-
Notifications
You must be signed in to change notification settings - Fork 0
/
test16-align.saty
37 lines (32 loc) · 1.25 KB
/
test16-align.saty
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
% -*- coding: utf-8 -*-
@require: stdja
@require: math
let-math \sums m1 m2 = ${\upper{\lower{\sum}{#m1}}{#m2}}
in
document (|
title = {Sample Document}; author = {gfn};
show-title = false; show-toc = false;
|) '<
+align[
${| \paren{x - \alpha} \paren{x - \beta}
|= x^2 - \paren{\alpha + \beta} x + \alpha \beta |};
${| |= x^2 + \frac{b}{a} x + \frac{c}{a} |};
];
+p{
Cauchy–Schwarzの不等式:
${\paren{\sums{i = 1}{n} x_i y_i}^2 \leq \paren{\sums{i = 1}{n} x_i^2} \paren{\sums{i = 1}{n} y_i^2}}
を示す.帰納法による.${n = 1, 2} のときは容易に確かめられる.${n = m + 1 \geq 3} のときは,
}
+align(
let sumsm = ${\sums{i = 1}{m}} in
[
${| \paren{\sums{i = 1}{m + 1} x_i y_i}^2
|= \paren{\sums{i = 1}{m} x_i y_i + x_{m + 1} y_{m + 1}}^2 |};
${| |\leq \paren{\sqrt{#sumsm x_i^2} \sqrt{#sumsm y_i^2} + x_{m + 1} y_{m + 1}}^2
| \math-skip!(2cm)\paren{\text!{${m} に関する帰納法の仮定より}} |};
${| |\leq \paren{#sumsm x_i^2 + x_{m + 1}^2} \paren{#sumsm y_i^2 + y_{m + 1}^2}
| \math-skip!(2cm)\paren{\text!{${n = 2} のときより}}|};
${| |= \paren{\sums{i = 1}{m + 1} x_i^2} \paren{\sums{i = 1}{m + 1} y_i^2} |}
]
);
>