Special techniques for solving integrals: examples and problems
"Indispensable techniques are provided to solve integrals; Examples from the book can be used in classwork or for home assignments; It can be a helpful supplement to calculus and advanced calculus courses; Students training for mathematical competitions (like the MIT integration bee) will find...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2022]
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "Indispensable techniques are provided to solve integrals; Examples from the book can be used in classwork or for home assignments; It can be a helpful supplement to calculus and advanced calculus courses; Students training for mathematical competitions (like the MIT integration bee) will find here many useful techniques and examples"-- |
| Beschreibung: | xiv, 386 Seiten Illustrationen |
| ISBN: | 9789811236259 9789811235757 9811235759 9811236259 |
Internformat
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| 100 | 1 | |a Boyadzhiev, Khristo N. |e Verfasser |0 (DE-588)1163298875 |4 aut | |
| 245 | 1 | 0 | |a Special techniques for solving integrals |b examples and problems |c Khristo N. Boyadzhiev (Ohio Northern University, USA) |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2022] | |
| 264 | 4 | |c © 2022 | |
| 300 | |a xiv, 386 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 520 | 3 | |a "Indispensable techniques are provided to solve integrals; Examples from the book can be used in classwork or for home assignments; It can be a helpful supplement to calculus and advanced calculus courses; Students training for mathematical competitions (like the MIT integration bee) will find here many useful techniques and examples"-- | |
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Datensatz im Suchindex
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| adam_text | Contents Preface ....................................................................................................................... vii About the Author........................................................................................................ 1. Special Substitutions............................................................................................. 1.1 Introduction.................................................................................................. 1.2 Euler Substitutions....................................................................................... 1.2.1 First Euler substitution......................................................................... 1.2.2 Second Euler substitution..................................................................... 1.2.3 Third Euler substitution........................................................................ 1.3 Abel’s Substitution....................................................................................... 1.4 The Differential Binomialand Chebyshev’s Theorem................................... 1.5 Hyperbolic Substitutions for Integrals........................................................... 1.6 General Trigonometric Substitution............................................................. 1.6.1 Restrictions and extensions................................................................ 1.7 Arithmetic-Geometric Mean and the Gauss Formula................................... 1.7.1 The arithmetic-geometric mean.......................................................... 1.7.2
The Gauss formula............................................................................. 1.8 Some Interesting Examples........................................................................... xi 1 1 2 3 3 4 10 12 17 22 25 26 26 27 30 2. Solving Integrals by Differentiation with Respect to a Parameter....................... 2.1 Introduction................................................................................................... 2.2 General Examples.......................................................................................... 2.3 Using Differential Equations........................................................................ 2.4 Advanced Techniques.................................................................................. 2.5 The Basel Problem and Related Integrals..................................................... 2.5.1 Introduction......................................................................................... 2.5.2 Special integrals with arctangents....................................................... 2.5.3 Several integrals with logarithms........................................................ 2.6 Some Theorems............................................................................................. 41 41 47 69 79 85 85 89 92 101 3. Solving Logarithmic Integrals by Using Fourier Series....................................... 3.1 Introduction................................................................................................... 3.2
Examples....................................................................................................... 3.3 A Binet Type Formula for the Log-Gamma Function.................................. 3.4 Some Theorems............................................................................................. 103 103 104 140 143 4. Evaluating Integrals by Laplace and Fourier Transforms. Integrals Related to Riemann’s Zeta Function............................................................................ 145 4.1 Introduction................................................................................................... 145 4.2 Laplace Transform........................................................................................ 146 xiii
xiv Special Techniques for Solving Integrals 4.3 A Tale of Two Integrals................................................................................ 158 4.4 Parseval’s Theorem....................................................................................... 164 4.5 Some Important Hyperbolic Integrals............................................................ 171 4.5.1 Expansion of the cotangent in partial fractions................................... 171 4.5.2 Evaluation of important hyperbolic integrals...................................... 176 4.6 Exponential Polynomials and Gamma Integrals........................................... 181 4.7 The Functional Equation of the Riemann Zeta Function.............................. 189 4.8 The Functional Equation for Euler’s Цѕ) Function...................................... 198 4.9 Euler’s Formula for Zeta(2n)........................................................................ 200 4.9.1 Bernoulli numbers............................................................................... 200 4.9.2 Euler numbers and Euler’s formula for L(2n + 1)............................... 204 5. Various Techniques.............................................................................................. 215 5.1 The Formula of Poisson................................................................................ 215 5.2 Frullani Integrals.......................................................................................... 219 5.3 A Special
