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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | (OCoLC)1002128300 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Alexander Isaev |

ISBN: | 9783319681702 3319681702 |

OCLC Number: | 1013820783 |

Description: | 1 online resource. |

Contents: | 1. Complex Numbers. The Fundamental Theorem of Algebra.- 2. R- and C-Differentiability -- 3 The Stereographic Projection. Conformal Maps. The Open Mapping Theorem.- 4. Conformal Maps (Continued). Möbius Transformations.- 5. Möbius Transformations (Continued). Generalised Circles. Symmetry.- 6. Domains Bounded by Pairs of Generalised Circles. Integration.- 7. Primitives Along Paths. Holomorphic Primitives on a Disk. Goursat’s Lemma.- 8. Proof of Lemma 7.2. Homotopy. The Riemann Mapping Theorem.- 9. Cauchy’s Independence of Homotopy Theorem. Jordan Domains.- 10. Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula.- 11. Morera’s Theorem. Power Series. Abel’s Theorem. Disk and Radius of Convergence.- 12. Power Series (Cont’d). Expansion of a Holomorphic Function. The Uniqueness Theorem -- 13. Liouville’s Theorem. Laurent Series. Isolated Singularities.- 14. Isolated Singularities (Continued). Poles and Zeroes. Isolated Singularities at infinity.- 15. Isolated Singularities at infinity (Continued). Residues. Cauchy’s Residue Theorem.- 16. Residues (Continued). Contour Integration. The Argument Principle 137.- 17. The Argument Principle (Cont’d). Rouché’s Theorem. The Maximum Modulus Principle.- 18. Schwarz’s Lemma. (Pre) Compactness. Montel’s Theorem. Hurwitz’s Theorem.- 19. Analytic Continuation.- 20. Analytic Continuation (Continued). The Monodromy Theorem.- 21. Proof of Theorem 8.3. Conformal Transformations of Simply- Connected Domains -- Index. |

Series Title: | Springer undergraduate mathematics series. |

Responsibility: | Alexander Isaev. |

### Abstract:

Setting the tone for the entire book, the material begins with a proof of the Fundamental Theorem of Algebra to demonstrate the power of complex numbers and concludes with a proof of another major milestone, the Riemann Mapping Theorem, which is rarely part of a one-semester undergraduate course.
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