2018年3月30日金曜日

学習環境

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。


  1. 0 1 1 x s dx = lim t 0 t 1 1 x s dx = lim t 0 1 s + 1 x - s + 1 t 1 = 1 s + 1 lim t 0 1 - t - s + 1 = 1 s + 1

    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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