Formula........................................................................................ 224 5.4 Miscellaneous Selected Integrals.................................................................. 239 5.5 Catalan’s Constant........................................................................................ 264 5.6 Summation of Series by Using Integrals...................................................... 278 5.7 Generating Functions for Harmonic and Skew-Harmonic Numbers........................................................................................................ 290 5.7.1 Harmonic numbers.............................................................................. 290 5.7.2 Skew-harmonic numbers..................................................................... 303 5.7.3 Double integrals related to the above series........................................ 311 5.7.4 Expansions of dilogarithms and trilogarithms..................................... 313 5.8 Fun with Lobachevsky.................................................................................. 316 5.9 More Special Functions................................................................................. 321 Appendix A. List of Solved Integrals.......................................................................... 333 References.................................................................................................................. 379 Index........................................................................................................................... 385
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| adam_txt |
Contents Preface . vii About the Author. 1. Special Substitutions. 1.1 Introduction. 1.2 Euler Substitutions. 1.2.1 First Euler substitution. 1.2.2 Second Euler substitution. 1.2.3 Third Euler substitution. 1.3 Abel’s Substitution. 1.4 The Differential Binomialand Chebyshev’s Theorem. 1.5 Hyperbolic Substitutions for Integrals. 1.6 General Trigonometric Substitution. 1.6.1 Restrictions and extensions. 1.7 Arithmetic-Geometric Mean and the Gauss Formula. 1.7.1 The arithmetic-geometric mean. 1.7.2
The Gauss formula. 1.8 Some Interesting Examples. xi 1 1 2 3 3 4 10 12 17 22 25 26 26 27 30 2. Solving Integrals by Differentiation with Respect to a Parameter. 2.1 Introduction. 2.2 General Examples. 2.3 Using Differential Equations. 2.4 Advanced Techniques. 2.5 The Basel Problem and Related Integrals. 2.5.1 Introduction. 2.5.2 Special integrals with arctangents. 2.5.3 Several integrals with logarithms. 2.6 Some Theorems. 41 41 47 69 79 85 85 89 92 101 3. Solving Logarithmic Integrals by Using Fourier Series. 3.1 Introduction. 3.2
Examples. 3.3 A Binet Type Formula for the Log-Gamma Function. 3.4 Some Theorems. 103 103 104 140 143 4. Evaluating Integrals by Laplace and Fourier Transforms. Integrals Related to Riemann’s Zeta Function. 145 4.1 Introduction. 145 4.2 Laplace Transform. 146 xiii
xiv Special Techniques for Solving Integrals 4.3 A Tale of Two Integrals. 158 4.4 Parseval’s Theorem. 164 4.5 Some Important Hyperbolic Integrals. 171 4.5.1 Expansion of the cotangent in partial fractions. 171 4.5.2 Evaluation of important hyperbolic integrals. 176 4.6 Exponential Polynomials and Gamma Integrals. 181 4.7 The Functional Equation of the Riemann Zeta Function. 189 4.8 The Functional Equation for Euler’s Цѕ) Function. 198 4.9 Euler’s Formula for Zeta(2n). 200 4.9.1 Bernoulli numbers. 200 4.9.2 Euler numbers and Euler’s formula for L(2n + 1). 204 5. Various Techniques. 215 5.1 The Formula of Poisson. 215 5.2 Frullani Integrals. 219 5.3 A Special
Formula. 224 5.4 Miscellaneous Selected Integrals. 239 5.5 Catalan’s Constant. 264 5.6 Summation of Series by Using Integrals. 278 5.7 Generating Functions for Harmonic and Skew-Harmonic Numbers. 290 5.7.1 Harmonic numbers. 290 5.7.2 Skew-harmonic numbers. 303 5.7.3 Double integrals related to the above series. 311 5.7.4 Expansions of dilogarithms and trilogarithms. 313 5.8 Fun with Lobachevsky. 316 5.9 More Special Functions. 321 Appendix A. List of Solved Integrals. 333 References. 379 Index. 385 |
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| id | DE-604.BV047686164 |
| illustrated | Illustrated |
| index_date | 2024-07-03T18:56:35Z |
| indexdate | 2024-07-10T09:19:12Z |
| institution | BVB |
| isbn | 9789811236259 9789811235757 9811235759 9811236259 |
| language | English |
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| physical | xiv, 386 Seiten Illustrationen |
| publishDate | 2022 |
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| publisher | World Scientific |
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| spelling | Boyadzhiev, Khristo N. Verfasser (DE-588)1163298875 aut Special techniques for solving integrals examples and problems Khristo N. Boyadzhiev (Ohio Northern University, USA) New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2022] © 2022 xiv, 386 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier "Indispensable techniques are provided to solve integrals; Examples from the book can be used in classwork or for home assignments; It can be a helpful supplement to calculus and advanced calculus courses; Students training for mathematical competitions (like the MIT integration bee) will find here many useful techniques and examples"-- Integralrechnung (DE-588)4027232-1 gnd rswk-swf Integrals Calculus Integralrechnung (DE-588)4027232-1 s DE-604 Erscheint auch als Online-Ausgabe 978-981-1235-76-4 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033070192&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boyadzhiev, Khristo N. Special techniques for solving integrals examples and problems Integralrechnung (DE-588)4027232-1 gnd |
| subject_GND | (DE-588)4027232-1 |
| title | Special techniques for solving integrals examples and problems |
| title_auth | Special techniques for solving integrals examples and problems |
| title_exact_search | Special techniques for solving integrals examples and problems |
| title_exact_search_txtP | Special techniques for solving integrals examples and problems |
| title_full | Special techniques for solving integrals examples and problems Khristo N. Boyadzhiev (Ohio Northern University, USA) |
| title_fullStr | Special techniques for solving integrals examples and problems Khristo N. Boyadzhiev (Ohio Northern University, USA) |
| title_full_unstemmed | Special techniques for solving integrals examples and problems Khristo N. Boyadzhiev (Ohio Northern University, USA) |
| title_short | Special techniques for solving integrals |
| title_sort | special techniques for solving integrals examples and problems |
| title_sub | examples and problems |
| topic | Integralrechnung (DE-588)4027232-1 gnd |
| topic_facet | Integralrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033070192&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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