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ویرایش: 2
نویسندگان: Steven G. Krantz
سری: Textbooks in Mathematics
ISBN (شابک) : 0367222671, 9780367222673
ناشر: Chapman and Hall/CRC
سال نشر: 2019
تعداد صفحات: 378
زبان: English
فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود)
حجم فایل: 2 مگابایت
کلمات کلیدی مربوط به کتاب متغیرهای مختلط: یک رویکرد فیزیکی با کاربردها (کتاب های درسی ریاضیات): ریاضیات، حساب دیفرانسیل و انتگرال، متغیر مختلط
در صورت تبدیل فایل کتاب Complex Variables: A Physical Approach with Applications (Textbooks in Mathematics) به فرمت های PDF، EPUB، AZW3، MOBI و یا DJVU می توانید به پشتیبان اطلاع دهید تا فایل مورد نظر را تبدیل نمایند.
توجه داشته باشید کتاب متغیرهای مختلط: یک رویکرد فیزیکی با کاربردها (کتاب های درسی ریاضیات) نسخه زبان اصلی می باشد و کتاب ترجمه شده به فارسی نمی باشد. وبسایت اینترنشنال لایبرری ارائه دهنده کتاب های زبان اصلی می باشد و هیچ گونه کتاب ترجمه شده یا نوشته شده به فارسی را ارائه نمی دهد.
ایده اعداد مختلط حداقل به 300 سال قبل از گاوس و اویلر برمی گردد. امروزه تحلیل پیچیده بخش مرکزی تفکر تحلیلی مدرن است. در مهندسی، فیزیک، ریاضیات، اخترفیزیک و بسیاری از زمینه های دیگر استفاده می شود. این کتاب ابزارهای قدرتمندی برای انجام تجزیه و تحلیل ریاضی فراهم می کند و اغلب پاسخ های خوشایند و غیرمنتظره ای به دست می دهد.
این کتاب موضوع تجزیه و تحلیل پیچیده را برای مخاطبان وسیعی قابل دسترس می کند. اعداد مختلط یک سیستم اعداد تا حدی مرموز هستند که به نظر می رسد از آبی بیرون آمده اند. برای دانشآموزان مهم است که ببینند این واقعاً مجموعهای بسیار ملموس از اشیاء است که کاربردهای بسیار ملموس و معنیداری دارد.
ویژگی ها:
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The idea of complex numbers dates back at least 300 years―to Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers.
This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications.
Features:
Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface to the Second Edition for the Instructor Preface to the Second Edition for the Student Preface to the First Edition 1: Basic Ideas 1.1 Complex Arithmetic 1.1.1 The Real Numbers 1.1.2 The Complex Numbers 1.1.3 Complex Conjugate Exercises 1.2 Algebraic and Geometric Properties 1.2.1 Modulus of a Complex Number 1.2.2 The Topology of the Complex Plane 1.2.3 The Complex Numbers as a Field 1.2.4 The Fundamental Theorem of Algebra Exercises 2: The Exponential and Applications 2.1 The Exponential Function 2.1.1 Laws of Exponentiation 2.1.2 The Polar Form of a Complex Number Exercises 2.1.3 Roots of Complex Numbers 2.1.4 The Argument of a Complex Number 2.1.5 Fundamental Inequalities Exercises 3: Holomorphic and Harmonic Functions 3.1 Holomorphic Functions 3.1.1 Continuously Differentiable and Ck Functions 3.1.2 The Cauchy–Riemann Equations 3.1.3 Derivatives 3.1.4 Definition of Holomorphic Function 3.1.5 Examples of Holomorphic Functions 3.1.6 The Complex Derivative 3.1.7 Alternative Terminology for Holomorphic Functions Exercises 3.2 Holomorphic and Harmonic Functions 3.2.1 Harmonic Functions 3.2.2 Holomorphic and Harmonic Functions Exercises 3.3 Complex Differentiability 3.3.1 Conformality Exercises 4: The Cauchy Theory 4.1 Real and Complex Line Integrals 4.1.1 Curves 4.1.2 Closed Curves 4.1.3 Differentiable and Ck Curves 4.1.4 Integrals on Curves 4.1.5 The Fundamental Theorem of Calculus along Curves 4.1.6 The Complex Line Integral 4.1.7 Properties of Integrals Exercises 4.2 The Cauchy Integral Theorem 4.2.1 The Cauchy Integral Theorem, Basic Form 4.2.2 More General Forms of the Cauchy Theorem 4.2.3 Deformability of Curves 4.2.4 Cauchy Integral Formula, Basic Form 4.2.5 More