## Linear Integral EquationsIn the ten years since the first edition of this book appeared, integral equations and integral operators have revealed more of their mathematical beauty and power to me. Therefore, I am pleased to have the opportunity to share some of these new insights with the readers of this book. As in the first edition, the main motivation is to present the fundamental theory of integral equations, some of their main applications, and the basic concepts of their numerical solution in a single volume. This is done from my own perspective of integral equations; I have made no attempt to include all of the recent developments. In addition to making corrections and adjustments throughout the text and updating the references, the following topics have been added: In Sec tion 4.3 the presentation of the Fredholm alternative in dual systems has been slightly simplified and in Section 5.3 the short presentation on the index of operators has been extended. The treatment of boundary value problems in potential theory now includes proofs of the jump relations for single-and double-layer potentials in Section 6.3 and the solution of the Dirichlet problem for the exterior of an arc in two dimensions (Section 7.6). The numerical analysis of the boundary integral equations in Sobolev space settings has been extended for both integral equations of the first kind in Section 13.4 and integral equations of the second kind in Section 12.4. |

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### Contents

III | 1 |

IV | 2 |

V | 5 |

VI | 6 |

VII | 9 |

VIII | 11 |

IX | 13 |

X | 15 |

LV | 175 |

LVI | 177 |

LVII | 178 |

LVIII | 179 |

LIX | 182 |

LX | 187 |

LXI | 192 |

LXII | 195 |

XI | 17 |

XII | 18 |

XIII | 20 |

XIV | 27 |

XV | 28 |

XVI | 34 |

XVII | 36 |

XVIII | 38 |

XIX | 39 |

XX | 41 |

XXI | 45 |

XXII | 50 |

XXIII | 53 |

XXIV | 55 |

XXV | 57 |

XXVI | 62 |

XXVII | 66 |

XXVIII | 67 |

XXIX | 74 |

XXX | 78 |

XXXI | 82 |

XXXII | 87 |

XXXIII | 92 |

XXXIV | 94 |

XXXV | 97 |

XXXVI | 105 |

XXXVII | 107 |

XXXVIII | 115 |

XXXIX | 120 |

XL | 124 |

XLI | 125 |

XLII | 135 |

XLIII | 142 |

XLIV | 151 |

XLV | 152 |

XLVI | 155 |

XLVII | 160 |

XLVIII | 162 |

XLIX | 163 |

L | 164 |

LI | 165 |

LII | 167 |

LIII | 168 |

LIV | 170 |

LXIII | 197 |

LXIV | 198 |

LXV | 201 |

LXVI | 206 |

LXVII | 213 |

LXVIII | 216 |

LXIX | 218 |

LXX | 223 |

LXXI | 225 |

LXXII | 232 |

LXXIII | 240 |

LXXIV | 245 |

LXXV | 247 |

LXXVI | 248 |

LXXVII | 251 |

LXXVIII | 255 |

LXXIX | 260 |

LXXX | 264 |

LXXXI | 265 |

LXXXII | 269 |

LXXXIII | 271 |

LXXXIV | 277 |

LXXXV | 281 |

LXXXVI | 289 |

LXXXVII | 290 |

LXXXVIII | 291 |

LXXXIX | 292 |

XC | 298 |

XCI | 301 |

XCII | 307 |

XCIII | 308 |

XCIV | 313 |

XCV | 315 |

XCVI | 317 |

XCVII | 318 |

XCVIII | 320 |

XCIX | 328 |

C | 333 |

CI | 341 |

CII | 345 |

CIII | 347 |

361 | |

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### Common terms and phrases

27r-periodic assume Banach space boundary condition boundary value problem bounded inverse bounded linear operator Cauchy Cauchy-Schwarz inequality collocation method compact linear operator compact operator constant continuous functions continuous kernel continuously differentiable Corollary defined degenerate kernel denote dense density derivative Dirichlet problem domain double-layer potential dual system eigenvalues elements error estimate exact solution exists exterior finite finite-dimensional following theorem Fredholm alternative function f Galerkin method given harmonic function Hence Hilbert space holomorphic homogeneous ill-posed implies injective integral equation integral operator iteration Lemma linear space linear system logarithmic Lz(dD mapping matrix maximum norm Neumann problem normed space nullspace numerical Nystrom method obtain orthogonal pointwise convergence projection method projection operator proof of Theorem quadrature regularization respect Riesz right-hand side satisfies scalar product second kind self-adjoint sequence single-layer potential Sobolev spaces solving subset subspace theory unique solution weakly singular kernels