## Harmonic Function TheoryThis is a book about harmonic functions in Euclidean space. Readers with a background in real and complex analysis at the beginning graduate level will feel comfortable with the material presented here. The authors have taken unusual care to motivate concepts and simplify proofs. Topics include: basic properties of harmonic functions, Poisson integrals, the Kelvin transform, spherical harmonics, harmonic Hardy spaces, harmonic Bergman spaces, the decomposition theorem, Laurent expansions, isolated singularities, and the Dirichlet problem. The new edition contains a completely rewritten chapter on spherical harmonics, a new section on extensions of Bocher¿s Theorem, new exercises and proofs, as well as revisions throughout to improve the text. A unique software package-designed by the authors and available by email-supplements the text for readers who wish to explore harmonic function theory on a computer. |

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### Contents

XPTER | 3 |

PTER | 7 |

itive Harmonic Functions | 45 |

PTER 4 | 58 |

PTER 5 | 73 |

Explicit Bases of 3mR and 9mS | 92 |

Exercises | 106 |

The Hilbert Space I12 B | 121 |

Positive Harmonic Functions on the Upper HalfSpace | 157 |

PTER 8 | 170 |

The Reproducing Kernel of the Upper HalfSpace | 185 |

Removable Sets for Bounded Harmonic Functions | 201 |

The Residue Theorem 2 13 | 217 |

The Perron Construction | 226 |

ENDIX | 239 |

255 | |

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### Common terms and phrases

assume ause B6cher’s Theorem ball Bloch space Borel measurable boundary data bounded harmonic function chapter compact subset compute continuous function converges uniformly Corollary decomposition deﬁned deﬁnition denote Dirichlet problem E C(S equals equation Exercise exists ﬁnd ﬁnite ﬁrst ﬁxed formula function f function on Q half-space harmonic ctions harmonic func harmonic on Q harmonic polynomials Harnack’s hence holomorphic function homogeneous polynomial hyperplane implies inequality inﬁnite inversion iPm(R isolated singularity IxI2 Kelvin transform Lemma linear Liouville’s Theorem maximum principle mean-value property monic function multi-index nontangential limit Poisson integral Poisson kernel positive and harmonic positive harmonic functions power series Proposition Prove Q E S Q is connected real analytic reﬂection removable singularity result satisﬁes Schwarz Lemma sequence Speciﬁcally subset of Q Suppose Q symmetric tt(x uniformly on compact unique zonal harmonics