2018年6月20日水曜日

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19.を取り組んでみる。



    1. - π π f x sin n x dx = - π 0 f x sin n x dx + 0 π f x sin n x dx = - 0 - π f x sin n x dx + 0 π f x sin n x dx = - 0 π f x sin n x dx + 0 π f x sin n x dx = 0 b n = 0

    2. c 0 = 1 2 π - π π f x dx = 0 - π π f x cos n x dx = - π 0 f x cos n x dx + 0 π f x cos n x dx = - 0 - π f x cos n x dx + 0 π f x cos n x dx = - 0 π f - x - cos n x dx + 0 π f x cos n x dx = - 0 π f x cos n x dx + 0 π f x cos n x dx = 0 a n = 0 - π π f x sin n x dx = - π 0 f x sin n x dx + 0 π f x sin n x dx = - 0 - π f x sin n x dx + 0 π f x sin n x dx = - 0 π f - x sin n x dx + 0 π f x sin n x dx = 0 π f x sin n x dx + 0 π f x sin n x dx = 2 0 π f x sin n x dx b n = 2 π 0 π f x sin n x dx

コード(Emacs)

Python 3

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

print('19.')
x = symbols('x')
n = symbols('n', positive=True, integer=True)

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


fs = [c0, an, bn]

for i, h in enumerate([x ** 2, x]):
    print(f'({chr(ord("a") + i)})')
    for j, t in zip(['c0', 'an', 'bn'], fs):
        print(f'{j}')
        g = t.subs({f: h})
        for s in [g, g.doit()]:
            pprint(s)
            print()
        print()
    print()

I = 2 / pi * Integral(x * sin(n * x), (x, 0, pi))
for t in [I, I.doit()]:
    pprint(t)
    print()

p = plot(x ** 2, x ** 2 * cos(2 * x), x ** 2 * sin(2 * x), x, x *
         cos(2 * x), x * sin(2 * x), xlim=(-5, 5), legend=True, show=False)
colors = ['red', 'green', 'blue', 'orange', 'brown', 'purple']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample26.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample26.py
19.
(a)
c0
π       
⌠       
⎮   2   
⎮  x  dx
⌡       
-π      
────────
  2⋅π   

 2
π 
──
3 


an
π                
⌠                
⎮   2            
⎮  x ⋅cos(n⋅x) dx
⌡                
-π               
─────────────────
        π        

      n
4⋅(-1) 
───────
    2  
   n   


bn
π                
⌠                
⎮   2            
⎮  x ⋅sin(n⋅x) dx
⌡                
-π               
─────────────────
        π        

0



(b)
c0
π      
⌠      
⎮  x dx
⌡      
-π     
───────
  2⋅π  

0


an
π               
⌠               
⎮  x⋅cos(n⋅x) dx
⌡               
-π              
────────────────
       π        

0


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

       n 
-2⋅(-1)  
─────────
    n    



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

       n 
-2⋅(-1)  
─────────
    n    

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<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="sample26.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, 'gray'],
                 [Math.PI, y1, Math.PI, y2, 'gray']],
        fns = [[(x) => x ** 2, 'red'],
               [(x) => x ** 2 * Math.cos(n0 * x), 'green'],
               [(x) => x ** 2 * Math.sin(n0 * x), 'blue'],
               [(x) => x, 'orange'],
               [(x) => x * Math.cos(n0 * x), 'brown'],
               [(x) => x * Math.sin(n0 * x), 'purple']];               

    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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