An introduction to integral transforms:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC PRESS
2023
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| Edition: | First issued in paperback |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 411 Seiten Illustrationen 24 cm |
| ISBN: | 1032653353 9781032653358 |
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Contents 1 FOURIER TRANSFORM 1 1.1 Introduction. 1 1.2 Classes of functions. 2 1.3 Fourier Series and Fourier Integral Formula. 2 1.4 Fourier Transforms. 6 Fourier sine and cosine Transforms. 7 1.5 Linearity property of Fourier Transforms. 8 1.6 Change of Scale property. 9 1.7 The Modulation theorem. 10 1.8 Evaluation of integrals by means of inversion theorems. 1.9 Fourier Transform of some particular functions. 1.4.1 . 11 13 1.10 Convolution or Faltung of two integrable functions. 20 1.11 Convolution or Falting or Faltung Theorem for FT. 21 1.12 Parseval’s relations for Fourier Transforms. 23 1.13 Fourier Transform of the derivative of a function. 26 1.14 Fourier Transform of some more useful functions. 30 1.15 Fourier Transforms of Rational Functions. 36 1.16 Other important examples concerning derivative of FT. . 37 1.17 The solution of Integral Equations of Convolution Type. . 47 1.18 Fourier Transform of Functions of several variables. 53 1.19 Application of Fourier Transform to Boundary Value Prob lems
. 55 2 FINITE FOURIER TRANSFORM 79 2.1 Introduction. 79
2.2 Finite Fourier cosine and sine Transforms. 79 2.3 Relation between finite Fourier Transform of the deriva tives of a function.81 2.4 Faltung or convolution theorems for finite Fourier Trans form. 82 2.5 Multiple Finite Fourier Transform. 85 2.6 Double Transforms ofpartial derivatives of functions. . 86 2.7 Application of finite Fourier Transforms to boundary value problems. 87 3 THE LAPLACE TRANSFORM 102 3.1 Introduction. 102 3.2 Definitions. 103 3.3 Sufficient conditions for existence of Laplace Transform. . 103 3.4 Linearity property of Laplace Transform. 104 3.5 Laplace transforms of some elementary functions. 105 3.6 First shift theorem. 107 3.7 Second shift theorem.107 3.8 The change of scale property. 107 3.9 Examples.108 3.10 Laplace Transform of derivatives of a function. 110 3.11 Laplace Transform of Integral of a
function. 112 3.12 Laplace Transform of tnf(t). 113 3.13 Laplace Transform of f(t)/t . 114 3.14 Laplace Transform of a periodic function. 115 3.15 The initial-value theorem and the final-value theorem of Laplace Transform. 116 3.16 Examples.117 3.17 Laplace Transform of some special functions. 121 3.18 The Convolution of two functions. 131 3.19 Applications. 132
4 THE INVERSE LAPLACE TRANSFORM AND APPLICATION 141 4.1 Introduction. 141 4.2 Calculation of Laplace inversion of some elementary func tions. 143 4.3 Method of expansion into partial fractions of the ratio of two polynomials . 145 4.4 The general evaluation technique of inverse Laplace trans form. 153 4.5 Inversion Formula from a different stand point : The Tricomi’s method.158 4.6 The Double Laplace Transform. 161 4.7 The iterative Laplace transform. 166 4.8 The Bilateral Laplace Transform. 166 4.9 Application of Laplace Transforms. 168 5 Hilbert and Stieltjes Transforms 220 5.1 Introduction. 220 5.2 Definition of Hilbert Transform. 220 5.3 Some Important properties of HilbertTransforms. 221 5.4 Relation between Hilbert Transform and FourierTransform. 225 5.5 Finite Hilbert Transform.226 5.6 One-sided Hilbert
Transform. 227 5.7 Asymptotic Expansions of one-sided Hilbert Transform. .228 5.8 The Stieltjes Transform.230 5.9 Some Deductions. 231 5.10 The Inverse Stieltjes Transform. 232 5.11 Relation between Hilbert Transform and Stieltjes Trans form. 234 6 Hankel Transforms 238 6.1 Introduction. 238
6.2 The Hankel Transform. 238 6.3 Elementary properties. 238 6.4 Inversion formula for Hankel Transform. 242 6.5 The Parseval Relation for HankelTransforms. 244 6.6 Illustrative Examples: . 245 7 Finite Hankel Transforms 260 7.1 Introduction. 260 7.2 Expansion of some functions in series involving cylinder functions : Fourier-Bessel Series. 260 7.3 The Finite Hankel Transform. 262 7.4 Illustrative Examples. 263 7.5 Finite Hankel Transform of order n in 0 x 1 of the derivatrive of a function.265 7.6 Finite Hankel Transform over 0 ^ x ^ 1 of order n of ΐ| + i , when p is the root of Jn(p) = 0. 266 7.7 Finite Hankel Transform of f"(x) + ^f'(x) — ^ f(x), where p is the root of Jn(p) = 0in0^x^l. 266 7.8 Other forms of finite Hankel Transforms. 267 7.9 Illustrations. 268 7.10 Application of finite Hankel Transforms. 269 8 The Mellin Transform 277 8.1