General Versions of the Cauchy Formula Exercises 4.3 Variants of the Cauchy Formula 4.4 The Limitations of the Cauchy Formula Exercises 5: Applications of the Cauchy Theory 5.1 The Derivatives of a Holomorphic Function 5.1.1 A Formula for the Derivative 5.1.2 The Cauchy Estimates 5.1.3 Entire Functions and Liouville’s Theorem 5.1.4 The Fundamental Theorem of Algebra 5.1.5 Sequences of Holomorphic Functions and Their Derivatives 5.1.6 The Power Series Representation of a Holomorphic Function 5.1.7 Table of Elementary Power Series Exercises 5.2 The Zeros of a Holomorphic Function 5.2.1 The Zero Set of a Holomorphic Function 5.2.2 Discrete Sets and Zero Sets 5.2.3 Uniqueness of Analytic Continuation Exercises 6: Isolated Singularities 6.1 Behavior Near an Isolated Singularity 6.1.1 Isolated Singularities 6.1.2 A Holomorphic Function on a Punctured Domain 6.1.3 Classification of Singularities 6.1.4 Removable Singularities, Poles, and Essential Singularities 6.1.5 The Riemann Removable Singularities Theorem 6.1.6 The Casorati–Weierstrass Theorem 6.1.7 Concluding Remarks Exercises 6.2 Expansion around Singular Points 6.2.1 Laurent Series 6.2.2 Convergence of a Doubly Infinite Series 6.2.3 Annulus of Convergence 6.2.4 Uniqueness of the Laurent Expansion 6.2.5 The Cauchy Integral Formula for an Annulus 6.2.6 Existence of Laurent Expansions 6.2.7 Holomorphic Functions with Isolated Singularities 6.2.8 Classification of Singularities in Terms of Laurent Series Exercises 7: Meromorphic Functions 7.1 Examples of Laurent Expansions 7.1.1 Principal Part of a Function 7.1.2 Algorithm for Calculating the Coefficients of the Laurent Expansion Exercises 7.2 Meromorphic Functions 7.2.1 Meromorphic Functions 7.2.2 Discrete Sets and Isolated Points 7.2.3 Definition of Meromorphic Function 7.2.4 Examples of Meromorphic Functions 7.2.5 Meromorphic Functions with Infinitely Many Poles 7.2.6 Singularities at Infinity 7.2.7 The Laurent Expansion at Infinity 7.2.8 Meromorphic at Infinity 7.2.9 Meromorphic Functions in the Extended Plane Exercises 8: The Calculus of Residues 8.1 Residues 8.1.1 Functions with Multiple Singularities 8.1.2 The Concept of Residue 8.1.3 The Residue Theorem 8.1.4 Residues 8.1.5 The Index or Winding Number of a Curve about a Point 8.1.6 Restatement of the Residue Theorem 8.1.7 Method for Calculating Residues 8.1.8 Summary Charts of Laurent Series and Residues Exercises 8.2 Applications to the Calculation of Integrals 8.2.1 The Evaluation of Definite Integrals 8.2.2 A Basic Example 8.2.3 Complexification of the Integrand 8.2.4 An Example with a More Subtle Choice of Contour 8.2.5 Making the Spurious Part of the Integral Disappear 8.2.6 The Use of the Logarithm 8.2.7 Summing a Series Using Residues 8.2.8 Summary Chart of Some Integration Techniques Exercises 9: The Argument Principle 9.1 Counting Zeros and Poles 9.1.1 Local Geometric Behavior of a Holomorphic Function 9.1.2 Locating the Zeros of a Holomorphic Function 9.1.3 Zero of Order n 9.1.4 Counting the Zeros of a Holomorphic Function 9.1.5 The Argument Principle 9.1.6 Location of Poles 9.1.7 The Argument Principle for Meromorphic Functions Exercises 9.2 Local Geometry of Functions 9.2.1 The Open Mapping Theorem Exercises 9.3 Further Results on Zeros 9.3.1 Rouché’s Theorem 9.3.2 A Typical Application of Rouché’s Theorem 9.3.3 Rouché’s Theorem and the Fundamental Theorem of Algebra 9.3.4 Hurwitz’s Theorem Exercises 10: The Maximum Principle 10.1 Local and Boundary Maxima 10.1.1 The Maximum Modulus Principle 10.1.2 Boundary Maximum Modulus Theorem 10.1.3 The