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
オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田 武(著)、東海大学出版会)の第II部(関数の定義(Definitions of Functions))、第5章(テイラー展開(Taylar's Expansion))、5.2(テイラー級数)、5.2.1(剰余項を求める)、5.2.2(テイラー級数の定義)、5.2.3(項別微分・項別積分)、問題2.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, summation, oo
print('2.')
x, n = symbols('x n')
fs = [(-1) ** n * x ** (2 * n),
(n + 1) * x ** n]
for i, f in enumerate(fs, 1):
print(f'[{i}]')
s = summation(f, (n, 0, oo))
pprint(s)
print()
入出力結果(Terminal, IPython)
$ ./sample2.py 2. [1] ⎧ 1 │ 2│ ⎪ ────── for │x │ < 1 ⎪ 2 ⎪ x + 1 ⎪ ⎪ ∞ ⎨ ___ ⎪ ╲ ⎪ ╲ n 2⋅n ⎪ ╱ (-1) ⋅x otherwise ⎪ ╱ ⎪ ‾‾‾ ⎩n = 0 [2] ⎛⎧ x ⎞ ⎛⎧ 1 ⎞ ⎜⎪───────── for │x│ < 1⎟ ⎜⎪ ────── for │x│ < 1⎟ ⎜⎪ 2 ⎟ ⎜⎪ -x + 1 ⎟ ⎜⎪(-x + 1) ⎟ ⎜⎪ ⎟ ⎜⎪ ⎟ ⎜⎪ ∞ ⎟ ⎜⎪ ∞ ⎟ ⎜⎪ ___ ⎟ ⎜⎨ ___ ⎟ + ⎜⎨ ╲ ⎟ ⎜⎪ ╲ ⎟ ⎜⎪ ╲ n ⎟ ⎜⎪ ╲ n ⎟ ⎜⎪ ╱ x otherwise ⎟ ⎜⎪ ╱ n⋅x otherwise ⎟ ⎜⎪ ╱ ⎟ ⎜⎪ ╱ ⎟ ⎜⎪ ‾‾‾ ⎟ ⎜⎪ ‾‾‾ ⎟ ⎜⎪n = 0 ⎟ ⎝⎩n = 0 ⎠ ⎝⎩ ⎠ $
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="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-2"> <label for="x2">x2 = </label> <input id="x2" type="number" value="2"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-2"> <label for="y2">y2 = </label> <input id="y2" type="number" value="2"> <br> <label for="n0">n = </label> <input id="n0" type="number" min="0" step="1" value="0"> <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="sample2.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 f = (x) => 1 / (1 + x ** 2),
g = (x) => 1 / (1 - x) ** 2,
fterm = (n) => (x) => (-1) ** n * x ** (2 * n),
gterm = (n) => (x) => (n + 1) * x ** 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 = parseInt(input_n0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [[-1, y1, -1, y2, 'brown'],
[1, y1, 1, y2, 'brown']],
f1 = (x) =>
range(0, n0 + 1)
.reduce((prev, next) => prev + fterm(next)(x), 0),
g1 = (x) =>
range(0, n0 + 1)
.reduce((prev, next) => prev + gterm(next)(x), 0),
fns = [[f, 'red'],
[f1, 'green'],
[g, 'blue'],
[g1, 'orange']],
fns1 = [],
fns2 = [];
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, color]);
}
}
});
fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(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')));
p(Math.sqrt(a0 + b0) - Math.sqrt(b0));
p(2 * Math.sqrt(a0 + b0) / a0 -
4 / (3 * a0 ** 2) * (Math.sqrt((a0 + b0) ** 3) - Math.sqrt(b0 ** 3)));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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