Introduction.277 8.2 Definition of Mellin Transform. 278 8.3 Mellin Transform of derivative of a function. 281 8.4 Mellin Transform of Integral of a function. 283 8.5 Mellin Inversion theorem.285 8.6 Convolution theorem of Mellin Transform. 286 8.7 Illustrative solved Examples. 287 8.8 Solution of Integral equations. 292
8.9 Application to Summation of Series. 293 8.10 The Generalised Mellin Transform. 295 8.11 Convolution of generalisedMellin Transform. 297 8.12 Finite Mellin Transform.297 9 Finite Laplace Transforms 9.1 302 Introduction. 302 9.2 Definition of Finite Laplace Transform. 302 9.3 Finite Laplace Transform of elementary functions. 304 9.4 Operational Properties. 307 9.5 The Initial Value and the Final Value Theorem. 311 9.6 Applications.312 10 Legendre Transforms 317 10.1 Introduction. 317 10.2 Definition of Legendre Transform. 317 10.3 Elementary properties of Legendre Transforms. 318 10.4 Operational Properties of Legendre Transforms. 323 10.5 Application to Boundary Value Problems. 325 11 The Kontorovich-Lebedev Transform 328 11.1 Introduction. 328 11.2 Definition of Kontorovich - Lebedev Transform. 328 11.3 Parseval Relation for Kontorovich-Lebedev Transforms. . 329 11.4
Illustrative Examples. 330 11.5 Boundary Value Problem in a wedge of finite thickness. . 332 12 The Mehler-Fock Transform 335 12.1 Introduction. 335 12.2 Fock’s Theorem (with weaker restriction). 335 12.3 Mehler-Fock Transform of zero order and its properties. . 337
12.4 Parseval type relation. 339 12.5 Mehler-Fock Transform of order m. 341 12.6 Application to Boundary Value Problems. 342 12.6.1 First Example .342 12.6.2 Second Example. 344 12.6.3 Third Example.345 12.6.4 Fourth Example .347 12.7 Application of Mehler-Fock Transform for solving dual integral equation. 348 13 Jacobi, Gegenbauer, Laguerre and Hermite Transforms 351 13.1 Introduction. 351 13.2 Definition of Jacobi Transform. 351 13.3 The Gegenbauer Transform. 355 13.4 Convolution Theorem . 356 13.5 Application of the Transforms. 357 13.6 The Laguerre Transform. 359 13.7 Operational properties.361 13.8 Hermite Transform. 364 13.9 Operational Properties. 366 13.10
Hermite Transform of derivative of a function.367 14 The Z-Transform 372 14.1 Introduction. 372 14.2 Z - Transform : Definition. 372 14.3 Some Operational Properties of Z-Transform. 376 14.4 Application of Z-Transforms. 383 Appendix Bibliography Index 390 405 407 |
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| illustrated | Illustrated |
| indexdate | 2025-02-12T07:01:31Z |
| institution | BVB |
| isbn | 1032653353 9781032653358 |
| language | English |
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| spelling | Patra, Baidyanath Verfasser (DE-588)1169875866 aut An introduction to integral transforms First issued in paperback Boca Raton ; London ; New York CRC PRESS 2023 411 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Integraltransformation (DE-588)4027235-7 gnd rswk-swf Integral transforms Transformations intégrales (DE-588)4151278-9 Einführung gnd-content Integraltransformation (DE-588)4027235-7 s DE-604 978-1-138-58803-5 hbk Erscheint auch als Online-Ausgabe 978-0-429-50358-0 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=035394258&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Patra, Baidyanath An introduction to integral transforms Integraltransformation (DE-588)4027235-7 gnd |
| subject_GND | (DE-588)4027235-7 (DE-588)4151278-9 |
| title | An introduction to integral transforms |
| title_auth | An introduction to integral transforms |
| title_exact_search | An introduction to integral transforms |
| title_full | An introduction to integral transforms |
| title_fullStr | An introduction to integral transforms |
| title_full_unstemmed | An introduction to integral transforms |
| title_short | An introduction to integral transforms |
| title_sort | an introduction to integral transforms |
| topic | Integraltransformation (DE-588)4027235-7 gnd |
| topic_facet | Integraltransformation Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035394258&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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