## Applied Hyperfunction TheoryGeneralized functions are now widely recognized as important mathematical tools for engineers and physicists. But they are considered to be inaccessible for non-specialists. To remedy this situation, this book gives an intelligible exposition of generalized functions based on Sato's hyperfunction, which is essentially the `boundary value of analytic functions'. An intuitive image -- hyperfunction = vortex layer -- is adopted, and only an elementary knowledge of complex function theory is assumed. The treatment is entirely self-contained. The first part of the book gives a detailed account of fundamental operations such as the four arithmetical operations applicable to hyperfunctions, namely differentiation, integration, and convolution, as well as Fourier transform. Fourier series are seen to be nothing but periodic hyperfunctions. In the second part, based on the general theory, the Hilbert transform and Poisson-Schwarz integral formula are treated and their application to integral equations is studied. A great number of formulas obtained in the course of treatment are summarized as tables in the appendix. In particular, those concerning convolution, the Hilbert transform and Fourier transform contain much new material. For mathematicians, mathematical physicists and engineers whose work involves generalized functions. |

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### Contents

INTRODUCTION | 1 |

2 Satos hyperfunction | 2 |

3 Aim | 3 |

4 Complex velocity and analytic function | 4 |

5 Distribution of vortices and hyperfunctions | 6 |

6 Ordinary functions and hyperfunctions | 8 |

OPERATIONS ON HYPERFUNCTIONS | 11 |

2 Linear combinations | 13 |

6 Convolution of two periodic hyperfunctions | 299 |

3 Properties of conjugate hyperfunctions | 307 |

6 Formulae of Hilbert transforms | 314 |

7 Standardtype generating function | 322 |

8 Hilbert type transforms | 324 |

9 Summary | 326 |

POISSONSCHWARZ INTEGRAL FORMULAE | 327 |

3 PoissonSchwarz integral formula for a circle | 329 |

3 Product of a hyperfunction and an analytic function | 14 |

4 Reinterpretation of ordinary functions as hyperfunctions | 15 |

5 Differentiation of hyperfunctions | 19 |

6 Definite integrals of hyperfunctions | 22 |

7 Summary | 23 |

viii | 25 |

10 Hz lz sgnz | 40 |

HYPERFUNCTIONS DEPENDING ON PARAMETERS | 53 |

8 Powertype hyperfunctions | 67 |

10 Calculation of pf integrals | 75 |

11 Summary | 81 |

3 Theorems about Fourier transformation | 88 |

5 Examples of calculations of Fourier transforms | 97 |

2 Tza | 103 |

6 FxmlogxnHx xmlogxn xmlogxnsgnx | 109 |

6 Upper lower powertype hyperfunctions and their Fourier trans | 126 |

4 Regularity of GQ Ftzl+z on the axis | 137 |

8 Sufficient conditions for the existence of convolution | 267 |

ill Fourier transforms of convolutions | 275 |

15 Summary | 282 |

4 Convolution of periodic hyperfunctions | 293 |

4 Generalization of the PoissonSchwarz integral formula | 331 |

5 RiemannHilbert problem | 337 |

6 Integral equations related to Hilbert transforms | 339 |

7 IZo ftt x1 dt gx oo x oo | 340 |

9 faftt x1 dt gx ax b | 343 |

10 HJx axfx + 0x | 348 |

11 i ftlogt xdt gx 1 x 1 | 352 |

12 Summary | 355 |

INTEGRAL EQUATIONS | 357 |

3 Solution of convolution equations | 358 |

4 Examples of application | 360 |

5 Alternative method | 362 |

6 Volterra integral equations | 363 |

7 Abels integral equation | 365 |

8 JJWe1dt gx x0 | 368 |

LAPLACE TRANSFORMS | 381 |

395 | |

Appendix E Uppertype and lowertype hyperfunctions | 409 |

Fourier transforms | 427 |

xvi | 435 |

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### Common terms and phrases

absolutely integrable analytic continuation analytic function regular arbitrary constant arbitrary hyperfunction axis calculation central hyperfunction chapter complex number contour convergence convolution corresponding definite integral derive differentiation digamma function domain entire function EXAMPLE exists expressed F_(z F+(x F+(z Ff{x fi(x Fi(z fi{x finite following theorem formal product formula Fourier series Fourier transform function F(z generalised 6-function gives Heaviside function Hence Hilbert transform hyper hyperfunction defined hyperfunction f(x imaginary infinite principal-value integral integer integral equation interval J-oo J—oo Laplace transform linear combination lower components lower hyperfunction obtain ordinary function Parseval's theorem periodic function periodic hyperfunction power-type hyperfunctions Proof real number reinterpreted respectively right hyperfunction sgnx similarly single-valued analytic function singular points solution standard generating function standard-type generating function Substituting Taking H.F. Theorem 12 upper and lower upper half-plane upper lower upper lower)-type hyperfunctions x-axis