学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.5(関数の近似、テイラーの定理)、関数の整級数展開(2)、問59.を取り組んでみる。
問題の仮定より以下のようになる。
補助定理より次のことが成り立つ。
よって、剰余項は0に収束するので、 と整級数展開される。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, Derivative, summation, exp, factorial, E, sin, cos
print('59.')
n = symbols('n')
x = 2
fs = [(E, 1 / factorial(n)),
(sin(x), (-1) ** (n - 1) * x * (2 * n - 1) / factorial(2 * n - 1)),
(cos(x), (-1) ** n * x ** (2 * n) / factorial(2 * n))]
for c, f in fs:
pprint(c)
pprint(f)
for n0 in range(10):
print(f'n = {n0}')
s = summation(f, (n, 0, n0))
pprint(s)
print(f'差: {float(c - s)}')
print()
入出力結果(Terminal, IPython)
$ ./sample59.py
59.
ℯ
1
──
n!
n = 0
1
差: 1.7182818284590453
n = 1
2
差: 0.7182818284590452
n = 2
5/2
差: 0.21828182845904523
n = 3
8/3
差: 0.05161516179237857
n = 4
65
──
24
差: 0.009948495125711903
n = 5
163
───
60
差: 0.0016151617923785687
n = 6
1957
────
720
差: 0.0002262729034896798
n = 7
685
───
252
差: 2.7860205076981392e-05
n = 8
109601
──────
40320
差: 3.0586177753940906e-06
n = 9
98641
─────
36288
差: 3.028858529955014e-07
sin(2)
n - 1
2⋅(-1) ⋅(2⋅n - 1)
─────────────────────
(2⋅n - 1)!
n = 0
0
差: 0.9092974268256817
n = 1
2
差: -1.0907025731743183
n = 2
1
差: -0.0907025731743183
n = 3
13
──
12
差: -0.17403590650765163
n = 4
389
───
360
差: -0.17125812872987387
n = 5
4357
────
4032
差: -0.17130773190447704
n = 6
1960649
───────
1814400
差: -0.17130718075809256
n = 7
258805669
─────────
239500800
差: -0.17130718493344396
n = 8
47102631757
───────────
43589145600
差: -0.17130718491050245
n = 9
11304631621681
──────────────
10461394944000
差: -0.17130718491059804
cos(2)
n 2⋅n
(-1) ⋅2
──────────
(2⋅n)!
n = 0
1
差: -1.4161468365471424
n = 1
-1
差: 0.5838531634528576
n = 2
-1/3
差: -0.08281350321380905
n = 3
-19
────
45
差: 0.006075385675079835
n = 4
-131
─────
315
差: -0.000273820674126514
n = 5
-5899
──────
14175
差: 8.366274727101539e-06
n = 6
-27809
───────
66825
差: -1.8484493512923474e-07
n = 7
-17714341
──────────
42567525
差: 3.0917607439690546e-09
n = 8
-265715113
───────────
638512875
差: -4.051752058434203e-11
n = 9
-3695855663
────────────
8881133625
差: 4.272933313886496e-13
$
HTML5
<div id="graph0"></div> <pre id="output0"></pre> <label for="r0">r = </label> <input id="r0" type="number" min="0" value="0.5"> <label for="dx">dx = </label> <input id="dx" type="number" min="0" step="0.0001" value="0.01"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-10"> <label for="x2">x2 = </label> <input id="x2" type="number" value="10"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-10"> <label for="y2">y2 = </label> <input id="y2" type="number" value="10"> <br> <label for="n0">n = </label> <input id="n0" type="number" min="0" value="1"> <button id="draw0">draw</button> <button id="clear0">clear</button> <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script> <script src="sample59.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
padding = 50,
btn0 = document.querySelector('#draw0'),
btn1 = document.querySelector('#clear0'),
input_r = document.querySelector('#r0'),
input_dx = document.querySelector('#dx'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
input_n0 = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_n0],
p = (x) => pre0.textContent += x + '\n',
range = (start, end, step=1) => {
let res = [];
for (let i = start; i < end; i += step) {
res.push(i);
}
return res;
};
let factorial = (n) => range(1, n + 1).reduce((x, y) => x * y, 1);
let term1 = (n) => (x) => x ** n / factorial(n),
term2 = (n) => (x) => (-1) ** (n - 1) * x ** (2 * n - 1) / factorial(2 * n - 1),
term3 = (n) => (x) => (-1) ** n * x ** (2 * n) / factorial(2 * n);
let draw = () => {
pre0.textContent = '';
let r = parseFloat(input_r.value),
dx = parseFloat(input_dx.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value),
n0 = parseFloat(input_n0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
f1 = (x) => range(0, n0 + 1).reduce((prev, i) => prev + term1(i)(x), 0),
f2 = (x) => range(1, n0 + 1).reduce((prev, i) => prev + term2(i)(x), 0),
f3 = (x) => range(0, n0 + 1).reduce((prev, i) => prev + term3(i)(x), 0),
fns = [[Math.exp, 'red'],
[Math.sin, 'green'],
[Math.cos, 'blue'],
[f1, 'orange'],
[f2, 'brown'],
[f3, 'purple']],
fns1 = [],
fns2 = [];
fns.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = fn(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, color]);
}
}
});
fns1.forEach((o) => {
let [fn, color] = o;
lines.push([x1, fn(x1), x2, fn(x2), color]);
});
fns2.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = fn(x);
lines.push([x1, g(x1), x2, g(x2), color]);
}
});
let xscale = d3.scaleLinear()
.domain([x1, x2])
.range([padding, width - padding]);
let yscale = d3.scaleLinear()
.domain([y1, y2])
.range([height - padding, padding]);
let xaxis = d3.axisBottom().scale(xscale);
let yaxis = d3.axisLeft().scale(yscale);
div0.innerHTML = '';
let svg = d3.select('#graph0')
.append('svg')
.attr('width', width)
.attr('height', height);
svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
.enter()
.append('line')
.attr('x1', (d) => xscale(d[0]))
.attr('y1', (d) => yscale(d[1]))
.attr('x2', (d) => xscale(d[2]))
.attr('y2', (d) => yscale(d[3]))
.attr('stroke', (d) => d[4] || 'black');
svg.selectAll('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d) => d[2] || 'green');
svg.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);
[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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