main :2021/9/7(9:48)
物理数学―量子力学のためのフーリエ解析・特殊関数―
柴田 尚和・是常 隆著
https://www.kyoritsu-pub.co.jp/bookdetail/9784320036161

目

次第 1 章 級数展開
1.1

1.2

1.3

1.4

級数による数の表現 ··························································

11.1.1 無限級数 ································································

2

1.1.2 無限級数の値 ·························································

2級数の収束判定法 ·····························································

41.2.1 ダランベルトの判定式 ·············································

41.2.2 絶対収束 ································································

4

級数を用いた関数の表現 ····················································

51.3.1 関数項級数 ·····························································

51.3.2 関数項級数の和 ······················································

51.3.3 一様収束 ································································

6ベキ級数を用いた関数の表現 ··············································

71.4.1 テイラー展開 ·························································

71.4.2 ベキ級数の収束半径 ················································

81.4.3 ベキ級数の性質 ······················································

81.4.4 テイラー展開の例 ···················································

9

第 2 章 フーリエ解析
2.1112フーリエ級数 ··································································· 122.1.1 フーリエ級数展開 ··················································· 122.1.2 三角関数の直交性 ··················································· 132.1.3 直交関数展開の展開係数 ·········································· 142.1.4 フーリエ展開係数 ··················································· 152.2

フーリエ級数の収束性 ······················································· 18

main :2021/9/7(9:48)
物理数学―量子力学のためのフーリエ解析・特殊関数―
柴田 尚和・是常 隆著
https://www.kyoritsu-pub.co.jp/bookdetail/9784320036161

vi

目

次

2.2.1 フーリエ級数の各点収束 ·········································· 19
2.2.2 パーセバルの等式 ··················································· 22
2.2.3 フーリエ級数の積分と微分の収束性 ··························· 242.3

様々なフーリエ級数 ·························································· 252.3.1 偶関数と奇関数のフーリエ級数 ································· 25
2.3.2 複素フーリエ級数展開 ············································· 262.3.3 周期 L の周期関数のフーリエ級数展開 ······················· 292.4

非周期関数のフーリエ級数展開とフーリエ変換 ····················· 302.5

フーリエ変換 ··································································· 322.6

フーリエ変換の完全性 ······················································· 34

2.7ガウス関数のフーリエ変換 ················································· 352.8

フーリエ変換と不確定性原理 ·············································· 38

2.9デルタ関数 ······································································ 41

2.9.1 デルタ関数の性質 ··················································· 41
2.9.2 デルタ関数の表現 ··················································· 43
2.10 微分のフーリエ変換 ·························································· 46
2.11 畳み込み積分 ··································································· 47

第 3 章 フーリエ変換を用いた微分方程式の解法
3.149

強制振動 ·········································································· 49

3.1.1 外力がないときの解 ················································ 49
3.1.2 外力があるときの解 ················································ 51
3.2

熱伝導 ············································································· 53

3.2.1 熱伝導方程式の導出 ················································ 53
3.2.2 無限に長い 1 次元棒の熱伝導 ···································· 54
3.2.3 有限の長さの 1 次元棒の熱伝導 ································· 57

第 4 章 ラプラス変換
4.161

ラプラス変換の定義 ·························································· 61

main :2021/9/7(9:48)
物理数学―量子力学のためのフーリエ解析・特殊関数―
柴田 尚和・是常 隆著
https://www.kyoritsu-pub.co.jp/bookdetail/9784320036161
目4.2

次vii

ラプラス変換の性質 ·························································· 62

4.2.1 ラプラス変換の線形性 ············································· 62
4.2.2 導関数のラプラス変換 ············································· 62
4.2.3 積分のラプラス変換 ················································ 63
4.3

ラプラス変換を用いた微分方程式の解法 ······························ 64

第 5 章 級数展開による微分方程式の解法

66

5.1

解が正則な場合 ································································ 66

5.2

解に特異点がある場合 ······················································· 67

第 6 章 ベッセル関数

726.1

2 次元極座標系でのラプラシアン ········································ 72

6.2

極座標でのヘルムホルツ方程式 ··········································· 74

6.3

ベッセル関数のベキ級数展開 ·············································· 75

6.4

円形膜 (太鼓) の振動 ························································· 79

6.4.1 振動が円対称になっている場合 ································· 83
6.4.2 振動が円対称になっていない場合 ······························ 84
6.5

ベッセル関数の母関数 ······················································· 86

6.6ガンマ関数 ······································································ 88

6.7

n が任意の実数の場合のベッセル関数 ································· 906.8

第 2 種，第 3 種のベッセル関数 ·········································· 93

6.9

ベッセル関数の漸化式 ······················································· 94

6.10 ベッセル関数の漸近形 ······················································· 96
6.11 球ベッセル関数 ································································ 97

第 7 章 ルジャンドル関数

102

7.1

ルジャンドルの微分方程式 ················································· 102

7.2

ルジャンドル関数とルジャンドル多項式 ······························ 103

7.3

ルジャンドル多項式の具体的な表式とロドリグの公式 ············ 106

main :

2021/9/7(9:48)
物理数学―量子力学のためのフーリエ解析・特殊関数―
柴田 尚和・是常 隆著
https://www.kyoritsu-pub.co.jp/bookdetail/9784320036161

viii

目次

7.4

ルジャンドル多項式の母関数 ·············································· 107

7.5

ルジャンドル多項式の直交性 ·············································· 111

7.6

ルジャンドル多項式の応用:ラプラス方程式 ·························· 113

7.7

ルジャンドルの陪微分方程式 ·············································· 116

7.8

ルジャンドル陪関数 ·························································· 119

7.9

ルジャンドル陪関数の直交性 ·············································· 120

7.10 球面調和関数 ··································································· 121
7.11 球面調和関数の直交性 ······················································· 123

第 8 章 エルミート多項式1288.1調和振動子 ······································································ 128

8.2

エルミート多項式 ····························································· 132

8.3

エルミート多項式の母関数 ················································· 133

8.4

エルミート多項式の直交性 ················································· 134

8.5

エルミート多項式の漸化式 ················································· 135

付録 A

微分方程式と解のまとめ

139

A.1

本書で扱った主な微分方程式 ············································· 139

A.2

微分方程式と解の関係 ······················································· 139

A.3

ヘルムホルツ方程式の一般解の各座標系での表現 ·················· 141

参考文献

144

問題の解答

145

索

165

引