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)
- Nebo(Windows アプリ)
- iPad Pro + Apple Pencil
- MyScript Nebo(iPad アプリ)
- 参考書籍
解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第10章(積分の性質)、4(広義の積分)、練習問題11.を取り組んでみる。
よって、この広義積分は存在する。
コード(Emacs)
Python 3
#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, plot, Rational
x, s = symbols('x, s')
f = 1 / x ** s
I = Integral(f, (x, 0, 1))
for t in [I, I.doit()]:
pprint(t)
print()
fs = []
for n in list(range(1, 10)) + [-Rational(1, 10)]:
s0 = Rational(1, n)
I0 = I.subs({s: s0})
fs.append(f.subs({s: s0}))
print(f's = {s0}')
for t in [I0, I0.doit()]:
pprint(t)
print()
print()
p = plot(*fs, (x, 0, 1), ylim=(0, 10), legend=True, show=False)
colors = ['red', 'green', 'blue', 'orange', 'pink',
'brown', 'yellow', 'purple', 'gray', 'black']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample11.svg')
入出力結果(Terminal, Jupyter(IPython))
$ ./sample11.py 1 ⌠ ⎮ -s ⎮ x dx ⌡ 0 1 ⌠ ⎮ -s ⎮ x dx ⌡ 0 s = 1 1 ⌠ ⎮ 1 ⎮ ─ dx ⎮ x ⌡ 0 ∞ s = 1/2 1 ⌠ ⎮ 1 ⎮ ── dx ⎮ √x ⌡ 0 2 s = 1/3 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 3 ___ ⎮ ╲╱ x ⌡ 0 3/2 s = 1/4 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 4 ___ ⎮ ╲╱ x ⌡ 0 4/3 s = 1/5 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 5 ___ ⎮ ╲╱ x ⌡ 0 5/4 s = 1/6 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 6 ___ ⎮ ╲╱ x ⌡ 0 6/5 s = 1/7 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 7 ___ ⎮ ╲╱ x ⌡ 0 7/6 s = 1/8 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 8 ___ ⎮ ╲╱ x ⌡ 0 8/7 s = 1/9 1 ⌠ ⎮ 1 ⎮ ───── dx ⎮ 9 ___ ⎮ ╲╱ x ⌡ 0 9/8 s = -10 1 ⌠ ⎮ 10 ⎮ x dx ⌡ 0 1/11 $
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="0.001" 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="s0">s = </label> <input id="s0" type="number" value="10"> <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="sample11.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_s0 = document.querySelector('#s0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_s0],
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 / x;
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),
s0 = parseFloat(input_s0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [[0, y1, 0, y2, 'blue'],
[1, y1, 1, y2, 'orange']],
g = (x) => 1 / x ** (1 / s0),
fns = [[f, 'red'],
[g, 'green']],
fns1 = [],
fns2 = [];
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);
points.push([x, y, color]);
}
});
fns1
.forEach((o) => {
let [f, color] = o;
lines.push([x1, f(x1), x2, f(x2), 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')));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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