Minimum Principle 10.1.4 The Maximum Principle on an Unbounded Domain Exercises 10.2 The Schwarz Lemma 10.2.1 Schwarz’s Lemma 10.2.2 The Schwarz–Pick Lemma Exercises 11: The Geometric Theory 11.1 The Idea of a Conformal Mapping 11.1.1 Conformal Mappings 11.1.2 Conformal Self-Maps of the Plane Exercises 11.2 Mappings of the Disc 11.2.1 Conformal Self-Maps of the Disc 11.2.2 Möbius Transformations 11.2.3 Self-Maps of the Disc Exercises 11.3 Linear Fractional Transformations 11.3.1 Linear Fractional Mappings 11.3.2 The Topology of the Extended Plane 11.3.3 The Riemann Sphere 11.3.4 Conformal Self-Maps of the Riemann Sphere 11.3.5 The Cayley Transform 11.3.6 Generalized Circles and Lines 11.3.7 The Cayley Transform Revisited 11.3.8 Summary Chart of Linear Fractional Transformations Exercises 11.4 The Riemann Mapping Theorem 11.4.1 The Concept of Homeomorphism 11.4.2 The Riemann Mapping Theorem 11.4.3 The Riemann Mapping Theorem: Second Formulation Exercises 11.5 Conformal Mappings of Annuli 11.5.1 Conformal Mappings of Annuli 11.5.2 Conformal Equivalence of Annuli 11.5.3 Classification of Planar Domains Exercises 11.6 A Compendium of Useful Conformal Mappings 12: Applications of Conformal Mapping 12.1 Conformal Mapping 12.1.1 The Study of Conformal Mappings 12.2 The Dirichlet Problem 12.2.1 The Dirichlet Problem 12.2.2 Physical Motivation for the Dirichlet Problem Exercises 12.3 Physical Examples 12.3.1 Steady-State Heat Distribution on a Lens-Shaped Region 12.3.2 Electrostatics on a Disc 12.3.3 Incompressible Fluid Flow around a Post Exercises 12.4 Numerical Techniques 12.4.1 Numerical Approximation of the Schwarz–Christoffel Mapping 12.4.2 Numerical Approximation to a Mapping onto a Smooth Domain Exercises 13: Harmonic Functions 13.1 Basic Properties of Harmonic Functions 13.1.1 The Laplace Equation 13.1.2 Definition of Harmonic Function 13.1.3 Real- and Complex-Valued Harmonic Functions 13.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions 13.1.5 Smoothness of Harmonic Functions Exercises 13.2 The Maximum Principle 13.2.1 The Maximum Principle for Harmonic Functions 13.2.2 The Minimum Principle for Harmonic Functions 13.2.3 The Maximum Principle 13.2.4 The Mean Value Property 13.2.5 Boundary Uniqueness for Harmonic Functions Exercises 13.3 The Poisson Integral Formula 13.3.1 The Poisson Integral 13.3.2 The Poisson Kernel 13.3.3 The Dirichlet Problem 13.3.4 The Solution of the Dirichlet Problem on the Disc 13.3.5 The Dirichlet Problem on a General Disc Exercises 14: The Fourier Theory 14.1 Fourier Series 14.1.1 Basic Definitions 14.1.2 A Remark on Intervals of Arbitrary Length 14.1.3 Calculating Fourier Coefficients 14.1.4 Calculating Fourier Coefficients Using Complex Analysis 14.1.5 Steady-State Heat Distribution 14.1.6 The Derivative and Fourier Series Exercises 14.2 The Fourier Transform 14.2.1 Basic Definitions 14.2.2 Some Fourier Transform Examples that Use Complex Variables 14.2.3 Solving a Differential Equation Using the Fourier Transform Exercises 15: Other Transforms 15.1 The Laplace Transform 15.1.1 Prologue 15.1.2 Solving a Differential Equation Using the Laplace Transform Exercises 15.2 The z-Transform 15.2.1 Basic Definitions 15.2.2 Population Growth by Means of the z-Transform Exercises 16: Boundary Value Problems 16.1 Fourier Methods 16.1.1 Remarks on Different Fourier Notations 16.1.2 The Dirichlet Problem on the Disc 16.1.3 The Poisson Integral 16.1.4 The Wave Equation Exercises Appendices Glossary List of Notation Table of Laplace Transforms A Guide to the Literature References Index