. --
Integration around a Branch Point. --
8. Conformal Mapping [174] --
Rotation of Tangents. --
Conformal Mapping. --
Examples. --
Conjugate Harmonic Functions. --
Inverse Functions. --
Transformation of Harmonic Functions. --
Transformation of Boundary Conditions. --
9. Applications of Conformal Mapping [189] --
Steady Temperatures. --
Steady Temperatures in a Wall. --
Temperatures in a Quadrant with Part of One Boundary Insulated. --
Electric Potential. --
Potential in a Cylindrical Space. --
Two-dimensional Fluid Flow. --
The Stream Function. --
Flow around a Corner. --
Flow around a Cylinder. --
10. The Schwarz-Christoffel Transformation [218] --
Mapping the Real Axis onto a Polygon. --
The Schwarz-Christoffel Transformation. --
Triangles and Rectangles. --
Degenerate Polygons. --
The Infinite Strip. --
Fluid Flow in a Channel through a Slit. --
Flow in a Channel with an Offset. --
Electrostatic Potential about an Edge of a Conducting Plate. --
11. Integral Formulas of Poisson Type [242] --
Poisson’s Integral Formula. --
A Dirichlet Problem for the Circle. --
Related Boundary Value Problems. --
Integral Formulas for the Half Plane. --
A Dirichlet Problem for the Half Plane. --
Neumann Problems for Circular Regions. --
A Neumann Problem for the Half Plane --
Further Theory of Functions [259] --
Analytic Continuation [259] --
Conditions under Which f(z) ≡ [0.] --
Permanence of Forms of Functional Identities. --
Uniqueness of Analytic Continuation. --
Examples. --
The Principle of Reflection. --
B. Singular Points and Zeros [268] --
Poles and Zeros. --
Essential Singular Points. --
The Number of Zeros and Poles. --
C. Riemann Surfaces [273] --
A Surface for the Function log z. --
A Surface for the Function z1/2. --
Surfaces for Other Irrational Functions. --
Appendixes [281] --
1. Bibliography [281] --
2. Table of Transformations of Regions [284] --
Index [293] --
MR, 22 #3793
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