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NM-hamphuc - TRNG AI HOC A LAT NHAP MON HAM PHC GIAO TRNH...

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TRÖÔØNG ÑAÏI HOÏC ÑAØ LAÏT F 7 G GIAÙO TRÌNH NHAÄP MOÂN HAØM PHÖÙC TAÏ LEÂ LÔÏI - 2004
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Nhaäp moân haøm phöùc Taï Leâ Lôïi Muïc luïc Chöông I. Soá phöùc - Haøm phöùc 1.1 Soá phöùc . .................................................................. 1 1.1.1 Ñònh nghóa . ........................................................... 1 1.1.2 Caùc pheùp toaùn. ....................................................... 1 1.1.3 Bieåu dieãn soá phöùc . ................................................... 2 1.1.4 Tính chaát . ............................................................ 3 1.1.5 Caên baäc n 3 1.1.6 Bieåu dieãn caàu. ........................................................ 4 1.2 Söï hoäi tuï . ................................................................. 5 1.2.1 Khoaûng caùch . 5 1.2.2 Daõy hoäi tuï. 5 1.2.3 Caùc taäp cô baûn trong C ............................................... 6 1.2.4 Caùc ñònh lyù cô baûn: Cantor, Heine-Borel. ............................. 7 1.3 Haøm phöùc - Tính lieân tuïc . ................................................. 7 1.3.1 Ñònh nghóa . 7 1.3.2 Haøm phöùc xem nhö pheùp bieán ñoåi treân R a 2 ........................... 8 1.3.3 Giôùi haïn haøm. 9 1.3.4 Haøm lieân tuïc. ......................................................... 9 1.3.5 Caùc ñònh lyù cô baûn cuûa haøm lieân tuïc: Cauchy, Cantor, Weiersrtass. ..... 9 1.3.6 Ñònh lyù cô baûn cuûa ñaïi soá . .......................................... 10 Chöông II. Chuoãi luõy thöøa - Haøm giaûi tích 2.1 Chuoãi luõy thöøa hình thöùc . 11 2.1.1 Chuoãi luõy thöøa hình thöùc . ............................................ 2.1.2 Ñaïi soá C [[ Z ]] caùc chuoãi hình thöùc . ................................... 2.1.3 Pheùp chia . .......................................................... 2.1.4 Ñaïo haøm hình thöùc . 12 2.1.5 Thay bieán . 13 2.1.6 Chuoãi ngöôïc . 14 2.1.7 Quan heä ñoàng dö modulo Z N vaø kyù hieäu O ( Z N ) ..................... 15 2.1.8 Haøm sinh . 16 2.2 Hoäi tuï ñeàu . .............................................................. 17 2.2.1 Chuoãi soá. 2.2.2 Daõy haøm - Söï hoäi tuï ñeàu. 2.2.3 Chuoãi haøm . 18 2.3 Chuoãi luõy thöøa hoäi tuï . .................................................... 19 2.3.1 Ñònh lyù Abel. Baùn kính hoäi tuï . ....................................... 2.3.2 Toång, tích chuoãi luõy thöøa hoäi tuï. ..................................... 21
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2.3.3 Thay bieán trong chuoãi luõy thöøa hoäi tuï. ................................ 21 2.3.4 Nghòch ñaûo cuûa chuoãi luõy thöøa hoäi tu ï................................. 22 2.3.5 Ñaïo haøm chuoãi luõy thöøa hoäi tuï . ...................................... 2.3.6 Chuoãi ngöôïc . ....................................................... 23 2.4 Moät soá haøm sô caáp . ...................................................... 24 2.4.1 Haøm tuyeán tính . ..................................................... 2.4.2 Haøm luõy thöøa . 2.4.3 Haøm muõ. ............................................................ 2.4.4 Caùc haøm löôïng giaùc. ................................................. 2.4.5 Logarithm phöùc - Nhaùnh ñôn trò cuûa haøm logarithm . .................. 26 2.4.6 Haøm luõy thöøa toång quaùt . ............................................ 2.5 Haøm giaûi tích. 27 2.5.1 Ñònh nghóa. .......................................................... 2.5.2 Chuoãi luõy thöøa hoäi tuï laø haøm giaûi tích . ............................... 28 2.5.3 Khoâng ñieåm cuûa haøm giaûi tích. ....................................... 29 2.5.4 Nguyeân lyù thaùc trieån giaûi tích. ........................................ 2.5.5 Cöïc ñieåm - Haøm phaân hình . Chöông III. Haøm chænh hình - Tích phaân Cauchy 3.0 AÙnh xaï tuyeán tính treân R 2 vaø treân C 31 3.0.1 Bieåu dieãn soá phöùc bôûi ma traän thöïc. .................................. 3.3.2 AÙnh xaï tuyeán tính baûo giaùc . ......................................... 3.1 Tính khaû vi phöùc - Haøm chænh hình . 32 3.1.1 Ñaïo haøm. ........................................................... 3.1.2 Ñieàu kieän Cauchy-Riemann. 3.1.3 Coâng thöùc tính ñaïo haøm. ............................................. 33 3.1.4 Haøm chænh hình . 34 3.1.5 Tính baûo giaùc . 3.1.6 Löôùi toïa ñoä . ......................................................... 35 3.2 Tích phaân ñöôøng . ........................................................ 3.2.1 Ñöôøng cong trong C 3.2.2 Tích phaân ñöôøng . .................................................... 37 3.2.3 Tính chaát cuûa tích phaân ñöôøng . 3.2.4 Nguyeân haøm - Coâng thöùc Newton-Leibniz- Ñònh lyù Morera . .......... 38 3.3 Ñònh lyù Cauchy . 40 3.3.1 Ñònh lyù Cauchy cho mieàn ñôn lieân.
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