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
解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題1-7.を取り組んでみる。
-
部分積分法を繰り返す。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, Integral, exp, sin, S, plot
print('1.')
x = symbols('x')
a = symbols('a', positive=True)
b = symbols('b', nonzero=True)
fs = [(exp(- a * x) * sin(b * x), (0, S.Infinity))]
for i, (f, (x1, x2)) in enumerate(fs, 6):
print(f'({i})')
I = Integral(f, (x, x1, x2))
pprint(I)
I = I.doit()
pprint(I)
print('factor:')
pprint(I.factor())
print('expand:')
pprint(I.expand())
f = fs[0][0]
p = plot(f.subs({a: 2, b: -3}), show=False, legend=True)
p.save('sample1_7.svg')
入出力結果(Terminal, IPython)
$ ./sample1_7.py 1. (6) ∞ ⌠ ⎮ -a⋅x ⎮ ℯ ⋅sin(b⋅x) dx ⌡ 0 ⎧ 1 │ ⎛ 2 ⎞│ ⎪ ────────── for │periodic_argument⎝polar_lift (b), ∞⎠│ = 0 ⎪ ⎛ 2 ⎞ ⎪ ⎜a ⎟ ⎪ b⋅⎜── + 1⎟ ⎪ ⎜ 2 ⎟ ⎪ ⎝b ⎠ ⎨ ⎪∞ ⎪⌠ ⎪⎮ -a⋅x ⎪⎮ ℯ ⋅sin(b⋅x) dx otherwise ⎪⌡ ⎪0 ⎩ factor: ⎧ 1 │ ⎛ 2 ⎞│ ⎪ ────── for │periodic_argument⎝polar_lift (b), ∞⎠│ = 0 ⎪ 2 ⎪ a ⎪ ── + b ⎪ b ⎨ ⎪∞ ⎪⌠ ⎪⎮ -a⋅x ⎪⎮ ℯ ⋅sin(b⋅x) dx otherwise ⎪⌡ ⎩0 expand: ⎧ 1 │ ⎛ 2 ⎞│ ⎪ ────── for │periodic_argument⎝polar_lift (b), ∞⎠│ = 0 ⎪ 2 ⎪ a ⎪ ── + b ⎪ b ⎨ ⎪∞ ⎪⌠ ⎪⎮ -a⋅x ⎪⎮ ℯ ⋅sin(b⋅x) dx otherwise ⎪⌡ ⎩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="-5"> <label for="x2">x2 = </label> <input id="x2" type="number" value="5"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-5"> <label for="y2">y2 = </label> <input id="y2" type="number" value="5"> <br> <label for="a0">a = </label> <input id="a0" type="number" min="0" value="1"> <label for="b0">b = </label> <input id="b0" type="number" min="0" value="-2"> <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="sample1_7.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_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0],
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) => x * Math.asin(x) / Math.sqrt(1 - x ** 2);
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),
a0 = parseFloat(input_a0.value),
b0 = parseFloat(input_b0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
y3 = b0 / (a0 ** 2 + b0 ** 2),
lines = [[x1, y3, x2, y3, 'blue']],
f = (x) => Math.exp(-a0 * x) * Math.sin(b0 * x),
g = (x) => (-a0 * Math.exp(-a0 * x) * Math.sin(b0 * x) - b0 * Math.exp(-a0 * x) * Math.cos(b0 * x)) / (a0 ** 2 + b0 ** 2),
h = (x) => g(x) - g(0),
fns = [[f, 'green'],
[h, 'orange']];
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]);
}
}
});
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('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.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.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);
p(fns.join('\n'));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
0 コメント:
コメントを投稿