수리물리

신상진

서울대학사,U.C.Berkeley박사(1989),
U.Florida연구원(~1992),
한양대교수(1992~현재),
한국물리학회의이사,부회장,
학회지JKPS편집위원장역임,
끈이론과다체이론연구논문170여편,
양자역학(경문사),역서:최초의3분(양문사),
양자암흑물질의제창자(1994),
홀로그래픽평균장이론(2021)과콘도응축(2023)의제안,
성봉물리학상수상(2025)

서문iii

기본적수학지식1
0.1함수와사상.....................................1
0.2실함수의미적분..................................2
0.3복소수의기본성질.................................5
0.4급수의수렴.....................................7

I구조와해석-선형대수,벡터해석,복소변수13

제1장선형대수학15
1.1R3..........................................16
1.2벡터공간......................................20
1.3선형사상과행렬..................................24
1.4행렬식.......................................33
1.5연산자(operator).................................42
1.6스펙트럼분해...................................51
1.7∗선형사상과인덱스정리.............................55

제2장벡터의미분과적분59
2.1벡터에작용하는미분연산자들..........................60
2.2벡터의적분.....................................63
2.3미분연산자들의의미...............................64
2.4가우스정리와스톡스정리............................67
2.5퍼텐셜이론.....................................71

제3장곡선좌표계의벡터해석79
3.1곡선좌표계와기저................................80
3.2곡선좌표계에서의미분연산자..........................83

제4장곡선좌표계의미분기하학89
4.1텐서와물리법칙..................................90
4.2공변미분과측지선................................93

제5장복소수와복소적분99
5.1복소함수의미분:코시-리만조건.........................100
5.2코시의적분정리..................................102
5.3코시의적분공식과그응용............................103
5.4복소함수에의한사상(mapping).........................109
5.5유수정리와그응용................................111
5.6해석성,분산관계,그리고인과율의동치성...................123
5.7안장점근사법...................................126
5.8해석함수의폴과제로...............................132

제6장감마함수137
6.1감마함수의세가지정의와그동등성......................138
6.2폴리감마함수....................................142
6.3베타함수......................................145

II미분방정식과특수함수론149

제7장상미분방정식151
7.1특이점에따른상미분방정식의분류.......................152
7.2급수해법......................................153
7.3론스키안과두번째해...............................157

제8장편미분방정식163
8.1편미분방정식의분류...............................163
8.2변수분리법과특수함수들의근원........................168

제9장함수공간의선형대수:스텀-리우빌이론171
9.1함수공간......................................172
9.2자기수반연산자..................................172
9.3고윳값과고윳값방정식..............................174
9.4에르미트연산자의성질..............................176
9.5직교다항식.....................................178

제10장그린함수185
10.1그린함수의정의..................................186
10.21차원에서의그린함수...............................189

제11장원함수:푸리에급수와푸리에적분195
11.1기본성질......................................196
11.2푸리에급수의응용................................199
11.3깁스현상......................................203
11.4푸리에적분.....................................205

제12장원주함수:베셀213
12.1제1종베셀함수...................................214
12.2제2종베셀함수:노이만함수...........................219
12.3한켈함수......................................223
12.4변형베셀함수Iν(x)와Kν(x)...........................226
12.5베셀함수의점근전개공식.............................229
12.6구면베셀함수....................................231

제13장구면함수:르장드르235
13.1르장드르생성함수.................................236
13.2연관르장드르함수................................241
13.3구면조화함수와각운동량연산자.........................244
13.4르장드르제2종해.................................251

제14장직교함수계253
14.1에르미트함수...................................254
14.2라게르함수.....................................258
14.3체비셰프다항식..................................265
14.4초기하함수.....................................269
14.5합류초기하함수..................................272

III대국적고려-적분변환과적분방정식,변분원리277

제15장적분변환279
15.1여러가지적분변환................................280
15.2라플라스변환...................................281

제16장적분방정식289
16.1미분방정식의적분방정식화............................290
16.2적분변환과생성함수...............................292
16.3적분방정식의해법.................................294
16.4힐베르트-슈미트이론...............................298

제17장변분이론303
17.1변분의개념과오일러의방정식..........................304
17.2고전역학의변분법적정식화...........................308
17.3구속조건과라그랑주승수법...........................311

IV대칭성319

제18장대칭성과군론321
18.1군의정의와행렬표현...............................322
18.2이산군과응용...................................326
18.3리군과리대수...................................330
18.4로렌츠군과특수상대론..............................337

제19장각운동량대수와그표현343
19.1각운동량의양자화.................................344
19.2스핀1/2.......................................351
19.3각운동량의합...................................353
19.4텐서연산자와위그너-에카르트정리.......................357
19.5SU(n)군의영타블로...............................360
19.6일반적리대수의표현이론............................362

V미분기하학367

제20장다양체의미분기하학369
20.1위상공간......................................370
20.2다양체.......................................375
20.3벡터,짝벡터와미분형식,텐서..........................379
20.4흐름과리미분...................................382

제21장미분형식과스톡스정리389
21.1외미분(exteriorderivative).............