2018年6月16日土曜日

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11章(積分の計算)、3(三角関数の積分)、練習問題18-(i).を取り組んでみる。



    1. c 0 = 1 2 π · 4 0 π 2 cos x dx = 2 π sin x 0 π 2 = 2 π
      n = 1

      のとき。

      - - π - π 2 cos 2 x dx = - - π - π 2 cos 2 x + 1 2 dx = - 1 2 sin 2 x 2 + x - π - π 2 = - 1 2 - π 2 + π = - π 4
      1 2 sin 2 x 2 + x - π 2 0 = π 4 1 2 sin 2 x 2 + x 0 π 2 = π 4 - 1 2 sin 2 x 2 + x π 2 π = - π 4
      a n = 0
      n 1

      のとき。

      - - π - π 2 cos x cos n x dx = - - π - π 2 cos 1 + n x + cos 1 - n x 2 dx = - 1 2 sin 1 + n x 1 + n + sin 1 - n x 1 - n - π - π 2 = - 1 2 - sin 1 + n · π 2 1 + n + - sin 1 - n π 2 1 - n = 1 2 sin 1 + n π 2 1 + n + sin 1 - n π 2 1 - n
      1 2 sin 1 + n x 1 + n + sin 1 - n x 1 - n - π 2 0 = 1 2 sin 1 + n π 2 1 + n + sin 1 - n π 2 1 - n
      1 2 sin 1 + n x 1 + n + sin 1 - n x 1 - n 0 π 2 = 1 2 sin 1 + n π 2 1 + n + sin 1 - n π 2 1 - n
      a n = 2 π sin 1 + n π 2 1 + n + sin 1 - n π 2 1 - n = 2 π 1 - n 2 sin 1 + n π 2 - n sin 1 + n π 2 + sin 1 - n π 2 + n sin 1 - n π 2 = 2 π 1 - n 2 2 cos n π 2 = 4 π 1 - n 2 cos n π 2
      n = 1 - - π - π 2 cos x sin x dx = - - π - π 2 sin 2 x 2 dx = 1 4 cos 2 x - π - π 2 = - 1 2 - 1 4 cos 2 x - π 2 0 = 0 - 1 4 cos 2 x 0 π 2 = 0 1 4 cos 2 x π 2 π = 1 2 b n = 0
      n 1 - - π - π 2 cos x sin n x dx = - - π - π 2 sin 1 + n x + sin 1 - n x 2 dx = 1 2 cos 1 + n x 1 + n + cos 1 - n x 1 - n - π - π 2 = 1 2 cos 1 + n x 1 + n + cos 1 - n x 1 - n π π 2
      - 1 2 cos 1 + n x 1 + n + cos 1 - n x 1 - n - π 2 0 - 1 2 cos 1 + n x 1 + n + cos 1 - n x 1 - n 0 π 2 1 2 cos 1 + n x 1 + n + cos 1 - n x 1 - n π 2 π b n = 0

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, pi, sin, cos, Integral, Function, plot

x = symbols('x')
n = symbols('n', positive=True, integer=True)

f = Function('f')(x)
c0 = 1 / (2 * pi) * (Integral(-f, (x, -pi, -pi / 2)) + Integral(f,  (x, -
                                                                     pi / 2, 0)) + Integral(f, (x, 0, pi / 2)) + Integral(-f, (x, pi / 2, pi)))
an = 1 / pi * (Integral(-f * cos(n * x), (x, -pi, -pi / 2)) +
               Integral(f * cos(n * x), (x, -pi / 2, 0)) +
               Integral(f * cos(n * x), (x, 0, pi / 2)) +
               Integral(-f * cos(n * x), (x, pi / 2, pi)))
bn = 1 / pi * (Integral(-f * sin(n * x), (x, -pi, -pi / 2)) +
               Integral(f * sin(n * x), (x, -pi / 2, 0)) +
               Integral(f * sin(n * x), (x, 0, pi / 2)) +
               Integral(-f * sin(n * x), (x, pi / 2, pi)))


fs = [c0, an, bn]

for t in fs:
    g = t.subs({f: cos(x)})
    for s in [g, g.doit(), g.doit().simplify()]:
        pprint(s)
        print()
    print()

p = plot(abs(cos(x)), abs(cos(x)) * cos(2 * x),
         abs(cos(x)) * sin(2 * x), legend=True, show=False)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color
p.save('sample23.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample22.py
0               π          
⌠               ⌠          
⎮  -sin(x) dx + ⎮ sin(x) dx
⌡               ⌡          
-π              0          
───────────────────────────
            2⋅π            

2
─
π

2
─
π


0                        π                   
⌠                        ⌠                   
⎮  -sin(x)⋅cos(n⋅x) dx + ⎮ sin(x)⋅cos(n⋅x) dx
⌡                        ⌡                   
-π                       0                   
─────────────────────────────────────────────
                      π                      

⎛⎧   0     for n = 1⎞   ⎛⎧  0     for n = 1⎞   ⎛⎧        0          for n = 1⎞
⎜⎪                  ⎟   ⎜⎪                 ⎟   ⎜⎪                            ⎟
⎜⎪     n            ⎟   ⎜⎪  1              ⎟   ⎜⎪      n                     ⎟
⎜⎨-(-1)             ⎟ - ⎜⎨──────  otherwise⎟ + ⎜⎨  (-1)       1              ⎟
⎜⎪───────  otherwise⎟   ⎜⎪ 2               ⎟   ⎜⎪- ────── - ──────  otherwise⎟
⎜⎪  2               ⎟   ⎜⎪n  - 1           ⎟   ⎜⎪   2        2               ⎟
⎝⎩ n  - 1           ⎠   ⎝⎩                 ⎠   ⎝⎩  n  - 1   n  - 1           ⎠
──────────────────────────────────────────────────────────────────────────────
                                      π                                       

⎧       0         for n = 1
⎪                          
⎪ ⎛      n    ⎞            
⎨-⎝2⋅(-1)  + 2⎠            
⎪───────────────  otherwise
⎪     ⎛ 2    ⎞             
⎩   π⋅⎝n  - 1⎠             


0                        π                   
⌠                        ⌠                   
⎮  -sin(x)⋅sin(n⋅x) dx + ⎮ sin(x)⋅sin(n⋅x) dx
⌡                        ⌡                   
-π                       0                   
─────────────────────────────────────────────
                      π                      

⎛⎧-π            ⎞   ⎛⎧π           ⎞
⎜⎪───  for n = 1⎟   ⎜⎪─  for n = 1⎟
⎜⎨ 2            ⎟ + ⎜⎨2           ⎟
⎜⎪              ⎟   ⎜⎪            ⎟
⎝⎩ 0   otherwise⎠   ⎝⎩0  otherwise⎠
───────────────────────────────────
                 π                 

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="0.001" 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="n0">n = </label>
<input id="n0" type="number" min="1" 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="sample23.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';
           
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 = [[-Math.PI, y1, -Math.PI, y2, 'orange'],
                 [Math.PI, y1, Math.PI, y2, 'brown']],
        fns = [[(x) => Math.abs(Math.cos(x)), 'red'],
               [(x) => Math.abs(Math.cos(x)) * Math.cos(n0 * x), 'green'],
               [(x) => Math.abs(Math.cos(x)) * Math.sin(n0 * x), 'blue']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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('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].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();








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